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import Cartesian3 from "./Cartesian3.js";
import Cartographic from "./Cartographic.js";
import Check from "./Check.js";
import defined from "./defined.js";
import DeveloperError from "./DeveloperError.js";
import CesiumMath from "./Math.js";
import scaleToGeodeticSurface from "./scaleToGeodeticSurface.js";
function initialize(ellipsoid, x, y, z) {
x = x ?? 0.0;
y = y ?? 0.0;
z = z ?? 0.0;
//>>includeStart('debug', pragmas.debug);
Check.typeOf.number.greaterThanOrEquals("x", x, 0.0);
Check.typeOf.number.greaterThanOrEquals("y", y, 0.0);
Check.typeOf.number.greaterThanOrEquals("z", z, 0.0);
//>>includeEnd('debug');
ellipsoid._radii = new Cartesian3(x, y, z);
ellipsoid._radiiSquared = new Cartesian3(x * x, y * y, z * z);
ellipsoid._radiiToTheFourth = new Cartesian3(
x * x * x * x,
y * y * y * y,
z * z * z * z,
);
ellipsoid._oneOverRadii = new Cartesian3(
x === 0.0 ? 0.0 : 1.0 / x,
y === 0.0 ? 0.0 : 1.0 / y,
z === 0.0 ? 0.0 : 1.0 / z,
);
ellipsoid._oneOverRadiiSquared = new Cartesian3(
x === 0.0 ? 0.0 : 1.0 / (x * x),
y === 0.0 ? 0.0 : 1.0 / (y * y),
z === 0.0 ? 0.0 : 1.0 / (z * z),
);
ellipsoid._minimumRadius = Math.min(x, y, z);
ellipsoid._maximumRadius = Math.max(x, y, z);
ellipsoid._centerToleranceSquared = CesiumMath.EPSILON1;
if (ellipsoid._radiiSquared.z !== 0) {
ellipsoid._squaredXOverSquaredZ =
ellipsoid._radiiSquared.x / ellipsoid._radiiSquared.z;
}
}
/**
* A quadratic surface defined in Cartesian coordinates by the equation
* <code>(x / a)^2 + (y / b)^2 + (z / c)^2 = 1</code>. Primarily used
* by Cesium to represent the shape of planetary bodies.
*
* Rather than constructing this object directly, one of the provided
* constants is normally used.
* @alias Ellipsoid
* @constructor
*
* @param {number} [x=0] The radius in the x direction.
* @param {number} [y=0] The radius in the y direction.
* @param {number} [z=0] The radius in the z direction.
*
* @exception {DeveloperError} All radii components must be greater than or equal to zero.
*
* @see Ellipsoid.fromCartesian3
* @see Ellipsoid.WGS84
* @see Ellipsoid.UNIT_SPHERE
*/
function Ellipsoid(x, y, z) {
this._radii = undefined;
this._radiiSquared = undefined;
this._radiiToTheFourth = undefined;
this._oneOverRadii = undefined;
this._oneOverRadiiSquared = undefined;
this._minimumRadius = undefined;
this._maximumRadius = undefined;
this._centerToleranceSquared = undefined;
this._squaredXOverSquaredZ = undefined;
initialize(this, x, y, z);
}
Object.defineProperties(Ellipsoid.prototype, {
/**
* Gets the radii of the ellipsoid.
* @memberof Ellipsoid.prototype
* @type {Cartesian3}
* @readonly
*/
radii: {
get: function () {
return this._radii;
},
},
/**
* Gets the squared radii of the ellipsoid.
* @memberof Ellipsoid.prototype
* @type {Cartesian3}
* @readonly
*/
radiiSquared: {
get: function () {
return this._radiiSquared;
},
},
/**
* Gets the radii of the ellipsoid raise to the fourth power.
* @memberof Ellipsoid.prototype
* @type {Cartesian3}
* @readonly
*/
radiiToTheFourth: {
get: function () {
return this._radiiToTheFourth;
},
},
/**
* Gets one over the radii of the ellipsoid.
* @memberof Ellipsoid.prototype
* @type {Cartesian3}
* @readonly
*/
oneOverRadii: {
get: function () {
return this._oneOverRadii;
},
},
/**
* Gets one over the squared radii of the ellipsoid.
