Ellipsoid.java
- /* Copyright 2002-2024 CS GROUP
- * Licensed to CS GROUP (CS) under one or more
- * contributor license agreements. See the NOTICE file distributed with
- * this work for additional information regarding copyright ownership.
- * CS licenses this file to You under the Apache License, Version 2.0
- * (the "License"); you may not use this file except in compliance with
- * the License. You may obtain a copy of the License at
- *
- * http://www.apache.org/licenses/LICENSE-2.0
- *
- * Unless required by applicable law or agreed to in writing, software
- * distributed under the License is distributed on an "AS IS" BASIS,
- * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
- * See the License for the specific language governing permissions and
- * limitations under the License.
- */
- package org.orekit.bodies;
- import java.io.Serializable;
- import org.hipparchus.CalculusFieldElement;
- import org.hipparchus.exception.MathRuntimeException;
- import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
- import org.hipparchus.geometry.euclidean.threed.Vector3D;
- import org.hipparchus.geometry.euclidean.twod.FieldVector2D;
- import org.hipparchus.geometry.euclidean.twod.Vector2D;
- import org.hipparchus.util.FastMath;
- import org.hipparchus.util.MathArrays;
- import org.hipparchus.util.Precision;
- import org.orekit.errors.OrekitException;
- import org.orekit.errors.OrekitMessages;
- import org.orekit.frames.Frame;
- /**
- * Modeling of a general three-axes ellipsoid.
- * @since 7.0
- * @author Luc Maisonobe
- */
- public class Ellipsoid implements Serializable {
- /** Serializable UID. */
- private static final long serialVersionUID = 20140924L;
- /** Frame at the ellipsoid center, aligned with principal axes. */
- private final Frame frame;
- /** First semi-axis length. */
- private final double a;
- /** Second semi-axis length. */
- private final double b;
- /** Third semi-axis length. */
- private final double c;
- /** Simple constructor.
- * @param frame at the ellipsoid center, aligned with principal axes
- * @param a first semi-axis length
- * @param b second semi-axis length
- * @param c third semi-axis length
- */
- public Ellipsoid(final Frame frame, final double a, final double b, final double c) {
- this.frame = frame;
- this.a = a;
- this.b = b;
- this.c = c;
- }
- /** Get the length of the first semi-axis.
- * @return length of the first semi-axis (m)
- */
- public double getA() {
- return a;
- }
- /** Get the length of the second semi-axis.
- * @return length of the second semi-axis (m)
- */
- public double getB() {
- return b;
- }
- /** Get the length of the third semi-axis.
- * @return length of the third semi-axis (m)
- */
- public double getC() {
- return c;
- }
- /** Get the ellipsoid central frame.
- * @return ellipsoid central frame
- */
- public Frame getFrame() {
- return frame;
- }
- /** Check if a point is inside the ellipsoid.
- * @param point point to check, in the ellipsoid frame
- * @return true if the point is inside the ellipsoid
- * (or exactly on ellipsoid surface)
- * @since 7.1
- */
- public boolean isInside(final Vector3D point) {
- final double scaledX = point.getX() / a;
- final double scaledY = point.getY() / b;
- final double scaledZ = point.getZ() / c;
- return scaledX * scaledX + scaledY * scaledY + scaledZ * scaledZ <= 1.0;
- }
- /** Check if a point is inside the ellipsoid.
- * @param point point to check, in the ellipsoid frame
- * @return true if the point is inside the ellipsoid
- * (or exactly on ellipsoid surface)
- * @param <T> the type of the field elements
- * @since 12.0
- */
- public <T extends CalculusFieldElement<T>> boolean isInside(final FieldVector3D<T> point) {
- final T scaledX = point.getX().divide(a);
- final T scaledY = point.getY().divide(b);
- final T scaledZ = point.getZ().divide(c);
- final T d2 = scaledX.multiply(scaledX).add(scaledY.multiply(scaledY)).add(scaledZ.multiply(scaledZ));
- return d2.subtract(1.0).getReal() <= 0.0;
- }
- /** Compute the 2D ellipse at the intersection of the 3D ellipsoid and a plane.
