cesiumDemo/Source/Workers/FrustumGeometry-93211505.js

/* This file is automatically rebuilt by the Cesium build process. */
define(['exports', './defined-26bd4a03', './Check-da037458', './defaultValue-f2e68450', './Math-fa6e45cb', './Cartesian2-2a723276', './defineProperties-6f7a50f2', './Transforms-65aba0a4', './ComponentDatatype-69643096', './GeometryAttribute-ed359d71', './GeometryAttributes-eecc9f43', './Plane-a1a3fd52', './VertexFormat-fbb91dc7'], function (exports, defined, Check, defaultValue, _Math, Cartesian2, defineProperties, Transforms, ComponentDatatype, GeometryAttribute, GeometryAttributes, Plane, VertexFormat) { 'use strict';

    /**
         * The culling volume defined by planes.
         *
         * @alias CullingVolume
         * @constructor
         *
         * @param {Cartesian4[]} [planes] An array of clipping planes.
         */
        function CullingVolume(planes) {
            /**
             * Each plane is represented by a Cartesian4 object, where the x, y, and z components
             * define the unit vector normal to the plane, and the w component is the distance of the
             * plane from the origin.
             * @type {Cartesian4[]}
             * @default []
             */
            this.planes = defaultValue.defaultValue(planes, []);
        }

        var faces = [new Cartesian2.Cartesian3(), new Cartesian2.Cartesian3(), new Cartesian2.Cartesian3()];
        Cartesian2.Cartesian3.clone(Cartesian2.Cartesian3.UNIT_X, faces[0]);
        Cartesian2.Cartesian3.clone(Cartesian2.Cartesian3.UNIT_Y, faces[1]);
        Cartesian2.Cartesian3.clone(Cartesian2.Cartesian3.UNIT_Z, faces[2]);

        var scratchPlaneCenter = new Cartesian2.Cartesian3();
        var scratchPlaneNormal = new Cartesian2.Cartesian3();
        var scratchPlane = new Plane.Plane(new Cartesian2.Cartesian3(1.0, 0.0, 0.0), 0.0);

        /**
         * Constructs a culling volume from a bounding sphere. Creates six planes that create a box containing the sphere.
         * The planes are aligned to the x, y, and z axes in world coordinates.
         *
         * @param {BoundingSphere} boundingSphere The bounding sphere used to create the culling volume.
         * @param {CullingVolume} [result] The object onto which to store the result.
         * @returns {CullingVolume} The culling volume created from the bounding sphere.
         */
        CullingVolume.fromBoundingSphere = function(boundingSphere, result) {
            //>>includeStart('debug', pragmas.debug);
            if (!defined.defined(boundingSphere)) {
                throw new Check.DeveloperError('boundingSphere is required.');
            }
            //>>includeEnd('debug');

            if (!defined.defined(result)) {
                result = new CullingVolume();
            }

            var length = faces.length;
            var planes = result.planes;
            planes.length = 2 * length;

            var center = boundingSphere.center;
            var radius = boundingSphere.radius;

            var planeIndex = 0;

            for (var i = 0; i < length; ++i) {
                var faceNormal = faces[i];

                var plane0 = planes[planeIndex];
                var plane1 = planes[planeIndex + 1];

                if (!defined.defined(plane0)) {
                    plane0 = planes[planeIndex] = new Transforms.Cartesian4();
                }
                if (!defined.defined(plane1)) {
                    plane1 = planes[planeIndex + 1] = new Transforms.Cartesian4();
                }

                Cartesian2.Cartesian3.multiplyByScalar(faceNormal, -radius, scratchPlaneCenter);
                Cartesian2.Cartesian3.add(center, scratchPlaneCenter, scratchPlaneCenter);

                plane0.x = faceNormal.x;
                plane0.y = faceNormal.y;
                plane0.z = faceNormal.z;
                plane0.w = -Cartesian2.Cartesian3.dot(faceNormal, scratchPlaneCenter);

                Cartesian2.Cartesian3.multiplyByScalar(faceNormal, radius, scratchPlaneCenter);
                Cartesian2.Cartesian3.add(center, scratchPlaneCenter, scratchPlaneCenter);

                plane1.x = -faceNormal.x;
                plane1.y = -faceNormal.y;
                plane1.z = -faceNormal.z;
                plane1.w = -Cartesian2.Cartesian3.dot(Cartesian2.Cartesian3.negate(faceNormal, scratchPlaneNormal), scratchPlaneCenter);

                planeIndex += 2;
            }

            return result;
        };

        /**
         * Determines whether a bounding volume intersects the culling volume.
         *
         * @param {Object} boundingVolume The bounding volume whose intersection with the culling volume is to be tested.
         * @returns {Intersect}  Intersect.OUTSIDE, Intersect.INTERSECTING, or Intersect.INSIDE.
         */
        CullingVolume.prototype.computeVisibility = function(boundingVolume) {
            //>>includeStart('debug', pragmas.debug);
            if (!defined.defined(boundingVolume)) {
                throw new Check.DeveloperError('boundingVolume is required.');
            }
            //>>includeEnd('debug');

            var planes = this.planes;
            var intersecting = false;
            for (var k = 0, len = planes.length; k < len; ++k) {
                var result = boundingVolume.intersectPlane(Plane.Plane.fromCartesian4(planes[k], scratchPlane));
                if (result === Transforms.Intersect.OUTSIDE) {
                    return Transforms.Intersect.OUTSIDE;
                } else if (result === Transforms.Intersect.INTERSECTING) {
                    intersecting = true;
                }
            }

            return intersecting ? Transforms.Intersect.INTERSECTING : Transforms.Intersect.INSIDE;
        };

        /**
         * Determines whether a bounding volume intersects the culling volume.
         *
         * @param {Object} boundingVolume The bounding volume whose intersection with the culling volume is to be tested.
         * @param {Number} parentPlaneMask A bit mask from the boundingVolume's parent's check against the same culling
         *                                 volume, such that if (planeMask & (1 << planeIndex) === 0), for k < 31, then
         *                                 the parent (and therefore this) volume is completely inside plane[planeIndex]
         *                                 and that plane check can be skipped.
         * @returns {Number} A plane mask as described above (which can be applied to this boundingVolume's children).
         *
         * @private
         */
        CullingVolume.prototype.computeVisibilityWithPlaneMask = function(boundingVolume, parentPlaneMask) {
            //>>includeStart('debug', pragmas.debug);
            if (!defined.defined(boundingVolume)) {
                throw new Check.DeveloperError('boundingVolume is required.');
            }
            if (!defined.defined(parentPlaneMask)) {
                throw new Check.DeveloperError('parentPlaneMask is required.');
            }
            //>>includeEnd('debug');

            if (parentPlaneMask === CullingVolume.MASK_OUTSIDE || parentPlaneMask === CullingVolume.MASK_INSIDE) {
                // parent is completely outside or completely inside, so this child is as well.
                return parentPlaneMask;
            }

            // Start with MASK_INSIDE (all zeros) so that after the loop, the return value can be compared with MASK_INSIDE.
            // (Because if there are fewer than 31 planes, the upper bits wont be changed.)
            var mask = CullingVolume.MASK_INSIDE;

            var planes = this.planes;
            for (var k = 0, len = planes.length; k < len; ++k) {
                // For k greater than 31 (since 31 is the maximum number of INSIDE/INTERSECTING bits we can store), skip the optimization.
                var flag = (k < 31) ? (1 << k) : 0;
                if (k < 31 && (parentPlaneMask & flag) === 0) {
                    // boundingVolume is known to be INSIDE this plane.
                    continue;
                }

                var result = boundingVolume.intersectPlane(Plane.Plane.fromCartesian4(planes[k], scratchPlane));
                if (result === Transforms.Intersect.OUTSIDE) {
                    return CullingVolume.MASK_OUTSIDE;
                } else if (result === Transforms.Intersect.INTERSECTING) {
                    mask |= flag;
                }
            }

            return mask;
        };

        /**
         * For plane masks (as used in {@link CullingVolume#computeVisibilityWithPlaneMask}), this special value
         * represents the case where the object bounding volume is entirely outside the culling volume.
         *
         * @type {Number}
         * @private
         */
        CullingVolume.MASK_OUTSIDE = 0xffffffff;

        /**
         * For plane masks (as used in {@link CullingVolume.prototype.computeVisibilityWithPlaneMask}), this value
         * represents the case where the object bounding volume is entirely inside the culling volume.
         *
         * @type {Number}
         * @private
         */
        CullingVolume.MASK_INSIDE = 0x00000000;

        /**
         * For plane masks (as used in {@link CullingVolume.prototype.computeVisibilityWithPlaneMask}), this value
         * represents the case where the object bounding volume (may) intersect all planes of the culling volume.
         *
         * @type {Number}
         * @private
         */
        CullingVolume.MASK_INDETERMINATE = 0x7fffffff;

    /**
         * The viewing frustum is defined by 6 planes.
         * Each plane is represented by a {@link Cartesian4} object, where the x, y, and z components
         * define the unit vector normal to the plane, and the w component is the distance of the
         * plane from the origin/camera position.
         *
         * @alias OrthographicOffCenterFrustum
         * @constructor
         *
         * @param {Object} [options] An object with the following properties:
         * @param {Number} [options.left] The left clipping plane distance.
         * @param {Number} [options.right] The right clipping plane distance.
         * @param {Number} [options.top] The top clipping plane distance.
         * @param {Number} [options.bottom] The bottom clipping plane distance.
         * @param {Number} [options.near=1.0] The near clipping plane distance.
         * @param {Number} [options.far=500000000.0] The far clipping plane distance.
         *
         * @example
         * var maxRadii = ellipsoid.maximumRadius;
         *
         * var frustum = new Cesium.OrthographicOffCenterFrustum();
         * frustum.right = maxRadii * Cesium.Math.PI;
         * frustum.left = -c.frustum.right;
         * frustum.top = c.frustum.right * (canvas.clientHeight / canvas.clientWidth);
         * frustum.bottom = -c.frustum.top;
         * frustum.near = 0.01 * maxRadii;
         * frustum.far = 50.0 * maxRadii;
         */
        function OrthographicOffCenterFrustum(options) {
            options = defaultValue.defaultValue(options, defaultValue.defaultValue.EMPTY_OBJECT);

            /**
             * The left clipping plane.
             * @type {Number}
             * @default undefined
             */
            this.left = options.left;
            this._left = undefined;

            /**
             * The right clipping plane.
             * @type {Number}
             * @default undefined
             */
            this.right = options.right;
            this._right = undefined;

            /**
             * The top clipping plane.
             * @type {Number}
             * @default undefined
             */
            this.top = options.top;
            this._top = undefined;

            /**
             * The bottom clipping plane.
             * @type {Number}
             * @default undefined
             */
            this.bottom = options.bottom;
            this._bottom = undefined;

            /**
             * The distance of the near plane.
             * @type {Number}
             * @default 1.0
             */
            this.near = defaultValue.defaultValue(options.near, 1.0);
            this._near = this.near;