* @memberof Ellipsoid.prototype
* @type {Cartesian3}
* @readonly
*/
oneOverRadiiSquared: {
get: function () {
return this._oneOverRadiiSquared;
},
},
/**
* Gets the minimum radius of the ellipsoid.
* @memberof Ellipsoid.prototype
* @type {number}
* @readonly
*/
minimumRadius: {
get: function () {
return this._minimumRadius;
},
},
/**
* Gets the maximum radius of the ellipsoid.
* @memberof Ellipsoid.prototype
* @type {number}
* @readonly
*/
maximumRadius: {
get: function () {
return this._maximumRadius;
},
},
});
/**
* Duplicates an Ellipsoid instance.
*
* @param {Ellipsoid} ellipsoid The ellipsoid to duplicate.
* @param {Ellipsoid} [result] The object onto which to store the result, or undefined if a new
* instance should be created.
* @returns {Ellipsoid} The cloned Ellipsoid. (Returns undefined if ellipsoid is undefined)
*/
Ellipsoid.clone = function (ellipsoid, result) {
Iif (!defined(ellipsoid)) {
return undefined;
}
const radii = ellipsoid._radii;
if (!defined(result)) {
return new Ellipsoid(radii.x, radii.y, radii.z);
}
Cartesian3.clone(radii, result._radii);
Cartesian3.clone(ellipsoid._radiiSquared, result._radiiSquared);
Cartesian3.clone(ellipsoid._radiiToTheFourth, result._radiiToTheFourth);
Cartesian3.clone(ellipsoid._oneOverRadii, result._oneOverRadii);
Cartesian3.clone(ellipsoid._oneOverRadiiSquared, result._oneOverRadiiSquared);
result._minimumRadius = ellipsoid._minimumRadius;
result._maximumRadius = ellipsoid._maximumRadius;
result._centerToleranceSquared = ellipsoid._centerToleranceSquared;
return result;
};
/**
* Computes an Ellipsoid from a Cartesian specifying the radii in x, y, and z directions.
*
* @param {Cartesian3} [cartesian=Cartesian3.ZERO] The ellipsoid's radius in the x, y, and z directions.
* @param {Ellipsoid} [result] The object onto which to store the result, or undefined if a new
* instance should be created.
* @returns {Ellipsoid} A new Ellipsoid instance.
*
* @exception {DeveloperError} All radii components must be greater than or equal to zero.
*
* @see Ellipsoid.WGS84
* @see Ellipsoid.UNIT_SPHERE
*/
Ellipsoid.fromCartesian3 = function (cartesian, result) {
if (!defined(result)) {
result = new Ellipsoid();
}
if (!defined(cartesian)) {
return result;
}
initialize(result, cartesian.x, cartesian.y, cartesian.z);
return result;
};
/**
* An Ellipsoid instance initialized to the WGS84 standard.
*
* @type {Ellipsoid}
* @constant
*/
Ellipsoid.WGS84 = Object.freeze(
new Ellipsoid(6378137.0, 6378137.0, 6356752.3142451793),
);
/**
* An Ellipsoid instance initialized to radii of (1.0, 1.0, 1.0).
*
* @type {Ellipsoid}
* @constant
*/
Ellipsoid.UNIT_SPHERE = Object.freeze(new Ellipsoid(1.0, 1.0, 1.0));
/**
* An Ellipsoid instance initialized to a sphere with the lunar radius.
*
* @type {Ellipsoid}
* @constant
*/
Ellipsoid.MOON = Object.freeze(
new Ellipsoid(
CesiumMath.LUNAR_RADIUS,
CesiumMath.LUNAR_RADIUS,
CesiumMath.LUNAR_RADIUS,
),
);
/**
* An Ellipsoid instance initialized to a sphere with the mean radii of Mars.
* Source: https://epsg.io/104905
*
* @type {Ellipsoid}
* @constant
*/
Ellipsoid.MARS = Object.freeze(new Ellipsoid(3396190.0, 3396190.0, 3376200.0));
Ellipsoid._default = Ellipsoid.WGS84;
Object.defineProperties(Ellipsoid, {
/**
* The default ellipsoid used when not otherwise specified.