- * @param planePoint point belonging to the plane, in the ellipsoid frame
- * @param planeNormal normal of the plane, in the ellipsoid frame
- * @return plane section or null if there are no intersections
- * @exception MathRuntimeException if the norm of planeNormal is null
- */
- public Ellipse getPlaneSection(final Vector3D planePoint, final Vector3D planeNormal)
- throws MathRuntimeException {
- // we define the points Q in the plane using two free variables τ and υ as:
- // Q = P + τ u + υ v
- // where u and v are two unit vectors belonging to the plane
- // Q belongs to the 3D ellipsoid so:
- // (xQ / a)² + (yQ / b)² + (zQ / c)² = 1
- // combining both equations, we get:
- // (xP² + 2 xP (τ xU + υ xV) + (τ xU + υ xV)²) / a²
- // + (yP² + 2 yP (τ yU + υ yV) + (τ yU + υ yV)²) / b²
- // + (zP² + 2 zP (τ zU + υ zV) + (τ zU + υ zV)²) / c²
- // = 1
- // which can be rewritten:
- // α τ² + β υ² + 2 γ τυ + 2 δ τ + 2 ε υ + ζ = 0
- // with
- // α = xU² / a² + yU² / b² + zU² / c² > 0
- // β = xV² / a² + yV² / b² + zV² / c² > 0
- // γ = xU xV / a² + yU yV / b² + zU zV / c²
- // δ = xP xU / a² + yP yU / b² + zP zU / c²
- // ε = xP xV / a² + yP yV / b² + zP zV / c²
- // ζ = xP² / a² + yP² / b² + zP² / c² - 1
- // this is the equation of a conic (here an ellipse)
- // Of course, we note that if the point P belongs to the ellipsoid
- // then ζ = 0 and the equation holds at point P since τ = 0 and υ = 0
- final Vector3D u = planeNormal.orthogonal();
- final Vector3D v = Vector3D.crossProduct(planeNormal, u).normalize();
- final double xUOa = u.getX() / a;
- final double yUOb = u.getY() / b;
- final double zUOc = u.getZ() / c;
- final double xVOa = v.getX() / a;
- final double yVOb = v.getY() / b;
- final double zVOc = v.getZ() / c;
- final double xPOa = planePoint.getX() / a;
- final double yPOb = planePoint.getY() / b;
- final double zPOc = planePoint.getZ() / c;
- final double alpha = xUOa * xUOa + yUOb * yUOb + zUOc * zUOc;
- final double beta = xVOa * xVOa + yVOb * yVOb + zVOc * zVOc;
- final double gamma = MathArrays.linearCombination(xUOa, xVOa, yUOb, yVOb, zUOc, zVOc);
- final double delta = MathArrays.linearCombination(xPOa, xUOa, yPOb, yUOb, zPOc, zUOc);
- final double epsilon = MathArrays.linearCombination(xPOa, xVOa, yPOb, yVOb, zPOc, zVOc);
- final double zeta = MathArrays.linearCombination(xPOa, xPOa, yPOb, yPOb, zPOc, zPOc, 1, -1);
- // reduce the general equation α τ² + β υ² + 2 γ τυ + 2 δ τ + 2 ε υ + ζ = 0
- // to canonical form (λ/l)² + (μ/m)² = 1
- // using a coordinates change
- // τ = τC + λ cosθ - μ sinθ
- // υ = υC + λ sinθ + μ cosθ
- // or equivalently
- // λ = (τ - τC) cosθ + (υ - υC) sinθ
- // μ = - (τ - τC) sinθ + (υ - υC) cosθ
- // τC and υC are the coordinates of the 2D ellipse center with respect to P
- // 2l and 2m and are the axes lengths (major or minor depending on which one is greatest)
- // θ is the angle of the 2D ellipse axis corresponding to axis with length 2l
- // choose θ in order to cancel the coupling term in λμ
- // expanding the general equation, we get: A λ² + B μ² + C λμ + D λ + E μ + F = 0
- // with C = 2[(β - α) cosθ sinθ + γ (cos²θ - sin²θ)]
- // hence the term is cancelled when θ = arctan(t), with γ t² + (α - β) t - γ = 0
- // As the solutions of the quadratic equation obey t₁t₂ = -1, they correspond to
- // angles θ in quadrature to each other. Selecting one solution or the other simply
- // exchanges the principal axes. As we don't care about which axis we want as the
- // first one, we select an arbitrary solution
- final double tanTheta;
- if (FastMath.abs(gamma) < Precision.SAFE_MIN) {
- tanTheta = 0.0;
- } else {
- final double bMA = beta - alpha;
- tanTheta = (bMA >= 0) ?