            /**
             * The distance of the far plane.
             * @type {Number}
             * @default 500000000.0;
             */
            this.far = defaultValue.defaultValue(options.far, 500000000.0);
            this._far = this.far;

            this._cullingVolume = new CullingVolume();
            this._orthographicMatrix = new Transforms.Matrix4();
        }

        function update(frustum) {
            //>>includeStart('debug', pragmas.debug);
            if (!defined.defined(frustum.right) || !defined.defined(frustum.left) ||
                !defined.defined(frustum.top) || !defined.defined(frustum.bottom) ||
                !defined.defined(frustum.near) || !defined.defined(frustum.far)) {
                throw new Check.DeveloperError('right, left, top, bottom, near, or far parameters are not set.');
            }
            //>>includeEnd('debug');

            if (frustum.top !== frustum._top || frustum.bottom !== frustum._bottom ||
                frustum.left !== frustum._left || frustum.right !== frustum._right ||
                frustum.near !== frustum._near || frustum.far !== frustum._far) {

                //>>includeStart('debug', pragmas.debug);
                if (frustum.left > frustum.right) {
                    throw new Check.DeveloperError('right must be greater than left.');
                }
                if (frustum.bottom > frustum.top) {
                    throw new Check.DeveloperError('top must be greater than bottom.');
                }
                if (frustum.near <= 0 || frustum.near > frustum.far) {
                    throw new Check.DeveloperError('near must be greater than zero and less than far.');
                }
                //>>includeEnd('debug');

                frustum._left = frustum.left;
                frustum._right = frustum.right;
                frustum._top = frustum.top;
                frustum._bottom = frustum.bottom;
                frustum._near = frustum.near;
                frustum._far = frustum.far;
                frustum._orthographicMatrix = Transforms.Matrix4.computeOrthographicOffCenter(frustum.left, frustum.right, frustum.bottom, frustum.top, frustum.near, frustum.far, frustum._orthographicMatrix);
            }
        }

        defineProperties.defineProperties(OrthographicOffCenterFrustum.prototype, {
            /**
             * Gets the orthographic projection matrix computed from the view frustum.
             * @memberof OrthographicOffCenterFrustum.prototype
             * @type {Matrix4}
             * @readonly
             */
            projectionMatrix : {
                get : function() {
                    update(this);
                    return this._orthographicMatrix;
                }
            }
        });

        var getPlanesRight = new Cartesian2.Cartesian3();
        var getPlanesNearCenter = new Cartesian2.Cartesian3();
        var getPlanesPoint = new Cartesian2.Cartesian3();
        var negateScratch = new Cartesian2.Cartesian3();

        /**
         * Creates a culling volume for this frustum.
         *
         * @param {Cartesian3} position The eye position.
         * @param {Cartesian3} direction The view direction.
         * @param {Cartesian3} up The up direction.
         * @returns {CullingVolume} A culling volume at the given position and orientation.
         *
         * @example
         * // Check if a bounding volume intersects the frustum.
         * var cullingVolume = frustum.computeCullingVolume(cameraPosition, cameraDirection, cameraUp);
         * var intersect = cullingVolume.computeVisibility(boundingVolume);
         */
        OrthographicOffCenterFrustum.prototype.computeCullingVolume = function(position, direction, up) {
            //>>includeStart('debug', pragmas.debug);
            if (!defined.defined(position)) {
                throw new Check.DeveloperError('position is required.');
            }
            if (!defined.defined(direction)) {
                throw new Check.DeveloperError('direction is required.');
            }
            if (!defined.defined(up)) {
                throw new Check.DeveloperError('up is required.');
            }
            //>>includeEnd('debug');

            var planes = this._cullingVolume.planes;
            var t = this.top;
            var b = this.bottom;
            var r = this.right;
            var l = this.left;
            var n = this.near;
            var f = this.far;

            var right = Cartesian2.Cartesian3.cross(direction, up, getPlanesRight);
            Cartesian2.Cartesian3.normalize(right, right);
            var nearCenter = getPlanesNearCenter;
            Cartesian2.Cartesian3.multiplyByScalar(direction, n, nearCenter);
            Cartesian2.Cartesian3.add(position, nearCenter, nearCenter);

            var point = getPlanesPoint;

            // Left plane
            Cartesian2.Cartesian3.multiplyByScalar(right, l, point);
            Cartesian2.Cartesian3.add(nearCenter, point, point);

            var plane = planes[0];
            if (!defined.defined(plane)) {
                plane = planes[0] = new Transforms.Cartesian4();
            }
            plane.x = right.x;
            plane.y = right.y;
            plane.z = right.z;
            plane.w = -Cartesian2.Cartesian3.dot(right, point);

            // Right plane
            Cartesian2.Cartesian3.multiplyByScalar(right, r, point);
            Cartesian2.Cartesian3.add(nearCenter, point, point);

            plane = planes[1];
            if (!defined.defined(plane)) {
                plane = planes[1] = new Transforms.Cartesian4();
            }
            plane.x = -right.x;
            plane.y = -right.y;
            plane.z = -right.z;
            plane.w = -Cartesian2.Cartesian3.dot(Cartesian2.Cartesian3.negate(right, negateScratch), point);

            // Bottom plane
            Cartesian2.Cartesian3.multiplyByScalar(up, b, point);
            Cartesian2.Cartesian3.add(nearCenter, point, point);

            plane = planes[2];
            if (!defined.defined(plane)) {
                plane = planes[2] = new Transforms.Cartesian4();
            }
            plane.x = up.x;
            plane.y = up.y;
            plane.z = up.z;
            plane.w = -Cartesian2.Cartesian3.dot(up, point);

            // Top plane
            Cartesian2.Cartesian3.multiplyByScalar(up, t, point);
            Cartesian2.Cartesian3.add(nearCenter, point, point);

            plane = planes[3];
            if (!defined.defined(plane)) {
                plane = planes[3] = new Transforms.Cartesian4();
            }
            plane.x = -up.x;
            plane.y = -up.y;
            plane.z = -up.z;
            plane.w = -Cartesian2.Cartesian3.dot(Cartesian2.Cartesian3.negate(up, negateScratch), point);

            // Near plane
            plane = planes[4];
            if (!defined.defined(plane)) {
                plane = planes[4] = new Transforms.Cartesian4();
            }
            plane.x = direction.x;
            plane.y = direction.y;
            plane.z = direction.z;
            plane.w = -Cartesian2.Cartesian3.dot(direction, nearCenter);

            // Far plane
            Cartesian2.Cartesian3.multiplyByScalar(direction, f, point);
            Cartesian2.Cartesian3.add(position, point, point);

            plane = planes[5];
            if (!defined.defined(plane)) {
                plane = planes[5] = new Transforms.Cartesian4();
            }
            plane.x = -direction.x;
            plane.y = -direction.y;
            plane.z = -direction.z;
            plane.w = -Cartesian2.Cartesian3.dot(Cartesian2.Cartesian3.negate(direction, negateScratch), point);

            return this._cullingVolume;
        };

        /**
         * Returns the pixel's width and height in meters.
         *
         * @param {Number} drawingBufferWidth The width of the drawing buffer.
         * @param {Number} drawingBufferHeight The height of the drawing buffer.
         * @param {Number} distance The distance to the near plane in meters.
         * @param {Number} pixelRatio The scaling factor from pixel space to coordinate space.
         * @param {Cartesian2} result The object onto which to store the result.
         * @returns {Cartesian2} The modified result parameter or a new instance of {@link Cartesian2} with the pixel's width and height in the x and y properties, respectively.
         *
         * @exception {DeveloperError} drawingBufferWidth must be greater than zero.
         * @exception {DeveloperError} drawingBufferHeight must be greater than zero.
         * @exception {DeveloperError} pixelRatio must be greater than zero.
         *
         * @example
         * // Example 1
         * // Get the width and height of a pixel.
         * var pixelSize = camera.frustum.getPixelDimensions(scene.drawingBufferWidth, scene.drawingBufferHeight, 0.0, scene.pixelRatio, new Cesium.Cartesian2());
         */
        OrthographicOffCenterFrustum.prototype.getPixelDimensions = function(drawingBufferWidth, drawingBufferHeight, distance, pixelRatio, result) {
            update(this);

            if (pixelRatio instanceof Cartesian2.Cartesian2) {
                result = pixelRatio;
                pixelRatio = 1.0;
                Transforms.deprecationWarning('getPixelDimensions-parameter-change', 'getPixelDimensions now takes a pixelRatio argument before the result argument in Cesium 1.63. The previous function definition will no longer work in 1.65.');
            }

            //>>includeStart('debug', pragmas.debug);
            if (!defined.defined(drawingBufferWidth) || !defined.defined(drawingBufferHeight)) {
                throw new Check.DeveloperError('Both drawingBufferWidth and drawingBufferHeight are required.');
            }
            if (drawingBufferWidth <= 0) {
                throw new Check.DeveloperError('drawingBufferWidth must be greater than zero.');
            }
            if (drawingBufferHeight <= 0) {
                throw new Check.DeveloperError('drawingBufferHeight must be greater than zero.');
            }
            if (!defined.defined(distance)) {
                throw new Check.DeveloperError('distance is required.');
            }
            if (!defined.defined(pixelRatio)) {
                throw new Check.DeveloperError('pixelRatio is required.');
            }
            if (pixelRatio <= 0) {
                throw new Check.DeveloperError('pixelRatio must be greater than zero.');
            }
            if (!defined.defined(result)) {
                throw new Check.DeveloperError('A result object is required.');
            }
            //>>includeEnd('debug');

            var frustumWidth = this.right - this.left;
            var frustumHeight = this.top - this.bottom;
            var pixelWidth = pixelRatio * frustumWidth / drawingBufferWidth;
            var pixelHeight = pixelRatio * frustumHeight / drawingBufferHeight;

            result.x = pixelWidth;
            result.y = pixelHeight;
            return result;
        };

        /**
         * Returns a duplicate of a OrthographicOffCenterFrustum instance.
         *
         * @param {OrthographicOffCenterFrustum} [result] The object onto which to store the result.
         * @returns {OrthographicOffCenterFrustum} The modified result parameter or a new OrthographicOffCenterFrustum instance if one was not provided.
         */
        OrthographicOffCenterFrustum.prototype.clone = function(result) {
            if (!defined.defined(result)) {
                result = new OrthographicOffCenterFrustum();
            }

            result.left = this.left;
            result.right = this.right;
            result.top = this.top;
            result.bottom = this.bottom;
            result.near = this.near;
            result.far = this.far;

            // force update of clone to compute matrices
            result._left = undefined;
            result._right = undefined;
            result._top = undefined;
            result._bottom = undefined;
            result._near = undefined;
            result._far = undefined;

            return result;
        };