* @memberof Ellipsoid
* @type {Ellipsoid}
* @example
* Cesium.Ellipsoid.default = Cesium.Ellipsoid.MOON;
*
* // Apollo 11 landing site
* const position = Cesium.Cartesian3.fromRadians(
* 0.67416,
* 23.47315,
* );
*/
default: {
get: function () {
return Ellipsoid._default;
},
set: function (value) {
//>>includeStart('debug', pragmas.debug);
Check.typeOf.object("value", value);
//>>includeEnd('debug');
Ellipsoid._default = value;
Cartesian3._ellipsoidRadiiSquared = value.radiiSquared;
Cartographic._ellipsoidOneOverRadii = value.oneOverRadii;
Cartographic._ellipsoidOneOverRadiiSquared = value.oneOverRadiiSquared;
Cartographic._ellipsoidCenterToleranceSquared =
value._centerToleranceSquared;
},
},
});
/**
* Duplicates an Ellipsoid instance.
*
* @param {Ellipsoid} [result] The object onto which to store the result, or undefined if a new
* instance should be created.
* @returns {Ellipsoid} The cloned Ellipsoid.
*/
Ellipsoid.prototype.clone = function (result) {
return Ellipsoid.clone(this, result);
};
/**
* The number of elements used to pack the object into an array.
* @type {number}
*/
Ellipsoid.packedLength = Cartesian3.packedLength;
/**
* Stores the provided instance into the provided array.
*
* @param {Ellipsoid} 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
*/
Ellipsoid.pack = function (value, array, startingIndex) {
//>>includeStart('debug', pragmas.debug);
Check.typeOf.object("value", value);
Check.defined("array", array);
//>>includeEnd('debug');
startingIndex = startingIndex ?? 0;
Cartesian3.pack(value._radii, array, startingIndex);
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 {Ellipsoid} [result] The object into which to store the result.
* @returns {Ellipsoid} The modified result parameter or a new Ellipsoid instance if one was not provided.
*/
Ellipsoid.unpack = function (array, startingIndex, result) {
//>>includeStart('debug', pragmas.debug);
Check.defined("array", array);
//>>includeEnd('debug');
startingIndex = startingIndex ?? 0;
const radii = Cartesian3.unpack(array, startingIndex);
return Ellipsoid.fromCartesian3(radii, result);
};
/**
* Computes the unit vector directed from the center of this ellipsoid toward the provided Cartesian position.
* @function
*
* @param {Cartesian3} cartesian The Cartesian for which to to determine the geocentric normal.
* @param {Cartesian3} [result] The object onto which to store the result.
* @returns {Cartesian3} The modified result parameter or a new Cartesian3 instance if none was provided.
*/
Ellipsoid.prototype.geocentricSurfaceNormal = Cartesian3.normalize;
/**
* Computes the normal of the plane tangent to the surface of the ellipsoid at the provided position.
*
* @param {Cartographic} cartographic The cartographic position for which to to determine the geodetic normal.
* @param {Cartesian3} [result] The object onto which to store the result.
* @returns {Cartesian3} The modified result parameter or a new Cartesian3 instance if none was provided.
*/
Ellipsoid.prototype.geodeticSurfaceNormalCartographic = function (
cartographic,
result,
) {
//>>includeStart('debug', pragmas.debug);
Check.typeOf.object("cartographic", cartographic);
//>>includeEnd('debug');
const longitude = cartographic.longitude;
const latitude = cartographic.latitude;
const cosLatitude = Math.cos(latitude);
const x = cosLatitude * Math.cos(longitude);
const y = cosLatitude * Math.sin(longitude);
const z = Math.sin(latitude);
if (!defined(result)) {
result = new Cartesian3();
}
result.x = x;
result.y = y;
result.z = z;
return Cartesian3.normalize(result, result);
};
/**
* Computes the normal of the plane tangent to the surface of the ellipsoid at the provided position.
*
* @param {Cartesian3} cartesian The Cartesian position for which to to determine the surface normal.
* @param {Cartesian3} [result] The object onto which to store the result.
* @returns {Cartesian3} The modified result parameter or a new Cartesian3 instance if none was provided, or undefined if a normal cannot be found.