- (-2 * gamma / (bMA + FastMath.sqrt(bMA * bMA + 4 * gamma * gamma))) :
- (-2 * gamma / (bMA - FastMath.sqrt(bMA * bMA + 4 * gamma * gamma)));
- }
- final double tan2 = tanTheta * tanTheta;
- final double cos2 = 1 / (1 + tan2);
- final double sin2 = tan2 * cos2;
- final double cosSin = tanTheta * cos2;
- final double cos = FastMath.sqrt(cos2);
- final double sin = tanTheta * cos;
- // choose τC and υC in order to cancel the linear terms in λ and μ
- // expanding the general equation, we get: A λ² + B μ² + C λμ + D λ + E μ + F = 0
- // with D = 2[ (α τC + γ υC + δ) cosθ + (γ τC + β υC + ε) sinθ]
- // E = 2[-(α τC + γ υC + δ) sinθ + (γ τC + β υC + ε) cosθ]
- // θ can be eliminated by combining the equations
- // D cosθ - E sinθ = 2[α τC + γ υC + δ]
- // E cosθ + D sinθ = 2[γ τC + β υC + ε]
- // hence the terms D and E are both cancelled (regardless of θ) when
- // τC = (β δ - γ ε) / (γ² - α β)
- // υC = (α ε - γ δ) / (γ² - α β)
- final double denom = MathArrays.linearCombination(gamma, gamma, -alpha, beta);
- final double tauC = MathArrays.linearCombination(beta, delta, -gamma, epsilon) / denom;
- final double nuC = MathArrays.linearCombination(alpha, epsilon, -gamma, delta) / denom;
- // compute l and m
- // expanding the general equation, we get: A λ² + B μ² + C λμ + D λ + E μ + F = 0
- // with A = α cos²θ + β sin²θ + 2 γ cosθ sinθ
- // B = α sin²θ + β cos²θ - 2 γ cosθ sinθ
- // F = α τC² + β υC² + 2 γ τC υC + 2 δ τC + 2 ε υC + ζ
- // hence we compute directly l = √(-F/A) and m = √(-F/B)
- final double twogcs = 2 * gamma * cosSin;
- final double bigA = alpha * cos2 + beta * sin2 + twogcs;
- final double bigB = alpha * sin2 + beta * cos2 - twogcs;
- final double bigF = (alpha * tauC + 2 * (gamma * nuC + delta)) * tauC +
- (beta * nuC + 2 * epsilon) * nuC + zeta;
- final double l = FastMath.sqrt(-bigF / bigA);
- final double m = FastMath.sqrt(-bigF / bigB);
- if (Double.isNaN(l + m)) {
- // the plane does not intersect the ellipsoid
- return null;
- }
- if (l > m) {
- return new Ellipse(new Vector3D(1, planePoint, tauC, u, nuC, v),
- new Vector3D( cos, u, sin, v),
- new Vector3D(-sin, u, cos, v),
- l, m, frame);
- } else {
- return new Ellipse(new Vector3D(1, planePoint, tauC, u, nuC, v),
- new Vector3D(sin, u, -cos, v),
- new Vector3D(cos, u, sin, v),
- m, l, frame);
- }
- }
- /** Compute the 2D ellipse at the intersection of the 3D ellipsoid and a plane.