        /**
         * Compares the provided OrthographicOffCenterFrustum componentwise and returns
         * <code>true</code> if they are equal, <code>false</code> otherwise.
         *
         * @param {OrthographicOffCenterFrustum} [other] The right hand side OrthographicOffCenterFrustum.
         * @returns {Boolean} <code>true</code> if they are equal, <code>false</code> otherwise.
         */
        OrthographicOffCenterFrustum.prototype.equals = function(other) {
            return (defined.defined(other) && other instanceof OrthographicOffCenterFrustum &&
                    this.right === other.right &&
                    this.left === other.left &&
                    this.top === other.top &&
                    this.bottom === other.bottom &&
                    this.near === other.near &&
                    this.far === other.far);
        };

        /**
         * Compares the provided OrthographicOffCenterFrustum componentwise and returns
         * <code>true</code> if they pass an absolute or relative tolerance test,
         * <code>false</code> otherwise.
         *
         * @param {OrthographicOffCenterFrustum} other The right hand side OrthographicOffCenterFrustum.
         * @param {Number} relativeEpsilon The relative epsilon tolerance to use for equality testing.
         * @param {Number} [absoluteEpsilon=relativeEpsilon] The absolute epsilon tolerance to use for equality testing.
         * @returns {Boolean} <code>true</code> if this and other are within the provided epsilon, <code>false</code> otherwise.
         */
        OrthographicOffCenterFrustum.prototype.equalsEpsilon = function(other, relativeEpsilon, absoluteEpsilon) {
            return (other === this) ||
                   (defined.defined(other) &&
                    other instanceof OrthographicOffCenterFrustum &&
                    _Math.CesiumMath.equalsEpsilon(this.right, other.right, relativeEpsilon, absoluteEpsilon) &&
                    _Math.CesiumMath.equalsEpsilon(this.left, other.left, relativeEpsilon, absoluteEpsilon) &&
                    _Math.CesiumMath.equalsEpsilon(this.top, other.top, relativeEpsilon, absoluteEpsilon) &&
                    _Math.CesiumMath.equalsEpsilon(this.bottom, other.bottom, relativeEpsilon, absoluteEpsilon) &&
                    _Math.CesiumMath.equalsEpsilon(this.near, other.near, relativeEpsilon, absoluteEpsilon) &&
                    _Math.CesiumMath.equalsEpsilon(this.far, other.far, relativeEpsilon, absoluteEpsilon));
        };

    /**
         * The viewing frustum is defined by 6 planes.
         * Each plane is represented by a {@link Cartesian4} object, where the x, y, and z components
         * define the unit vector normal to the plane, and the w component is the distance of the
         * plane from the origin/camera position.
         *
         * @alias OrthographicFrustum
         * @constructor
         *
         * @param {Object} [options] An object with the following properties:
         * @param {Number} [options.width] The width of the frustum in meters.
         * @param {Number} [options.aspectRatio] The aspect ratio of the frustum's width to it's height.
         * @param {Number} [options.near=1.0] The distance of the near plane.
         * @param {Number} [options.far=500000000.0] The distance of the far plane.
         *
         * @example
         * var maxRadii = ellipsoid.maximumRadius;
         *
         * var frustum = new Cesium.OrthographicFrustum();
         * frustum.near = 0.01 * maxRadii;
         * frustum.far = 50.0 * maxRadii;
         */
        function OrthographicFrustum(options) {
            options = defaultValue.defaultValue(options, defaultValue.defaultValue.EMPTY_OBJECT);

            this._offCenterFrustum = new OrthographicOffCenterFrustum();

            /**
             * The horizontal width of the frustum in meters.
             * @type {Number}
             * @default undefined
             */
            this.width = options.width;
            this._width = undefined;

            /**
             * The aspect ratio of the frustum's width to it's height.
             * @type {Number}
             * @default undefined
             */
            this.aspectRatio = options.aspectRatio;
            this._aspectRatio = undefined;

            /**
             * The distance of the near plane.
             * @type {Number}
             * @default 1.0
             */
            this.near = defaultValue.defaultValue(options.near, 1.0);
            this._near = this.near;

            /**
             * The distance of the far plane.
             * @type {Number}
             * @default 500000000.0;
             */
            this.far = defaultValue.defaultValue(options.far, 500000000.0);
            this._far = this.far;
        }

        /**
         * The number of elements used to pack the object into an array.
         * @type {Number}
         */
        OrthographicFrustum.packedLength = 4;

        /**
         * Stores the provided instance into the provided array.
         *
         * @param {OrthographicFrustum} value The value to pack.
         * @param {Number[]} array The array to pack into.
         * @param {Number} [startingIndex=0] The index into the array at which to start packing the elements.
         *
         * @returns {Number[]} The array that was packed into
         */
        OrthographicFrustum.pack = function(value, array, startingIndex) {
            //>>includeStart('debug', pragmas.debug);
            Check.Check.typeOf.object('value', value);
            Check.Check.defined('array', array);
            //>>includeEnd('debug');

            startingIndex = defaultValue.defaultValue(startingIndex, 0);

            array[startingIndex++] = value.width;
            array[startingIndex++] = value.aspectRatio;
            array[startingIndex++] = value.near;
            array[startingIndex] = value.far;

            return array;
        };

        /**
         * Retrieves an instance from a packed array.
         *
         * @param {Number[]} array The packed array.
         * @param {Number} [startingIndex=0] The starting index of the element to be unpacked.
         * @param {OrthographicFrustum} [result] The object into which to store the result.
         * @returns {OrthographicFrustum} The modified result parameter or a new OrthographicFrustum instance if one was not provided.
         */
        OrthographicFrustum.unpack = function(array, startingIndex, result) {
            //>>includeStart('debug', pragmas.debug);
            Check.Check.defined('array', array);
            //>>includeEnd('debug');

            startingIndex = defaultValue.defaultValue(startingIndex, 0);

            if (!defined.defined(result)) {
                result = new OrthographicFrustum();
            }

            result.width = array[startingIndex++];
            result.aspectRatio = array[startingIndex++];
            result.near = array[startingIndex++];
            result.far = array[startingIndex];

            return result;
        };

        function update$1(frustum) {
            //>>includeStart('debug', pragmas.debug);
            if (!defined.defined(frustum.width) || !defined.defined(frustum.aspectRatio) || !defined.defined(frustum.near) || !defined.defined(frustum.far)) {
                throw new Check.DeveloperError('width, aspectRatio, near, or far parameters are not set.');
            }
            //>>includeEnd('debug');

            var f = frustum._offCenterFrustum;

            if (frustum.width !== frustum._width || frustum.aspectRatio !== frustum._aspectRatio ||
                frustum.near !== frustum._near || frustum.far !== frustum._far) {
                //>>includeStart('debug', pragmas.debug);
                if (frustum.aspectRatio < 0) {
                    throw new Check.DeveloperError('aspectRatio must be positive.');
                }
                if (frustum.near < 0 || frustum.near > frustum.far) {
                    throw new Check.DeveloperError('near must be greater than zero and less than far.');
                }
                //>>includeEnd('debug');

                frustum._aspectRatio = frustum.aspectRatio;
                frustum._width = frustum.width;
                frustum._near = frustum.near;
                frustum._far = frustum.far;

                var ratio = 1.0 / frustum.aspectRatio;
                f.right = frustum.width * 0.5;
                f.left = -f.right;
                f.top = ratio * f.right;
                f.bottom = -f.top;
                f.near = frustum.near;
                f.far = frustum.far;

            }
        }

        defineProperties.defineProperties(OrthographicFrustum.prototype, {
            /**
             * Gets the orthographic projection matrix computed from the view frustum.
             * @memberof OrthographicFrustum.prototype
             * @type {Matrix4}
             * @readonly
             */
            projectionMatrix : {
                get : function() {
                    update$1(this);
                    return this._offCenterFrustum.projectionMatrix;
                }
            }

        });

        /**
         * Creates a culling volume for this frustum.
         *
         * @param {Cartesian3} position The eye position.
         * @param {Cartesian3} direction The view direction.
         * @param {Cartesian3} up The up direction.
         * @returns {CullingVolume} A culling volume at the given position and orientation.
         *
         * @example
         * // Check if a bounding volume intersects the frustum.
         * var cullingVolume = frustum.computeCullingVolume(cameraPosition, cameraDirection, cameraUp);
         * var intersect = cullingVolume.computeVisibility(boundingVolume);
         */
        OrthographicFrustum.prototype.computeCullingVolume = function(position, direction, up) {
            update$1(this);
            return this._offCenterFrustum.computeCullingVolume(position, direction, up);
        };

        /**
         * Returns the pixel's width and height in meters.
         *
         * @param {Number} drawingBufferWidth The width of the drawing buffer.
         * @param {Number} drawingBufferHeight The height of the drawing buffer.
         * @param {Number} distance The distance to the near plane in meters.
         * @param {Number} pixelRatio The scaling factor from pixel space to coordinate space.
         * @param {Cartesian2} result The object onto which to store the result.
         * @returns {Cartesian2} The modified result parameter or a new instance of {@link Cartesian2} with the pixel's width and height in the x and y properties, respectively.
         *
         * @exception {DeveloperError} drawingBufferWidth must be greater than zero.
         * @exception {DeveloperError} drawingBufferHeight must be greater than zero.
         * @exception {DeveloperError} pixelRatio must be greater than zero.
         *
         * @example
         * // Example 1
         * // Get the width and height of a pixel.
         * var pixelSize = camera.frustum.getPixelDimensions(scene.drawingBufferWidth, scene.drawingBufferHeight, 0.0, scene.pixelRatio, new Cesium.Cartesian2());
         */
        OrthographicFrustum.prototype.getPixelDimensions = function(drawingBufferWidth, drawingBufferHeight, distance, pixelRatio, result) {
            update$1(this);

            if (pixelRatio instanceof Cartesian2.Cartesian2) {
                result = pixelRatio;
                pixelRatio = 1.0;
                Transforms.deprecationWarning('getPixelDimensions-parameter-change', 'getPixelDimensions now takes a pixelRatio argument before the result argument in Cesium 1.63. The previous function definition will no longer work in 1.65.');
            }
            return this._offCenterFrustum.getPixelDimensions(drawingBufferWidth, drawingBufferHeight, distance, pixelRatio, result);
        };

        /**
         * Returns a duplicate of a OrthographicFrustum instance.
         *
         * @param {OrthographicFrustum} [result] The object onto which to store the result.
         * @returns {OrthographicFrustum} The modified result parameter or a new OrthographicFrustum instance if one was not provided.
         */
        OrthographicFrustum.prototype.clone = function(result) {
            if (!defined.defined(result)) {
                result = new OrthographicFrustum();
            }

            result.aspectRatio = this.aspectRatio;
            result.width = this.width;
            result.near = this.near;
            result.far = this.far;

            // force update of clone to compute matrices
            result._aspectRatio = undefined;
            result._width = undefined;
            result._near = undefined;
            result._far = undefined;

            this._offCenterFrustum.clone(result._offCenterFrustum);

            return result;
        };