*/
Ellipsoid.prototype.geodeticSurfaceNormal = function (cartesian, result) {
//>>includeStart('debug', pragmas.debug);
Check.typeOf.object("cartesian", cartesian);
Iif (isNaN(cartesian.x) || isNaN(cartesian.y) || isNaN(cartesian.z)) {
throw new DeveloperError("cartesian has a NaN component");
}
//>>includeEnd('debug');
if (
Cartesian3.equalsEpsilon(cartesian, Cartesian3.ZERO, CesiumMath.EPSILON14)
) {
return undefined;
}
if (!defined(result)) {
result = new Cartesian3();
}
result = Cartesian3.multiplyComponents(
cartesian,
this._oneOverRadiiSquared,
result,
);
return Cartesian3.normalize(result, result);
};
const cartographicToCartesianNormal = new Cartesian3();
const cartographicToCartesianK = new Cartesian3();
/**
* Converts the provided cartographic to Cartesian representation.
*
* @param {Cartographic} cartographic The cartographic position.
* @param {Cartesian3} [result] The object onto which to store the result.
* @returns {Cartesian3} The modified result parameter or a new Cartesian3 instance if none was provided.
*
* @example
* //Create a Cartographic and determine it's Cartesian representation on a WGS84 ellipsoid.
* const position = new Cesium.Cartographic(Cesium.Math.toRadians(21), Cesium.Math.toRadians(78), 5000);
* const cartesianPosition = Cesium.Ellipsoid.WGS84.cartographicToCartesian(position);
*/
Ellipsoid.prototype.cartographicToCartesian = function (cartographic, result) {
//`cartographic is required` is thrown from geodeticSurfaceNormalCartographic.
const n = cartographicToCartesianNormal;
const k = cartographicToCartesianK;
this.geodeticSurfaceNormalCartographic(cartographic, n);
Cartesian3.multiplyComponents(this._radiiSquared, n, k);
const gamma = Math.sqrt(Cartesian3.dot(n, k));
Cartesian3.divideByScalar(k, gamma, k);
Cartesian3.multiplyByScalar(n, cartographic.height, n);
if (!defined(result)) {
result = new Cartesian3();
}
return Cartesian3.add(k, n, result);
};
/**
* Converts the provided array of cartographics to an array of Cartesians.
*
* @param {Cartographic[]} cartographics An array of cartographic positions.
* @param {Cartesian3[]} [result] The object onto which to store the result.
* @returns {Cartesian3[]} The modified result parameter or a new Array instance if none was provided.
*
* @example
* //Convert an array of Cartographics and determine their Cartesian representation on a WGS84 ellipsoid.
* const positions = [new Cesium.Cartographic(Cesium.Math.toRadians(21), Cesium.Math.toRadians(78), 0),
* new Cesium.Cartographic(Cesium.Math.toRadians(21.321), Cesium.Math.toRadians(78.123), 100),
* new Cesium.Cartographic(Cesium.Math.toRadians(21.645), Cesium.Math.toRadians(78.456), 250)];
* const cartesianPositions = Cesium.Ellipsoid.WGS84.cartographicArrayToCartesianArray(positions);
*/
Ellipsoid.prototype.cartographicArrayToCartesianArray = function (
cartographics,
result,
) {
//>>includeStart('debug', pragmas.debug);
Check.defined("cartographics", cartographics);
//>>includeEnd('debug');
const length = cartographics.length;
if (!defined(result)) {
result = new Array(length);
} else {
result.length = length;
}
for (let i = 0; i < length; i++) {
result[i] = this.cartographicToCartesian(cartographics[i], result[i]);
}
return result;
};
const cartesianToCartographicN = new Cartesian3();
const cartesianToCartographicP = new Cartesian3();
const cartesianToCartographicH = new Cartesian3();
/**
* Converts the provided cartesian to cartographic representation.
* The cartesian is undefined at the center of the ellipsoid.
*
* @param {Cartesian3} cartesian The Cartesian position to convert to cartographic representation.
* @param {Cartographic} [result] The object onto which to store the result.
* @returns {Cartographic} The modified result parameter, new Cartographic instance if none was provided, or undefined if the cartesian is at the center of the ellipsoid.
*
* @example
* //Create a Cartesian and determine it's Cartographic representation on a WGS84 ellipsoid.