- * @param planePoint point belonging to the plane, in the ellipsoid frame
- * @param planeNormal normal of the plane, in the ellipsoid frame
- * @return plane section or null if there are no intersections
- * @exception MathRuntimeException if the norm of planeNormal is null
- * @param <T> the type of the field elements
- * @since 12.0
- */
- public <T extends CalculusFieldElement<T>> FieldEllipse<T> getPlaneSection(final FieldVector3D<T> planePoint, final FieldVector3D<T> planeNormal)
- throws MathRuntimeException {
- final T zero = planePoint.getX().getField().getZero();
- final T one = planePoint.getX().getField().getOne();
- // we define the points Q in the plane using two free variables τ and υ as:
- // Q = P + τ u + υ v
- // where u and v are two unit vectors belonging to the plane
- // Q belongs to the 3D ellipsoid so:
- // (xQ / a)² + (yQ / b)² + (zQ / c)² = 1
- // combining both equations, we get:
- // (xP² + 2 xP (τ xU + υ xV) + (τ xU + υ xV)²) / a²
- // + (yP² + 2 yP (τ yU + υ yV) + (τ yU + υ yV)²) / b²
- // + (zP² + 2 zP (τ zU + υ zV) + (τ zU + υ zV)²) / c²
- // = 1
- // which can be rewritten:
- // α τ² + β υ² + 2 γ τυ + 2 δ τ + 2 ε υ + ζ = 0
- // with
- // α = xU² / a² + yU² / b² + zU² / c² > 0
- // β = xV² / a² + yV² / b² + zV² / c² > 0
- // γ = xU xV / a² + yU yV / b² + zU zV / c²
- // δ = xP xU / a² + yP yU / b² + zP zU / c²
- // ε = xP xV / a² + yP yV / b² + zP zV / c²
- // ζ = xP² / a² + yP² / b² + zP² / c² - 1
- // this is the equation of a conic (here an ellipse)
- // Of course, we note that if the point P belongs to the ellipsoid
- // then ζ = 0 and the equation holds at point P since τ = 0 and υ = 0
- final FieldVector3D<T> u = planeNormal.orthogonal();
- final FieldVector3D<T> v = FieldVector3D.crossProduct(planeNormal, u).normalize();
- final T xUOa = u.getX().divide(a);
- final T yUOb = u.getY().divide(b);
- final T zUOc = u.getZ().divide(c);
- final T xVOa = v.getX().divide(a);
- final T yVOb = v.getY().divide(b);
- final T zVOc = v.getZ().divide(c);
- final T xPOa = planePoint.getX().divide(a);
- final T yPOb = planePoint.getY().divide(b);
- final T zPOc = planePoint.getZ().divide(c);
- final T alpha = xUOa.multiply(xUOa).add(yUOb.multiply(yUOb)).add(zUOc.multiply(zUOc));
- final T beta = xVOa.multiply(xVOa).add(yVOb.multiply(yVOb)).add(zVOc.multiply(zVOc));
- final T gamma = alpha.linearCombination(xUOa, xVOa, yUOb, yVOb, zUOc, zVOc);
- final T delta = alpha.linearCombination(xPOa, xUOa, yPOb, yUOb, zPOc, zUOc);
- final T epsilon = alpha.linearCombination(xPOa, xVOa, yPOb, yVOb, zPOc, zVOc);
- final T zeta = alpha.linearCombination(xPOa, xPOa, yPOb, yPOb, zPOc, zPOc, one, one.negate());
- // reduce the general equation α τ² + β υ² + 2 γ τυ + 2 δ τ + 2 ε υ + ζ = 0
- // to canonical form (λ/l)² + (μ/m)² = 1
- // using a coordinates change
- // τ = τC + λ cosθ - μ sinθ
- // υ = υC + λ sinθ + μ cosθ
- // or equivalently
- // λ = (τ - τC) cosθ + (υ - υC) sinθ
- // μ = - (τ - τC) sinθ + (υ - υC) cosθ
- // τC and υC are the coordinates of the 2D ellipse center with respect to P
- // 2l and 2m and are the axes lengths (major or minor depending on which one is greatest)
- // θ is the angle of the 2D ellipse axis corresponding to axis with length 2l
- // choose θ in order to cancel the coupling term in λμ
- // expanding the general equation, we get: A λ² + B μ² + C λμ + D λ + E μ + F = 0
- // with C = 2[(β - α) cosθ sinθ + γ (cos²θ - sin²θ)]
- // hence the term is cancelled when θ = arctan(t), with γ t² + (α - β) t - γ = 0
- // As the solutions of the quadratic equation obey t₁t₂ = -1, they correspond to
- // angles θ in quadrature to each other. Selecting one solution or the other simply
- // exchanges the principal axes. As we don't care about which axis we want as the
- // first one, we select an arbitrary solution
- final T tanTheta;
- if (FastMath.abs(gamma.getReal()) < Precision.SAFE_MIN) {
- tanTheta = zero;
- } else {
- final T bMA = beta.subtract(alpha);
- tanTheta = (bMA.getReal() >= 0) ?