        /**
         * Compares the provided OrthographicFrustum componentwise and returns
         * <code>true</code> if they are equal, <code>false</code> otherwise.
         *
         * @param {OrthographicFrustum} [other] The right hand side OrthographicFrustum.
         * @returns {Boolean} <code>true</code> if they are equal, <code>false</code> otherwise.
         */
        OrthographicFrustum.prototype.equals = function(other) {
            if (!defined.defined(other) || !(other instanceof OrthographicFrustum)) {
                return false;
            }

            update$1(this);
            update$1(other);

            return (this.width === other.width &&
                    this.aspectRatio === other.aspectRatio &&
                    this._offCenterFrustum.equals(other._offCenterFrustum));
        };

        /**
         * Compares the provided OrthographicFrustum componentwise and returns
         * <code>true</code> if they pass an absolute or relative tolerance test,
         * <code>false</code> otherwise.
         *
         * @param {OrthographicFrustum} other The right hand side OrthographicFrustum.
         * @param {Number} relativeEpsilon The relative epsilon tolerance to use for equality testing.
         * @param {Number} [absoluteEpsilon=relativeEpsilon] The absolute epsilon tolerance to use for equality testing.
         * @returns {Boolean} <code>true</code> if this and other are within the provided epsilon, <code>false</code> otherwise.
         */
        OrthographicFrustum.prototype.equalsEpsilon = function(other, relativeEpsilon, absoluteEpsilon) {
            if (!defined.defined(other) || !(other instanceof OrthographicFrustum)) {
                return false;
            }

            update$1(this);
            update$1(other);

            return (_Math.CesiumMath.equalsEpsilon(this.width, other.width, relativeEpsilon, absoluteEpsilon) &&
                    _Math.CesiumMath.equalsEpsilon(this.aspectRatio, other.aspectRatio, relativeEpsilon, absoluteEpsilon) &&
                    this._offCenterFrustum.equalsEpsilon(other._offCenterFrustum, relativeEpsilon, absoluteEpsilon));
        };

    /**
         * The viewing frustum is defined by 6 planes.
         * Each plane is represented by a {@link Cartesian4} object, where the x, y, and z components
         * define the unit vector normal to the plane, and the w component is the distance of the
         * plane from the origin/camera position.
         *
         * @alias PerspectiveOffCenterFrustum
         * @constructor
         *
         * @param {Object} [options] An object with the following properties:
         * @param {Number} [options.left] The left clipping plane distance.
         * @param {Number} [options.right] The right clipping plane distance.
         * @param {Number} [options.top] The top clipping plane distance.
         * @param {Number} [options.bottom] The bottom clipping plane distance.
         * @param {Number} [options.near=1.0] The near clipping plane distance.
         * @param {Number} [options.far=500000000.0] The far clipping plane distance.
         *
         * @example
         * var frustum = new Cesium.PerspectiveOffCenterFrustum({
         *     left : -1.0,
         *     right : 1.0,
         *     top : 1.0,
         *     bottom : -1.0,
         *     near : 1.0,
         *     far : 100.0
         * });
         *
         * @see PerspectiveFrustum
         */
        function PerspectiveOffCenterFrustum(options) {
            options = defaultValue.defaultValue(options, defaultValue.defaultValue.EMPTY_OBJECT);

            /**
             * Defines the left clipping plane.
             * @type {Number}
             * @default undefined
             */
            this.left = options.left;
            this._left = undefined;

            /**
             * Defines the right clipping plane.
             * @type {Number}
             * @default undefined
             */
            this.right = options.right;
            this._right = undefined;

            /**
             * Defines the top clipping plane.
             * @type {Number}
             * @default undefined
             */
            this.top = options.top;
            this._top = undefined;

            /**
             * Defines the bottom clipping plane.
             * @type {Number}
             * @default undefined
             */
            this.bottom = options.bottom;
            this._bottom = undefined;

            /**
             * The distance of the near plane.
             * @type {Number}
             * @default 1.0
             */
            this.near = defaultValue.defaultValue(options.near, 1.0);
            this._near = this.near;

            /**
             * The distance of the far plane.
             * @type {Number}
             * @default 500000000.0
             */
            this.far = defaultValue.defaultValue(options.far, 500000000.0);
            this._far = this.far;

            this._cullingVolume = new CullingVolume();
            this._perspectiveMatrix = new Transforms.Matrix4();
            this._infinitePerspective = new Transforms.Matrix4();
        }

        function update$2(frustum) {
            //>>includeStart('debug', pragmas.debug);
            if (!defined.defined(frustum.right) || !defined.defined(frustum.left) ||
                !defined.defined(frustum.top) || !defined.defined(frustum.bottom) ||
                !defined.defined(frustum.near) || !defined.defined(frustum.far)) {
                throw new Check.DeveloperError('right, left, top, bottom, near, or far parameters are not set.');
            }
            //>>includeEnd('debug');

            var t = frustum.top;
            var b = frustum.bottom;
            var r = frustum.right;
            var l = frustum.left;
            var n = frustum.near;
            var f = frustum.far;

            if (t !== frustum._top || b !== frustum._bottom ||
                l !== frustum._left || r !== frustum._right ||
                n !== frustum._near || f !== frustum._far) {

                //>>includeStart('debug', pragmas.debug);
                if (frustum.near <= 0 || frustum.near > frustum.far) {
                    throw new Check.DeveloperError('near must be greater than zero and less than far.');
                }
                //>>includeEnd('debug');

                frustum._left = l;
                frustum._right = r;
                frustum._top = t;
                frustum._bottom = b;
                frustum._near = n;
                frustum._far = f;
                frustum._perspectiveMatrix = Transforms.Matrix4.computePerspectiveOffCenter(l, r, b, t, n, f, frustum._perspectiveMatrix);
                frustum._infinitePerspective = Transforms.Matrix4.computeInfinitePerspectiveOffCenter(l, r, b, t, n, frustum._infinitePerspective);
            }
        }

        defineProperties.defineProperties(PerspectiveOffCenterFrustum.prototype, {
            /**
             * Gets the perspective projection matrix computed from the view frustum.
             * @memberof PerspectiveOffCenterFrustum.prototype
             * @type {Matrix4}
             * @readonly
             *
             * @see PerspectiveOffCenterFrustum#infiniteProjectionMatrix
             */
            projectionMatrix : {
                get : function() {
                    update$2(this);
                    return this._perspectiveMatrix;
                }
            },

            /**
             * Gets the perspective projection matrix computed from the view frustum with an infinite far plane.
             * @memberof PerspectiveOffCenterFrustum.prototype
             * @type {Matrix4}
             * @readonly
             *
             * @see PerspectiveOffCenterFrustum#projectionMatrix
             */
            infiniteProjectionMatrix : {
                get : function() {
                    update$2(this);
                    return this._infinitePerspective;
                }
            }
        });

        var getPlanesRight$1 = new Cartesian2.Cartesian3();
        var getPlanesNearCenter$1 = new Cartesian2.Cartesian3();
        var getPlanesFarCenter = new Cartesian2.Cartesian3();
        var getPlanesNormal = new Cartesian2.Cartesian3();
        /**
         * Creates a culling volume for this frustum.
         *
         * @param {Cartesian3} position The eye position.
         * @param {Cartesian3} direction The view direction.
         * @param {Cartesian3} up The up direction.
         * @returns {CullingVolume} A culling volume at the given position and orientation.
         *
         * @example
         * // Check if a bounding volume intersects the frustum.
         * var cullingVolume = frustum.computeCullingVolume(cameraPosition, cameraDirection, cameraUp);
         * var intersect = cullingVolume.computeVisibility(boundingVolume);
         */
        PerspectiveOffCenterFrustum.prototype.computeCullingVolume = function(position, direction, up) {
            //>>includeStart('debug', pragmas.debug);
            if (!defined.defined(position)) {
                throw new Check.DeveloperError('position is required.');
            }

            if (!defined.defined(direction)) {
                throw new Check.DeveloperError('direction is required.');
            }

            if (!defined.defined(up)) {
                throw new Check.DeveloperError('up is required.');
            }
            //>>includeEnd('debug');

            var planes = this._cullingVolume.planes;

            var t = this.top;
            var b = this.bottom;
            var r = this.right;
            var l = this.left;
            var n = this.near;
            var f = this.far;

            var right = Cartesian2.Cartesian3.cross(direction, up, getPlanesRight$1);

            var nearCenter = getPlanesNearCenter$1;
            Cartesian2.Cartesian3.multiplyByScalar(direction, n, nearCenter);
            Cartesian2.Cartesian3.add(position, nearCenter, nearCenter);

            var farCenter = getPlanesFarCenter;
            Cartesian2.Cartesian3.multiplyByScalar(direction, f, farCenter);
            Cartesian2.Cartesian3.add(position, farCenter, farCenter);

            var normal = getPlanesNormal;

            //Left plane computation
            Cartesian2.Cartesian3.multiplyByScalar(right, l, normal);
            Cartesian2.Cartesian3.add(nearCenter, normal, normal);
            Cartesian2.Cartesian3.subtract(normal, position, normal);
            Cartesian2.Cartesian3.normalize(normal, normal);
            Cartesian2.Cartesian3.cross(normal, up, normal);
            Cartesian2.Cartesian3.normalize(normal, normal);

            var plane = planes[0];
            if (!defined.defined(plane)) {
                plane = planes[0] = new Transforms.Cartesian4();
            }
            plane.x = normal.x;
            plane.y = normal.y;
            plane.z = normal.z;
            plane.w = -Cartesian2.Cartesian3.dot(normal, position);

            //Right plane computation
            Cartesian2.Cartesian3.multiplyByScalar(right, r, normal);
            Cartesian2.Cartesian3.add(nearCenter, normal, normal);
            Cartesian2.Cartesian3.subtract(normal, position, normal);
            Cartesian2.Cartesian3.cross(up, normal, normal);
            Cartesian2.Cartesian3.normalize(normal, normal);

            plane = planes[1];
            if (!defined.defined(plane)) {
                plane = planes[1] = new Transforms.Cartesian4();
            }
            plane.x = normal.x;
            plane.y = normal.y;
            plane.z = normal.z;
            plane.w = -Cartesian2.Cartesian3.dot(normal, position);

            //Bottom plane computation
            Cartesian2.Cartesian3.multiplyByScalar(up, b, normal);
            Cartesian2.Cartesian3.add(nearCenter, normal, normal);
            Cartesian2.Cartesian3.subtract(normal, position, normal);
            Cartesian2.Cartesian3.cross(right, normal, normal);
            Cartesian2.Cartesian3.normalize(normal, normal);

            plane = planes[2];
            if (!defined.defined(plane)) {
                plane = planes[2] = new Transforms.Cartesian4();
            }
            plane.x = normal.x;
            plane.y = normal.y;
            plane.z = normal.z;
            plane.w = -Cartesian2.Cartesian3.dot(normal, position);