* const position = new Cesium.Cartesian3(17832.12, 83234.52, 952313.73);
* const cartographicPosition = Cesium.Ellipsoid.WGS84.cartesianToCartographic(position);
*/
Ellipsoid.prototype.cartesianToCartographic = function (cartesian, result) {
//`cartesian is required.` is thrown from scaleToGeodeticSurface
const p = this.scaleToGeodeticSurface(cartesian, cartesianToCartographicP);
if (!defined(p)) {
return undefined;
}
const n = this.geodeticSurfaceNormal(p, cartesianToCartographicN);
const h = Cartesian3.subtract(cartesian, p, cartesianToCartographicH);
const longitude = Math.atan2(n.y, n.x);
const latitude = Math.asin(n.z);
const height =
CesiumMath.sign(Cartesian3.dot(h, cartesian)) * Cartesian3.magnitude(h);
if (!defined(result)) {
return new Cartographic(longitude, latitude, height);
}
result.longitude = longitude;
result.latitude = latitude;
result.height = height;
return result;
};
/**
* Converts the provided array of cartesians to an array of cartographics.
*
* @param {Cartesian3[]} cartesians An array of Cartesian positions.
* @param {Cartographic[]} [result] The object onto which to store the result.
* @returns {Cartographic[]} The modified result parameter or a new Array instance if none was provided.
*
* @example
* //Create an array of Cartesians and determine their Cartographic representation on a WGS84 ellipsoid.
* const positions = [new Cesium.Cartesian3(17832.12, 83234.52, 952313.73),
* new Cesium.Cartesian3(17832.13, 83234.53, 952313.73),
* new Cesium.Cartesian3(17832.14, 83234.54, 952313.73)]
* const cartographicPositions = Cesium.Ellipsoid.WGS84.cartesianArrayToCartographicArray(positions);
*/
Ellipsoid.prototype.cartesianArrayToCartographicArray = function (
cartesians,
result,
) {
//>>includeStart('debug', pragmas.debug);
Check.defined("cartesians", cartesians);
//>>includeEnd('debug');
const length = cartesians.length;
if (!defined(result)) {
result = new Array(length);
} else {
result.length = length;
}
for (let i = 0; i < length; ++i) {
result[i] = this.cartesianToCartographic(cartesians[i], result[i]);
}
return result;
};
/**
* Scales the provided Cartesian position along the geodetic surface normal
* so that it is on the surface of this ellipsoid. If the position is
* at the center of the ellipsoid, this function returns undefined.
*
* @param {Cartesian3} cartesian The Cartesian position to scale.
* @param {Cartesian3} [result] The object onto which to store the result.
* @returns {Cartesian3} The modified result parameter, a new Cartesian3 instance if none was provided, or undefined if the position is at the center.
*/
Ellipsoid.prototype.scaleToGeodeticSurface = function (cartesian, result) {
return scaleToGeodeticSurface(
cartesian,
this._oneOverRadii,
this._oneOverRadiiSquared,
this._centerToleranceSquared,
result,
);
};
/**
* Scales the provided Cartesian position along the geocentric surface normal
* so that it is on the surface of this ellipsoid.
*
* @param {Cartesian3} cartesian The Cartesian position to scale.
* @param {Cartesian3} [result] The object onto which to store the result.
* @returns {Cartesian3} The modified result parameter or a new Cartesian3 instance if none was provided.
*/
Ellipsoid.prototype.scaleToGeocentricSurface = function (cartesian, result) {
//>>includeStart('debug', pragmas.debug);
Check.typeOf.object("cartesian", cartesian);
//>>includeEnd('debug');
if (!defined(result)) {
result = new Cartesian3();
}
const positionX = cartesian.x;
const positionY = cartesian.y;
const positionZ = cartesian.z;
const oneOverRadiiSquared = this._oneOverRadiiSquared;
const beta =
1.0 /
Math.sqrt(
positionX * positionX * oneOverRadiiSquared.x +
positionY * positionY * oneOverRadiiSquared.y +
positionZ * positionZ * oneOverRadiiSquared.z,
);
return Cartesian3.multiplyByScalar(cartesian, beta, result);
};
/**
* Transforms a Cartesian X, Y, Z position to the ellipsoid-scaled space by multiplying
* its components by the result of {@link Ellipsoid#oneOverRadii}.
*
* @param {Cartesian3} position The position to transform.
* @param {Cartesian3} [result] The position to which to copy the result, or undefined to create and
* return a new instance.
* @returns {Cartesian3} The position expressed in the scaled space. The returned instance is the
* one passed as the result parameter if it is not undefined, or a new instance of it is.