            //Top plane computation
            Cartesian2.Cartesian3.multiplyByScalar(up, t, normal);
            Cartesian2.Cartesian3.add(nearCenter, normal, normal);
            Cartesian2.Cartesian3.subtract(normal, position, normal);
            Cartesian2.Cartesian3.cross(normal, right, normal);
            Cartesian2.Cartesian3.normalize(normal, normal);

            plane = planes[3];
            if (!defined.defined(plane)) {
                plane = planes[3] = new Transforms.Cartesian4();
            }
            plane.x = normal.x;
            plane.y = normal.y;
            plane.z = normal.z;
            plane.w = -Cartesian2.Cartesian3.dot(normal, position);

            //Near plane computation
            plane = planes[4];
            if (!defined.defined(plane)) {
                plane = planes[4] = new Transforms.Cartesian4();
            }
            plane.x = direction.x;
            plane.y = direction.y;
            plane.z = direction.z;
            plane.w = -Cartesian2.Cartesian3.dot(direction, nearCenter);

            //Far plane computation
            Cartesian2.Cartesian3.negate(direction, normal);

            plane = planes[5];
            if (!defined.defined(plane)) {
                plane = planes[5] = new Transforms.Cartesian4();
            }
            plane.x = normal.x;
            plane.y = normal.y;
            plane.z = normal.z;
            plane.w = -Cartesian2.Cartesian3.dot(normal, farCenter);

            return this._cullingVolume;
        };

        /**
         * Returns the pixel's width and height in meters.
         *
         * @param {Number} drawingBufferWidth The width of the drawing buffer.
         * @param {Number} drawingBufferHeight The height of the drawing buffer.
         * @param {Number} distance The distance to the near plane in meters.
         * @param {Number} pixelRatio The scaling factor from pixel space to coordinate space.
         * @param {Cartesian2} result The object onto which to store the result.
         * @returns {Cartesian2} The modified result parameter or a new instance of {@link Cartesian2} with the pixel's width and height in the x and y properties, respectively.
         *
         * @exception {DeveloperError} drawingBufferWidth must be greater than zero.
         * @exception {DeveloperError} drawingBufferHeight must be greater than zero.
         * @exception {DeveloperError} pixelRatio must be greater than zero.
         *
         * @example
         * // Example 1
         * // Get the width and height of a pixel.
         * var pixelSize = camera.frustum.getPixelDimensions(scene.drawingBufferWidth, scene.drawingBufferHeight, 1.0, scene.pixelRatio, new Cesium.Cartesian2());
         *
         * @example
         * // Example 2
         * // Get the width and height of a pixel if the near plane was set to 'distance'.
         * // For example, get the size of a pixel of an image on a billboard.
         * var position = camera.position;
         * var direction = camera.direction;
         * var toCenter = Cesium.Cartesian3.subtract(primitive.boundingVolume.center, position, new Cesium.Cartesian3());      // vector from camera to a primitive
         * var toCenterProj = Cesium.Cartesian3.multiplyByScalar(direction, Cesium.Cartesian3.dot(direction, toCenter), new Cesium.Cartesian3()); // project vector onto camera direction vector
         * var distance = Cesium.Cartesian3.magnitude(toCenterProj);
         * var pixelSize = camera.frustum.getPixelDimensions(scene.drawingBufferWidth, scene.drawingBufferHeight, distance, scene.pixelRatio, new Cesium.Cartesian2());
         */
        PerspectiveOffCenterFrustum.prototype.getPixelDimensions = function(drawingBufferWidth, drawingBufferHeight, distance, pixelRatio, result) {
            update$2(this);

            if (pixelRatio instanceof Cartesian2.Cartesian2) {
                result = pixelRatio;
                pixelRatio = 1.0;
                Transforms.deprecationWarning('getPixelDimensions-parameter-change', 'getPixelDimensions now takes a pixelRatio argument before the result argument in Cesium 1.63. The previous function definition will no longer work in 1.65.');
            }

            //>>includeStart('debug', pragmas.debug);
            if (!defined.defined(drawingBufferWidth) || !defined.defined(drawingBufferHeight)) {
                throw new Check.DeveloperError('Both drawingBufferWidth and drawingBufferHeight are required.');
            }
            if (drawingBufferWidth <= 0) {
                throw new Check.DeveloperError('drawingBufferWidth must be greater than zero.');
            }
            if (drawingBufferHeight <= 0) {
                throw new Check.DeveloperError('drawingBufferHeight must be greater than zero.');
            }
            if (!defined.defined(distance)) {
                throw new Check.DeveloperError('distance is required.');
            }
            if (!defined.defined(pixelRatio)) {
                throw new Check.DeveloperError('pixelRatio is required');
            }
            if (pixelRatio <= 0) {
                throw new Check.DeveloperError('pixelRatio must be greater than zero.');
            }
            if (!defined.defined(result)) {
                throw new Check.DeveloperError('A result object is required.');
            }
            //>>includeEnd('debug');

            var inverseNear = 1.0 / this.near;
            var tanTheta = this.top * inverseNear;
            var pixelHeight = 2.0 * pixelRatio * distance * tanTheta / drawingBufferHeight;
            tanTheta = this.right * inverseNear;
            var pixelWidth = 2.0 * pixelRatio * distance * tanTheta / drawingBufferWidth;

            result.x = pixelWidth;
            result.y = pixelHeight;
            return result;
        };

        /**
         * Returns a duplicate of a PerspectiveOffCenterFrustum instance.
         *
         * @param {PerspectiveOffCenterFrustum} [result] The object onto which to store the result.
         * @returns {PerspectiveOffCenterFrustum} The modified result parameter or a new PerspectiveFrustum instance if one was not provided.
         */
        PerspectiveOffCenterFrustum.prototype.clone = function(result) {
            if (!defined.defined(result)) {
                result = new PerspectiveOffCenterFrustum();
            }

            result.right = this.right;
            result.left = this.left;
            result.top = this.top;
            result.bottom = this.bottom;
            result.near = this.near;
            result.far = this.far;

            // force update of clone to compute matrices
            result._left = undefined;
            result._right = undefined;
            result._top = undefined;
            result._bottom = undefined;
            result._near = undefined;
            result._far = undefined;

            return result;
        };

        /**
         * Compares the provided PerspectiveOffCenterFrustum componentwise and returns
         * <code>true</code> if they are equal, <code>false</code> otherwise.
         *
         * @param {PerspectiveOffCenterFrustum} [other] The right hand side PerspectiveOffCenterFrustum.
         * @returns {Boolean} <code>true</code> if they are equal, <code>false</code> otherwise.
         */
        PerspectiveOffCenterFrustum.prototype.equals = function(other) {
            return (defined.defined(other) && other instanceof PerspectiveOffCenterFrustum &&
                    this.right === other.right &&
                    this.left === other.left &&
                    this.top === other.top &&
                    this.bottom === other.bottom &&
                    this.near === other.near &&
                    this.far === other.far);
        };

        /**
         * Compares the provided PerspectiveOffCenterFrustum componentwise and returns
         * <code>true</code> if they pass an absolute or relative tolerance test,
         * <code>false</code> otherwise.
         *
         * @param {PerspectiveOffCenterFrustum} other The right hand side PerspectiveOffCenterFrustum.
         * @param {Number} relativeEpsilon The relative epsilon tolerance to use for equality testing.
         * @param {Number} [absoluteEpsilon=relativeEpsilon] The absolute epsilon tolerance to use for equality testing.
         * @returns {Boolean} <code>true</code> if this and other are within the provided epsilon, <code>false</code> otherwise.
         */
        PerspectiveOffCenterFrustum.prototype.equalsEpsilon = function(other, relativeEpsilon, absoluteEpsilon) {
            return (other === this) ||
                   (defined.defined(other) &&
                    other instanceof PerspectiveOffCenterFrustum &&
                    _Math.CesiumMath.equalsEpsilon(this.right, other.right, relativeEpsilon, absoluteEpsilon) &&
                    _Math.CesiumMath.equalsEpsilon(this.left, other.left, relativeEpsilon, absoluteEpsilon) &&
                    _Math.CesiumMath.equalsEpsilon(this.top, other.top, relativeEpsilon, absoluteEpsilon) &&
                    _Math.CesiumMath.equalsEpsilon(this.bottom, other.bottom, relativeEpsilon, absoluteEpsilon) &&
                    _Math.CesiumMath.equalsEpsilon(this.near, other.near, relativeEpsilon, absoluteEpsilon) &&
                    _Math.CesiumMath.equalsEpsilon(this.far, other.far, relativeEpsilon, absoluteEpsilon));
        };

    /**
         * The viewing frustum is defined by 6 planes.
         * Each plane is represented by a {@link Cartesian4} object, where the x, y, and z components
         * define the unit vector normal to the plane, and the w component is the distance of the
         * plane from the origin/camera position.
         *
         * @alias PerspectiveFrustum
         * @constructor
         *
         * @param {Object} [options] An object with the following properties:
         * @param {Number} [options.fov] The angle of the field of view (FOV), in radians.
         * @param {Number} [options.aspectRatio] The aspect ratio of the frustum's width to it's height.
         * @param {Number} [options.near=1.0] The distance of the near plane.
         * @param {Number} [options.far=500000000.0] The distance of the far plane.
         * @param {Number} [options.xOffset=0.0] The offset in the x direction.
         * @param {Number} [options.yOffset=0.0] The offset in the y direction.
         *
         * @example
         * var frustum = new Cesium.PerspectiveFrustum({
         *     fov : Cesium.Math.PI_OVER_THREE,
         *     aspectRatio : canvas.clientWidth / canvas.clientHeight
         *     near : 1.0,
         *     far : 1000.0
         * });
         *
         * @see PerspectiveOffCenterFrustum
         */
        function PerspectiveFrustum(options) {
            options = defaultValue.defaultValue(options, defaultValue.defaultValue.EMPTY_OBJECT);

            this._offCenterFrustum = new PerspectiveOffCenterFrustum();

            /**
             * The angle of the field of view (FOV), in radians.  This angle will be used
             * as the horizontal FOV if the width is greater than the height, otherwise
             * it will be the vertical FOV.
             * @type {Number}
             * @default undefined
             */
            this.fov = options.fov;
            this._fov = undefined;
            this._fovy = undefined;

            this._sseDenominator = undefined;

            /**
             * The aspect ratio of the frustum's width to it's height.
             * @type {Number}
             * @default undefined
             */
            this.aspectRatio = options.aspectRatio;
            this._aspectRatio = undefined;

            /**
             * The distance of the near plane.
             * @type {Number}
             * @default 1.0
             */
            this.near = defaultValue.defaultValue(options.near, 1.0);
            this._near = this.near;

            /**
             * The distance of the far plane.
             * @type {Number}
             * @default 500000000.0
             */
            this.far = defaultValue.defaultValue(options.far, 500000000.0);
            this._far = this.far;