*/
Ellipsoid.prototype.transformPositionToScaledSpace = function (
position,
result,
) {
if (!defined(result)) {
result = new Cartesian3();
}
return Cartesian3.multiplyComponents(position, this._oneOverRadii, result);
};
/**
* Transforms a Cartesian X, Y, Z position from the ellipsoid-scaled space by multiplying
* its components by the result of {@link Ellipsoid#radii}.
*
* @param {Cartesian3} position The position to transform.
* @param {Cartesian3} [result] The position to which to copy the result, or undefined to create and
* return a new instance.
* @returns {Cartesian3} The position expressed in the unscaled space. The returned instance is the
* one passed as the result parameter if it is not undefined, or a new instance of it is.
*/
Ellipsoid.prototype.transformPositionFromScaledSpace = function (
position,
result,
) {
if (!defined(result)) {
result = new Cartesian3();
}
return Cartesian3.multiplyComponents(position, this._radii, result);
};
/**
* Compares this Ellipsoid against the provided Ellipsoid componentwise and returns
* <code>true</code> if they are equal, <code>false</code> otherwise.
*
* @param {Ellipsoid} [right] The other Ellipsoid.
* @returns {boolean} <code>true</code> if they are equal, <code>false</code> otherwise.
*/
Ellipsoid.prototype.equals = function (right) {
return (
this === right ||
(defined(right) && Cartesian3.equals(this._radii, right._radii))
);
};
/**
* Creates a string representing this Ellipsoid in the format '(radii.x, radii.y, radii.z)'.
*
* @returns {string} A string representing this ellipsoid in the format '(radii.x, radii.y, radii.z)'.
*/
Ellipsoid.prototype.toString = function () {
return this._radii.toString();
};
/**
* Computes a point which is the intersection of the surface normal with the z-axis.
*
* @param {Cartesian3} position the position. must be on the surface of the ellipsoid.
* @param {number} [buffer = 0.0] A buffer to subtract from the ellipsoid size when checking if the point is inside the ellipsoid.
* In earth case, with common earth datums, there is no need for this buffer since the intersection point is always (relatively) very close to the center.
* In WGS84 datum, intersection point is at max z = +-42841.31151331382 (0.673% of z-axis).
* Intersection point could be outside the ellipsoid if the ratio of MajorAxis / AxisOfRotation is bigger than the square root of 2
* @param {Cartesian3} [result] The cartesian to which to copy the result, or undefined to create and
* return a new instance.
* @returns {Cartesian3 | undefined} the intersection point if it's inside the ellipsoid, undefined otherwise
*
* @exception {DeveloperError} position is required.
* @exception {DeveloperError} Ellipsoid must be an ellipsoid of revolution (radii.x == radii.y).
* @exception {DeveloperError} Ellipsoid.radii.z must be greater than 0.
*/
Ellipsoid.prototype.getSurfaceNormalIntersectionWithZAxis = function (
position,
buffer,
result,
) {
//>>includeStart('debug', pragmas.debug);
Check.typeOf.object("position", position);
if (
!CesiumMath.equalsEpsilon(
this._radii.x,
this._radii.y,
CesiumMath.EPSILON15,
)
) {
throw new DeveloperError(
"Ellipsoid must be an ellipsoid of revolution (radii.x == radii.y)",
);
}
Check.typeOf.number.greaterThan("Ellipsoid.radii.z", this._radii.z, 0);
//>>includeEnd('debug');
buffer = buffer ?? 0.0;
const squaredXOverSquaredZ = this._squaredXOverSquaredZ;
if (!defined(result)) {
result = new Cartesian3();
}
result.x = 0.0;
result.y = 0.0;
result.z = position.z * (1 - squaredXOverSquaredZ);
if (Math.abs(result.z) >= this._radii.z - buffer) {
return undefined;
}
return result;
};
const scratchEndpoint = new Cartesian3();
/**
* Computes the ellipsoid curvatures at a given position on the surface.
*
* @param {Cartesian3} surfacePosition The position on the ellipsoid surface where curvatures will be calculated.
* @param {Cartesian2} [result] The cartesian to which to copy the result, or undefined to create and return a new instance.
* @returns {Cartesian2} The local curvature of the ellipsoid surface at the provided position, in east and north directions.
*
* @exception {DeveloperError} position is required.