            /**
             * Offsets the frustum in the x direction.
             * @type {Number}
             * @default 0.0
             */
            this.xOffset = defaultValue.defaultValue(options.xOffset, 0.0);
            this._xOffset = this.xOffset;

            /**
             * Offsets the frustum in the y direction.
             * @type {Number}
             * @default 0.0
             */
            this.yOffset = defaultValue.defaultValue(options.yOffset, 0.0);
            this._yOffset = this.yOffset;
        }

        /**
         * The number of elements used to pack the object into an array.
         * @type {Number}
         */
        PerspectiveFrustum.packedLength = 6;

        /**
         * Stores the provided instance into the provided array.
         *
         * @param {PerspectiveFrustum} value The value to pack.
         * @param {Number[]} array The array to pack into.
         * @param {Number} [startingIndex=0] The index into the array at which to start packing the elements.
         *
         * @returns {Number[]} The array that was packed into
         */
        PerspectiveFrustum.pack = function(value, array, startingIndex) {
            //>>includeStart('debug', pragmas.debug);
            Check.Check.typeOf.object('value', value);
            Check.Check.defined('array', array);
            //>>includeEnd('debug');

            startingIndex = defaultValue.defaultValue(startingIndex, 0);

            array[startingIndex++] = value.fov;
            array[startingIndex++] = value.aspectRatio;
            array[startingIndex++] = value.near;
            array[startingIndex++] = value.far;
            array[startingIndex++] = value.xOffset;
            array[startingIndex] = value.yOffset;

            return array;
        };

        /**
         * Retrieves an instance from a packed array.
         *
         * @param {Number[]} array The packed array.
         * @param {Number} [startingIndex=0] The starting index of the element to be unpacked.
         * @param {PerspectiveFrustum} [result] The object into which to store the result.
         * @returns {PerspectiveFrustum} The modified result parameter or a new PerspectiveFrustum instance if one was not provided.
         */
        PerspectiveFrustum.unpack = function(array, startingIndex, result) {
            //>>includeStart('debug', pragmas.debug);
            Check.Check.defined('array', array);
            //>>includeEnd('debug');

            startingIndex = defaultValue.defaultValue(startingIndex, 0);

            if (!defined.defined(result)) {
                result = new PerspectiveFrustum();
            }

            result.fov = array[startingIndex++];
            result.aspectRatio = array[startingIndex++];
            result.near = array[startingIndex++];
            result.far = array[startingIndex++];
            result.xOffset = array[startingIndex++];
            result.yOffset = array[startingIndex];

            return result;
        };

        function update$3(frustum) {
            //>>includeStart('debug', pragmas.debug);
            if (!defined.defined(frustum.fov) || !defined.defined(frustum.aspectRatio) || !defined.defined(frustum.near) || !defined.defined(frustum.far)) {
                throw new Check.DeveloperError('fov, aspectRatio, near, or far parameters are not set.');
            }
            //>>includeEnd('debug');

            var f = frustum._offCenterFrustum;

            if (frustum.fov !== frustum._fov || frustum.aspectRatio !== frustum._aspectRatio ||
                frustum.near !== frustum._near || frustum.far !== frustum._far ||
                frustum.xOffset !== frustum._xOffset || frustum.yOffset !== frustum._yOffset) {
                //>>includeStart('debug', pragmas.debug);
                if (frustum.fov < 0 || frustum.fov >= Math.PI) {
                    throw new Check.DeveloperError('fov must be in the range [0, PI).');
                }

                if (frustum.aspectRatio < 0) {
                    throw new Check.DeveloperError('aspectRatio must be positive.');
                }

                if (frustum.near < 0 || frustum.near > frustum.far) {
                    throw new Check.DeveloperError('near must be greater than zero and less than far.');
                }
                //>>includeEnd('debug');

                frustum._aspectRatio = frustum.aspectRatio;
                frustum._fov = frustum.fov;
                frustum._fovy = (frustum.aspectRatio <= 1) ? frustum.fov : Math.atan(Math.tan(frustum.fov * 0.5) / frustum.aspectRatio) * 2.0;
                frustum._near = frustum.near;
                frustum._far = frustum.far;
                frustum._sseDenominator = 2.0 * Math.tan(0.5 * frustum._fovy);
                frustum._xOffset = frustum.xOffset;
                frustum._yOffset = frustum.yOffset;

                f.top = frustum.near * Math.tan(0.5 * frustum._fovy);
                f.bottom = -f.top;
                f.right = frustum.aspectRatio * f.top;
                f.left = -f.right;
                f.near = frustum.near;
                f.far = frustum.far;

                f.right += frustum.xOffset;
                f.left += frustum.xOffset;
                f.top += frustum.yOffset;
                f.bottom += frustum.yOffset;
            }
        }

        defineProperties.defineProperties(PerspectiveFrustum.prototype, {
            /**
             * Gets the perspective projection matrix computed from the view frustum.
             * @memberof PerspectiveFrustum.prototype
             * @type {Matrix4}
             * @readonly
             *
             * @see PerspectiveFrustum#infiniteProjectionMatrix
             */
            projectionMatrix : {
                get : function() {
                    update$3(this);
                    return this._offCenterFrustum.projectionMatrix;
                }
            },

            /**
             * The perspective projection matrix computed from the view frustum with an infinite far plane.
             * @memberof PerspectiveFrustum.prototype
             * @type {Matrix4}
             * @readonly
             *
             * @see PerspectiveFrustum#projectionMatrix
             */
            infiniteProjectionMatrix : {
                get : function() {
                    update$3(this);
                    return this._offCenterFrustum.infiniteProjectionMatrix;
                }
            },

            /**
             * Gets the angle of the vertical field of view, in radians.
             * @memberof PerspectiveFrustum.prototype
             * @type {Number}
             * @readonly
             * @default undefined
             */
            fovy : {
                get : function() {
                    update$3(this);
                    return this._fovy;
                }
            },

            /**
             * @readonly
             * @private
             */
            sseDenominator : {
                get : function () {
                    update$3(this);
                    return this._sseDenominator;
                }
            }
        });

        /**
         * Creates a culling volume for this frustum.
         *
         * @param {Cartesian3} position The eye position.
         * @param {Cartesian3} direction The view direction.
         * @param {Cartesian3} up The up direction.
         * @returns {CullingVolume} A culling volume at the given position and orientation.
         *
         * @example
         * // Check if a bounding volume intersects the frustum.
         * var cullingVolume = frustum.computeCullingVolume(cameraPosition, cameraDirection, cameraUp);
         * var intersect = cullingVolume.computeVisibility(boundingVolume);
         */
        PerspectiveFrustum.prototype.computeCullingVolume = function(position, direction, up) {
            update$3(this);
            return this._offCenterFrustum.computeCullingVolume(position, direction, up);
        };

        /**
         * Returns the pixel's width and height in meters.
         *
         * @param {Number} drawingBufferWidth The width of the drawing buffer.
         * @param {Number} drawingBufferHeight The height of the drawing buffer.
         * @param {Number} distance The distance to the near plane in meters.
         * @param {Number} pixelRatio The scaling factor from pixel space to coordinate space.
         * @param {Cartesian2} result The object onto which to store the result.
         * @returns {Cartesian2} The modified result parameter or a new instance of {@link Cartesian2} with the pixel's width and height in the x and y properties, respectively.
         *
         * @exception {DeveloperError} drawingBufferWidth must be greater than zero.
         * @exception {DeveloperError} drawingBufferHeight must be greater than zero.
         * @exception {DeveloperError} pixelRatio must be greater than zero.
         *
         * @example
         * // Example 1
         * // Get the width and height of a pixel.
         * var pixelSize = camera.frustum.getPixelDimensions(scene.drawingBufferWidth, scene.drawingBufferHeight, 1.0, scene.pixelRatio, new Cesium.Cartesian2());
         *
         * @example
         * // Example 2
         * // Get the width and height of a pixel if the near plane was set to 'distance'.
         * // For example, get the size of a pixel of an image on a billboard.
         * var position = camera.position;
         * var direction = camera.direction;
         * var toCenter = Cesium.Cartesian3.subtract(primitive.boundingVolume.center, position, new Cesium.Cartesian3());      // vector from camera to a primitive
         * var toCenterProj = Cesium.Cartesian3.multiplyByScalar(direction, Cesium.Cartesian3.dot(direction, toCenter), new Cesium.Cartesian3()); // project vector onto camera direction vector
         * var distance = Cesium.Cartesian3.magnitude(toCenterProj);
         * var pixelSize = camera.frustum.getPixelDimensions(scene.drawingBufferWidth, scene.drawingBufferHeight, distance, scene.pixelRatio, new Cesium.Cartesian2());
         */
        PerspectiveFrustum.prototype.getPixelDimensions = function(drawingBufferWidth, drawingBufferHeight, distance, pixelRatio, result) {
            update$3(this);

            if (pixelRatio instanceof Cartesian2.Cartesian2) {
                result = pixelRatio;
                pixelRatio = 1.0;
                Transforms.deprecationWarning('getPixelDimensions-parameter-change', 'getPixelDimensions now takes a pixelRatio argument before the result argument in Cesium 1.63. The previous function definition will no longer work in 1.65.');
            }

            return this._offCenterFrustum.getPixelDimensions(drawingBufferWidth, drawingBufferHeight, distance, pixelRatio, result);
        };

        /**
         * Returns a duplicate of a PerspectiveFrustum instance.
         *
         * @param {PerspectiveFrustum} [result] The object onto which to store the result.
         * @returns {PerspectiveFrustum} The modified result parameter or a new PerspectiveFrustum instance if one was not provided.
         */
        PerspectiveFrustum.prototype.clone = function(result) {
            if (!defined.defined(result)) {
                result = new PerspectiveFrustum();
            }

            result.aspectRatio = this.aspectRatio;
            result.fov = this.fov;
            result.near = this.near;
            result.far = this.far;

            // force update of clone to compute matrices
            result._aspectRatio = undefined;
            result._fov = undefined;
            result._near = undefined;
            result._far = undefined;

            this._offCenterFrustum.clone(result._offCenterFrustum);

            return result;
        };

        /**
         * Compares the provided PerspectiveFrustum componentwise and returns
         * <code>true</code> if they are equal, <code>false</code> otherwise.
         *
         * @param {PerspectiveFrustum} [other] The right hand side PerspectiveFrustum.
         * @returns {Boolean} <code>true</code> if they are equal, <code>false</code> otherwise.
         */
        PerspectiveFrustum.prototype.equals = function(other) {
            if (!defined.defined(other) || !(other instanceof PerspectiveFrustum)) {
                return false;
            }

            update$3(this);
            update$3(other);

            return (this.fov === other.fov &&
                    this.aspectRatio === other.aspectRatio &&
                    this._offCenterFrustum.equals(other._offCenterFrustum));
        };

        /**
         * Compares the provided PerspectiveFrustum componentwise and returns
         * <code>true</code> if they pass an absolute or relative tolerance test,
         * <code>false</code> otherwise.
         *
         * @param {PerspectiveFrustum} other The right hand side PerspectiveFrustum.
         * @param {Number} relativeEpsilon The relative epsilon tolerance to use for equality testing.
         * @param {Number} [absoluteEpsilon=relativeEpsilon] The absolute epsilon tolerance to use for equality testing.
         * @returns {Boolean} <code>true</code> if this and other are within the provided epsilon, <code>false</code> otherwise.
         */
        PerspectiveFrustum.prototype.equalsEpsilon = function(other, relativeEpsilon, absoluteEpsilon) {
            if (!defined.defined(other) || !(other instanceof PerspectiveFrustum)) {
                return false;
            }

            update$3(this);
            update$3(other);

            return (_Math.CesiumMath.equalsEpsilon(this.fov, other.fov, relativeEpsilon, absoluteEpsilon) &&
                    _Math.CesiumMath.equalsEpsilon(this.aspectRatio, other.aspectRatio, relativeEpsilon, absoluteEpsilon) &&
                    this._offCenterFrustum.equalsEpsilon(other._offCenterFrustum, relativeEpsilon, absoluteEpsilon));
        };

    var PERSPECTIVE = 0;
        var ORTHOGRAPHIC = 1;

        /**
         * Describes a frustum at the given the origin and orientation.
         *
         * @alias FrustumGeometry
         * @constructor
         *
         * @param {Object} options Object with the following properties:
         * @param {PerspectiveFrustum|OrthographicFrustum} options.frustum The frustum.
         * @param {Cartesian3} options.origin The origin of the frustum.
         * @param {Quaternion} options.orientation The orientation of the frustum.
         * @param {VertexFormat} [options.vertexFormat=VertexFormat.DEFAULT] The vertex attributes to be computed.
         */
        function FrustumGeometry(options) {
            //>>includeStart('debug', pragmas.debug);
            Check.Check.typeOf.object('options', options);
            Check.Check.typeOf.object('options.frustum', options.frustum);
            Check.Check.typeOf.object('options.origin', options.origin);
            Check.Check.typeOf.object('options.orientation', options.orientation);
            //>>includeEnd('debug');

            var frustum = options.frustum;
            var orientation = options.orientation;
            var origin = options.origin;
            var vertexFormat = defaultValue.defaultValue(options.vertexFormat, VertexFormat.VertexFormat.DEFAULT);

            // This is private because it is used by DebugCameraPrimitive to draw a multi-frustum by
            // creating multiple FrustumGeometrys. This way the near plane of one frustum doesn't overlap
            // the far plane of another.
            var drawNearPlane = defaultValue.defaultValue(options._drawNearPlane, true);

            var frustumType;
            var frustumPackedLength;
            if (frustum instanceof PerspectiveFrustum) {
                frustumType = PERSPECTIVE;
                frustumPackedLength = PerspectiveFrustum.packedLength;
            } else if (frustum instanceof OrthographicFrustum) {
                frustumType = ORTHOGRAPHIC;
                frustumPackedLength = OrthographicFrustum.packedLength;
            }

            this._frustumType = frustumType;
            this._frustum = frustum.clone();
            this._origin = Cartesian2.Cartesian3.clone(origin);
            this._orientation = Transforms.Quaternion.clone(orientation);
            this._drawNearPlane = drawNearPlane;
            this._vertexFormat = vertexFormat;
            this._workerName = 'createFrustumGeometry';

            /**
             * The number of elements used to pack the object into an array.
             * @type {Number}
             */
            this.packedLength = 2 + frustumPackedLength + Cartesian2.Cartesian3.packedLength + Transforms.Quaternion.packedLength + VertexFormat.VertexFormat.packedLength;
        }

        /**
         * Stores the provided instance into the provided array.
         *
         * @param {FrustumGeometry} value The value to pack.
         * @param {Number[]} array The array to pack into.
         * @param {Number} [startingIndex=0] The index into the array at which to start packing the elements.
         *
         * @returns {Number[]} The array that was packed into
         */
        FrustumGeometry.pack = function(value, array, startingIndex) {
            //>>includeStart('debug', pragmas.debug);
            Check.Check.typeOf.object('value', value);
            Check.Check.defined('array', array);
            //>>includeEnd('debug');

            startingIndex = defaultValue.defaultValue(startingIndex, 0);

            var frustumType = value._frustumType;
            var frustum = value._frustum;

            array[startingIndex++] = frustumType;

            if (frustumType === PERSPECTIVE) {
                PerspectiveFrustum.pack(frustum, array, startingIndex);
                startingIndex += PerspectiveFrustum.packedLength;
            } else {
                OrthographicFrustum.pack(frustum, array, startingIndex);
                startingIndex += OrthographicFrustum.packedLength;
            }

            Cartesian2.Cartesian3.pack(value._origin, array, startingIndex);
            startingIndex += Cartesian2.Cartesian3.packedLength;
            Transforms.Quaternion.pack(value._orientation, array, startingIndex);
            startingIndex += Transforms.Quaternion.packedLength;
            VertexFormat.VertexFormat.pack(value._vertexFormat, array, startingIndex);
            startingIndex += VertexFormat.VertexFormat.packedLength;
            array[startingIndex] = value._drawNearPlane ? 1.0 : 0.0;

            return array;
        };

        var scratchPackPerspective = new PerspectiveFrustum();
        var scratchPackOrthographic = new OrthographicFrustum();
        var scratchPackQuaternion = new Transforms.Quaternion();
        var scratchPackorigin = new Cartesian2.Cartesian3();
        var scratchVertexFormat = new VertexFormat.VertexFormat();

        /**
         * Retrieves an instance from a packed array.
         *
         * @param {Number[]} array The packed array.
         * @param {Number} [startingIndex=0] The starting index of the element to be unpacked.
         * @param {FrustumGeometry} [result] The object into which to store the result.
         */
        FrustumGeometry.unpack = function(array, startingIndex, result) {
            //>>includeStart('debug', pragmas.debug);
            Check.Check.defined('array', array);
            //>>includeEnd('debug');

            startingIndex = defaultValue.defaultValue(startingIndex, 0);

            var frustumType = array[startingIndex++];

            var frustum;
            if (frustumType === PERSPECTIVE) {
                frustum = PerspectiveFrustum.unpack(array, startingIndex, scratchPackPerspective);
                startingIndex += PerspectiveFrustum.packedLength;
            } else {
                frustum = OrthographicFrustum.unpack(array, startingIndex, scratchPackOrthographic);
                startingIndex += OrthographicFrustum.packedLength;
            }

            var origin = Cartesian2.Cartesian3.unpack(array, startingIndex, scratchPackorigin);
            startingIndex += Cartesian2.Cartesian3.packedLength;
            var orientation = Transforms.Quaternion.unpack(array, startingIndex, scratchPackQuaternion);
            startingIndex += Transforms.Quaternion.packedLength;
            var vertexFormat = VertexFormat.VertexFormat.unpack(array, startingIndex, scratchVertexFormat);
            startingIndex += VertexFormat.VertexFormat.packedLength;
            var drawNearPlane = array[startingIndex] === 1.0;

            if (!defined.defined(result)) {
                return new FrustumGeometry({
                    frustum : frustum,
                    origin : origin,
                    orientation : orientation,
                    vertexFormat : vertexFormat,
                    _drawNearPlane : drawNearPlane
                });
            }

            var frustumResult = frustumType === result._frustumType ? result._frustum : undefined;
            result._frustum = frustum.clone(frustumResult);

            result._frustumType = frustumType;
            result._origin = Cartesian2.Cartesian3.clone(origin, result._origin);
            result._orientation = Transforms.Quaternion.clone(orientation, result._orientation);
            result._vertexFormat = VertexFormat.VertexFormat.clone(vertexFormat, result._vertexFormat);
            result._drawNearPlane = drawNearPlane;

            return result;
        };

        function getAttributes(offset, normals, tangents, bitangents, st, normal, tangent, bitangent) {
            var stOffset = offset / 3 * 2;

            for (var i = 0; i < 4; ++i) {
                if (defined.defined(normals)) {
                    normals[offset] = normal.x;
                    normals[offset + 1] = normal.y;
                    normals[offset + 2] = normal.z;
                }
                if (defined.defined(tangents)) {
                    tangents[offset] = tangent.x;
                    tangents[offset + 1] = tangent.y;
                    tangents[offset + 2] = tangent.z;
                }
                if (defined.defined(bitangents)) {
                    bitangents[offset] = bitangent.x;
                    bitangents[offset + 1] = bitangent.y;
                    bitangents[offset + 2] = bitangent.z;
                }
                offset += 3;
            }

            st[stOffset] = 0.0;
            st[stOffset + 1] = 0.0;
            st[stOffset + 2] = 1.0;
            st[stOffset + 3] = 0.0;
            st[stOffset + 4] = 1.0;
            st[stOffset + 5] = 1.0;
            st[stOffset + 6] = 0.0;
            st[stOffset + 7] = 1.0;
        }

        var scratchRotationMatrix = new Transforms.Matrix3();
        var scratchViewMatrix = new Transforms.Matrix4();
        var scratchInverseMatrix = new Transforms.Matrix4();

        var scratchXDirection = new Cartesian2.Cartesian3();
        var scratchYDirection = new Cartesian2.Cartesian3();
        var scratchZDirection = new Cartesian2.Cartesian3();
        var scratchNegativeX = new Cartesian2.Cartesian3();
        var scratchNegativeY = new Cartesian2.Cartesian3();
        var scratchNegativeZ = new Cartesian2.Cartesian3();

        var frustumSplits = new Array(3);

        var frustumCornersNDC = new Array(4);
        frustumCornersNDC[0] = new Transforms.Cartesian4(-1.0, -1.0, 1.0, 1.0);
        frustumCornersNDC[1] = new Transforms.Cartesian4(1.0, -1.0, 1.0, 1.0);
        frustumCornersNDC[2] = new Transforms.Cartesian4(1.0, 1.0, 1.0, 1.0);
        frustumCornersNDC[3] = new Transforms.Cartesian4(-1.0, 1.0, 1.0, 1.0);

        var scratchFrustumCorners = new Array(4);
        for (var i = 0; i < 4; ++i) {
            scratchFrustumCorners[i] = new Transforms.Cartesian4();
        }

        FrustumGeometry._computeNearFarPlanes = function(origin, orientation, frustumType, frustum, positions, xDirection, yDirection, zDirection) {
            var rotationMatrix = Transforms.Matrix3.fromQuaternion(orientation, scratchRotationMatrix);
            var x = defaultValue.defaultValue(xDirection, scratchXDirection);
            var y = defaultValue.defaultValue(yDirection, scratchYDirection);
            var z = defaultValue.defaultValue(zDirection, scratchZDirection);

            x = Transforms.Matrix3.getColumn(rotationMatrix, 0, x);
            y = Transforms.Matrix3.getColumn(rotationMatrix, 1, y);
            z = Transforms.Matrix3.getColumn(rotationMatrix, 2, z);