*/
Ellipsoid.prototype.getLocalCurvature = function (surfacePosition, result) {
//>>includeStart('debug', pragmas.debug);
Check.typeOf.object("surfacePosition", surfacePosition);
//>>includeEnd('debug');
if (!defined(result)) {
result = new Cartesian2();
}
const primeVerticalEndpoint = this.getSurfaceNormalIntersectionWithZAxis(
surfacePosition,
0.0,
scratchEndpoint,
);
const primeVerticalRadius = Cartesian3.distance(
surfacePosition,
primeVerticalEndpoint,
);
// meridional radius = (1 - e^2) * primeVerticalRadius^3 / a^2
// where 1 - e^2 = b^2 / a^2,
// so meridional = b^2 * primeVerticalRadius^3 / a^4
// = (b * primeVerticalRadius / a^2)^2 * primeVertical
const radiusRatio =
(this.minimumRadius * primeVerticalRadius) / this.maximumRadius ** 2;
const meridionalRadius = primeVerticalRadius * radiusRatio ** 2;
return Cartesian2.fromElements(
1.0 / primeVerticalRadius,
1.0 / meridionalRadius,
result,
);
};
const abscissas = [
0.14887433898163, 0.43339539412925, 0.67940956829902, 0.86506336668898,
0.97390652851717, 0.0,
];
const weights = [
0.29552422471475, 0.26926671930999, 0.21908636251598, 0.14945134915058,
0.066671344308684, 0.0,
];
/**
* Compute the 10th order Gauss-Legendre Quadrature of the given definite integral.
*
* @param {number} a The lower bound for the integration.
* @param {number} b The upper bound for the integration.
* @param {Ellipsoid~RealValuedScalarFunction} func The function to integrate.
* @returns {number} The value of the integral of the given function over the given domain.
*
* @private
*/
function gaussLegendreQuadrature(a, b, func) {
//>>includeStart('debug', pragmas.debug);
Check.typeOf.number("a", a);
Check.typeOf.number("b", b);
Check.typeOf.func("func", func);
//>>includeEnd('debug');
// The range is half of the normal range since the five weights add to one (ten weights add to two).
// The values of the abscissas are multiplied by two to account for this.
const xMean = 0.5 * (b + a);
const xRange = 0.5 * (b - a);
let sum = 0.0;
for (let i = 0; i < 5; i++) {
const dx = xRange * abscissas[i];
sum += weights[i] * (func(xMean + dx) + func(xMean - dx));
}
// Scale the sum to the range of x.
sum *= xRange;
return sum;
}
/**
* A real valued scalar function.
* @callback Ellipsoid~RealValuedScalarFunction
*
* @param {number} x The value used to evaluate the function.
* @returns {number} The value of the function at x.
*
* @private
*/
/**
* Computes an approximation of the surface area of a rectangle on the surface of an ellipsoid using
* Gauss-Legendre 10th order quadrature.
*
* @param {Rectangle} rectangle The rectangle used for computing the surface area.
* @returns {number} The approximate area of the rectangle on the surface of this ellipsoid.
*/
Ellipsoid.prototype.surfaceArea = function (rectangle) {
//>>includeStart('debug', pragmas.debug);
Check.typeOf.object("rectangle", rectangle);
//>>includeEnd('debug');
const minLongitude = rectangle.west;
let maxLongitude = rectangle.east;
const minLatitude = rectangle.south;
const maxLatitude = rectangle.north;
while (maxLongitude < minLongitude) {
maxLongitude += CesiumMath.TWO_PI;
}
const radiiSquared = this._radiiSquared;
const a2 = radiiSquared.x;
const b2 = radiiSquared.y;
const c2 = radiiSquared.z;
const a2b2 = a2 * b2;
return gaussLegendreQuadrature(minLatitude, maxLatitude, function (lat) {
// phi represents the angle measured from the north pole
// sin(phi) = sin(pi / 2 - lat) = cos(lat), cos(phi) is similar
const sinPhi = Math.cos(lat);
const cosPhi = Math.sin(lat);
return (
Math.cos(lat) *
gaussLegendreQuadrature(minLongitude, maxLongitude, function (lon) {
const cosTheta = Math.cos(lon);
const sinTheta = Math.sin(lon);
return Math.sqrt(
a2b2 * cosPhi * cosPhi +
c2 *
(b2 * cosTheta * cosTheta + a2 * sinTheta * sinTheta) *
sinPhi *
sinPhi,
);
})
);
});
};
export default Ellipsoid;
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