            Cartesian2.Cartesian3.normalize(x, x);
            Cartesian2.Cartesian3.normalize(y, y);
            Cartesian2.Cartesian3.normalize(z, z);

            Cartesian2.Cartesian3.negate(x, x);

            var view = Transforms.Matrix4.computeView(origin, z, y, x, scratchViewMatrix);

            var inverseView;
            var inverseViewProjection;
            if (frustumType === PERSPECTIVE) {
                var projection = frustum.projectionMatrix;
                var viewProjection = Transforms.Matrix4.multiply(projection, view, scratchInverseMatrix);
                inverseViewProjection = Transforms.Matrix4.inverse(viewProjection, scratchInverseMatrix);
            } else {
                inverseView = Transforms.Matrix4.inverseTransformation(view, scratchInverseMatrix);
            }

            if (defined.defined(inverseViewProjection)) {
                frustumSplits[0] = frustum.near;
                frustumSplits[1] = frustum.far;
            } else {
                frustumSplits[0] = 0.0;
                frustumSplits[1] = frustum.near;
                frustumSplits[2] = frustum.far;
            }

            for (var i = 0; i < 2; ++i) {
                for (var j = 0; j < 4; ++j) {
                    var corner = Transforms.Cartesian4.clone(frustumCornersNDC[j], scratchFrustumCorners[j]);

                    if (!defined.defined(inverseViewProjection)) {
                        if (defined.defined(frustum._offCenterFrustum)) {
                            frustum = frustum._offCenterFrustum;
                        }

                        var near = frustumSplits[i];
                        var far = frustumSplits[i + 1];

                        corner.x = (corner.x * (frustum.right - frustum.left) + frustum.left + frustum.right) * 0.5;
                        corner.y = (corner.y * (frustum.top - frustum.bottom) + frustum.bottom + frustum.top) * 0.5;
                        corner.z = (corner.z * (near - far) - near - far) * 0.5;
                        corner.w = 1.0;

                        Transforms.Matrix4.multiplyByVector(inverseView, corner, corner);
                    } else {
                        corner = Transforms.Matrix4.multiplyByVector(inverseViewProjection, corner, corner);

                        // Reverse perspective divide
                        var w = 1.0 / corner.w;
                        Cartesian2.Cartesian3.multiplyByScalar(corner, w, corner);

                        Cartesian2.Cartesian3.subtract(corner, origin, corner);
                        Cartesian2.Cartesian3.normalize(corner, corner);

                        var fac = Cartesian2.Cartesian3.dot(z, corner);
                        Cartesian2.Cartesian3.multiplyByScalar(corner, frustumSplits[i] / fac, corner);
                        Cartesian2.Cartesian3.add(corner, origin, corner);
                    }

                    positions[12 * i + j * 3] = corner.x;
                    positions[12 * i + j * 3 + 1] = corner.y;
                    positions[12 * i + j * 3 + 2] = corner.z;
                }
            }
        };

        /**
         * Computes the geometric representation of a frustum, including its vertices, indices, and a bounding sphere.
         *
         * @param {FrustumGeometry} frustumGeometry A description of the frustum.
         * @returns {Geometry|undefined} The computed vertices and indices.
         */
        FrustumGeometry.createGeometry = function(frustumGeometry) {
            var frustumType = frustumGeometry._frustumType;
            var frustum = frustumGeometry._frustum;
            var origin = frustumGeometry._origin;
            var orientation = frustumGeometry._orientation;
            var drawNearPlane = frustumGeometry._drawNearPlane;
            var vertexFormat = frustumGeometry._vertexFormat;

            var numberOfPlanes = drawNearPlane ? 6 : 5;
            var positions = new Float64Array(3 * 4 * 6);
            FrustumGeometry._computeNearFarPlanes(origin, orientation, frustumType, frustum, positions);

            // -x plane
            var offset = 3 * 4 * 2;
            positions[offset]      = positions[3 * 4];
            positions[offset + 1]  = positions[3 * 4 + 1];
            positions[offset + 2]  = positions[3 * 4 + 2];
            positions[offset + 3]  = positions[0];
            positions[offset + 4]  = positions[1];
            positions[offset + 5]  = positions[2];
            positions[offset + 6]  = positions[3 * 3];
            positions[offset + 7]  = positions[3 * 3 + 1];
            positions[offset + 8]  = positions[3 * 3 + 2];
            positions[offset + 9]  = positions[3 * 7];
            positions[offset + 10] = positions[3 * 7 + 1];
            positions[offset + 11] = positions[3 * 7 + 2];

            // -y plane
            offset += 3 * 4;
            positions[offset]      = positions[3 * 5];
            positions[offset + 1]  = positions[3 * 5 + 1];
            positions[offset + 2]  = positions[3 * 5 + 2];
            positions[offset + 3]  = positions[3];
            positions[offset + 4]  = positions[3 + 1];
            positions[offset + 5]  = positions[3 + 2];
            positions[offset + 6]  = positions[0];
            positions[offset + 7]  = positions[1];
            positions[offset + 8]  = positions[2];
            positions[offset + 9]  = positions[3 * 4];
            positions[offset + 10] = positions[3 * 4 + 1];
            positions[offset + 11] = positions[3 * 4 + 2];

            // +x plane
            offset += 3 * 4;
            positions[offset]      = positions[3];
            positions[offset + 1]  = positions[3 + 1];
            positions[offset + 2]  = positions[3 + 2];
            positions[offset + 3]  = positions[3 * 5];
            positions[offset + 4]  = positions[3 * 5 + 1];
            positions[offset + 5]  = positions[3 * 5 + 2];
            positions[offset + 6]  = positions[3 * 6];
            positions[offset + 7]  = positions[3 * 6 + 1];
            positions[offset + 8]  = positions[3 * 6 + 2];
            positions[offset + 9]  = positions[3 * 2];
            positions[offset + 10] = positions[3 * 2 + 1];
            positions[offset + 11] = positions[3 * 2 + 2];

            // +y plane
            offset += 3 * 4;
            positions[offset]      = positions[3 * 2];
            positions[offset + 1]  = positions[3 * 2 + 1];
            positions[offset + 2]  = positions[3 * 2 + 2];
            positions[offset + 3]  = positions[3 * 6];
            positions[offset + 4]  = positions[3 * 6 + 1];
            positions[offset + 5]  = positions[3 * 6 + 2];
            positions[offset + 6]  = positions[3 * 7];
            positions[offset + 7]  = positions[3 * 7 + 1];
            positions[offset + 8]  = positions[3 * 7 + 2];
            positions[offset + 9]  = positions[3 * 3];
            positions[offset + 10] = positions[3 * 3 + 1];
            positions[offset + 11] = positions[3 * 3 + 2];

            if (!drawNearPlane) {
                positions = positions.subarray(3 * 4);
            }

            var attributes = new GeometryAttributes.GeometryAttributes({
                position : new GeometryAttribute.GeometryAttribute({
                    componentDatatype : ComponentDatatype.ComponentDatatype.DOUBLE,
                    componentsPerAttribute : 3,
                    values : positions
                })
            });

            if (defined.defined(vertexFormat.normal) || defined.defined(vertexFormat.tangent) || defined.defined(vertexFormat.bitangent) || defined.defined(vertexFormat.st)) {
                var normals = defined.defined(vertexFormat.normal) ? new Float32Array(3 * 4 * numberOfPlanes) : undefined;
                var tangents = defined.defined(vertexFormat.tangent) ? new Float32Array(3 * 4 * numberOfPlanes) : undefined;
                var bitangents = defined.defined(vertexFormat.bitangent) ? new Float32Array(3 * 4 * numberOfPlanes) : undefined;
                var st = defined.defined(vertexFormat.st) ? new Float32Array(2 * 4 * numberOfPlanes) : undefined;

                var x = scratchXDirection;
                var y = scratchYDirection;
                var z = scratchZDirection;

                var negativeX = Cartesian2.Cartesian3.negate(x, scratchNegativeX);
                var negativeY = Cartesian2.Cartesian3.negate(y, scratchNegativeY);
                var negativeZ = Cartesian2.Cartesian3.negate(z, scratchNegativeZ);

                offset = 0;
                if (drawNearPlane) {
                    getAttributes(offset, normals, tangents, bitangents, st, negativeZ, x, y); // near
                    offset += 3 * 4;
                }
                getAttributes(offset, normals, tangents, bitangents, st, z, negativeX, y); // far
                offset += 3 * 4;
                getAttributes(offset, normals, tangents, bitangents, st, negativeX, negativeZ, y); // -x
                offset += 3 * 4;
                getAttributes(offset, normals, tangents, bitangents, st, negativeY, negativeZ, negativeX); // -y
                offset += 3 * 4;
                getAttributes(offset, normals, tangents, bitangents, st, x, z, y); // +x
                offset += 3 * 4;
                getAttributes(offset, normals, tangents, bitangents, st, y, z, negativeX); // +y

                if (defined.defined(normals)) {
                    attributes.normal = new GeometryAttribute.GeometryAttribute({
                        componentDatatype : ComponentDatatype.ComponentDatatype.FLOAT,
                        componentsPerAttribute : 3,
                        values : normals
                    });
                }
                if (defined.defined(tangents)) {
                    attributes.tangent = new GeometryAttribute.GeometryAttribute({
                        componentDatatype : ComponentDatatype.ComponentDatatype.FLOAT,
                        componentsPerAttribute : 3,
                        values : tangents
                    });
                }
                if (defined.defined(bitangents)) {
                    attributes.bitangent = new GeometryAttribute.GeometryAttribute({
                        componentDatatype : ComponentDatatype.ComponentDatatype.FLOAT,
                        componentsPerAttribute : 3,
                        values : bitangents
                    });
                }
                if (defined.defined(st)) {
                    attributes.st = new GeometryAttribute.GeometryAttribute({
                        componentDatatype : ComponentDatatype.ComponentDatatype.FLOAT,
                        componentsPerAttribute : 2,
                        values : st
                    });
                }
            }

            var indices = new Uint16Array(6 * numberOfPlanes);
            for (var i = 0; i < numberOfPlanes; ++i) {
                var indexOffset = i * 6;
                var index = i * 4;

                indices[indexOffset] = index;
                indices[indexOffset + 1] = index + 1;
                indices[indexOffset + 2] = index + 2;
                indices[indexOffset + 3] = index;
                indices[indexOffset + 4] = index + 2;
                indices[indexOffset + 5] = index + 3;
            }

            return new GeometryAttribute.Geometry({
                attributes : attributes,
                indices : indices,
                primitiveType : GeometryAttribute.PrimitiveType.TRIANGLES,
                boundingSphere : Transforms.BoundingSphere.fromVertices(positions)
            });
        };

    exports.FrustumGeometry = FrustumGeometry;
    exports.OrthographicFrustum = OrthographicFrustum;
    exports.PerspectiveFrustum = PerspectiveFrustum;

});