- gamma.multiply(-2).divide(bMA.add(FastMath.sqrt(bMA.multiply(bMA).add(gamma.multiply(gamma).multiply(4))))) :
- gamma.multiply(-2).divide(bMA.subtract(FastMath.sqrt(bMA.multiply(bMA).add(gamma.multiply(gamma).multiply(4)))));
- }
- final T tan2 = tanTheta.multiply(tanTheta);
- final T cos2 = tan2.add(1).reciprocal();
- final T sin2 = tan2.multiply(cos2);
- final T cosSin = tanTheta.multiply(cos2);
- final T cos = FastMath.sqrt(cos2);
- final T sin = tanTheta.multiply(cos);
- // choose τC and υC in order to cancel the linear terms in λ and μ
- // expanding the general equation, we get: A λ² + B μ² + C λμ + D λ + E μ + F = 0
- // with D = 2[ (α τC + γ υC + δ) cosθ + (γ τC + β υC + ε) sinθ]
- // E = 2[-(α τC + γ υC + δ) sinθ + (γ τC + β υC + ε) cosθ]
- // θ can be eliminated by combining the equations
- // D cosθ - E sinθ = 2[α τC + γ υC + δ]
- // E cosθ + D sinθ = 2[γ τC + β υC + ε]
- // hence the terms D and E are both cancelled (regardless of θ) when
- // τC = (β δ - γ ε) / (γ² - α β)
- // υC = (α ε - γ δ) / (γ² - α β)
- final T invDenom = gamma.linearCombination(gamma, gamma, alpha.negate(), beta).reciprocal();
- final T tauC = gamma.linearCombination(beta, delta, gamma.negate(), epsilon).multiply(invDenom);
- final T nuC = gamma.linearCombination(alpha, epsilon, gamma.negate(), delta).multiply(invDenom);
- // compute l and m
- // expanding the general equation, we get: A λ² + B μ² + C λμ + D λ + E μ + F = 0
- // with A = α cos²θ + β sin²θ + 2 γ cosθ sinθ
- // B = α sin²θ + β cos²θ - 2 γ cosθ sinθ
- // F = α τC² + β υC² + 2 γ τC υC + 2 δ τC + 2 ε υC + ζ
- // hence we compute directly l = √(-F/A) and m = √(-F/B)
- final T twogcs = gamma.multiply(cosSin).multiply(2);
- final T bigA = alpha.multiply(cos2).add(beta.multiply(sin2)).add(twogcs);
- final T bigB = alpha.multiply(sin2).add(beta.multiply(cos2)).subtract(twogcs);
- final T bigFN = alpha.multiply(tauC).add(gamma.multiply(nuC).add(delta).multiply(2)).multiply(tauC).
- add(beta.multiply(nuC).add(epsilon.multiply(2)).multiply(nuC)).
- add(zeta).
- negate();
- final T l = FastMath.sqrt(bigFN.divide(bigA));
- final T m = FastMath.sqrt(bigFN.divide(bigB));
- if (l.add(m).isNaN()) {
- // the plane does not intersect the ellipsoid
- return null;
- }
- if (l.subtract(m).getReal() > 0) {
- return new FieldEllipse<>(new FieldVector3D<>(tauC.getField().getOne(), planePoint, tauC, u, nuC, v),
- new FieldVector3D<>(cos, u, sin, v),
- new FieldVector3D<>(sin.negate(), u, cos, v),
- l, m, frame);
- } else {
- return new FieldEllipse<>(new FieldVector3D<>(tauC.getField().getOne(), planePoint, tauC, u, nuC, v),
- new FieldVector3D<>(sin, u, cos.negate(), v),
- new FieldVector3D<>(cos, u, sin, v),
- m, l, frame);
- }
- }
- /** Find a point on ellipsoid limb, as seen by an external observer.
- * @param observer observer position in ellipsoid frame
- * @param outside point outside ellipsoid in ellipsoid frame, defining the phase around limb
- * @return point on ellipsoid limb
- * @exception MathRuntimeException if ellipsoid center, observer and outside
- * points are aligned
- * @since 7.1
- */
- public Vector3D pointOnLimb(final Vector3D observer, final Vector3D outside)
- throws MathRuntimeException {
- // There is no limb if we are inside the ellipsoid
- if (isInside(observer)) {
- throw new OrekitException(OrekitMessages.POINT_INSIDE_ELLIPSOID);
- }
- // Cut the ellipsoid, to find an elliptical plane section
- final Vector3D normal = Vector3D.crossProduct(observer, outside);
- final Ellipse section = getPlaneSection(Vector3D.ZERO, normal);
- // the point on limb is tangential to the ellipse
- // if T(xt, yt) is an ellipse point at which the tangent is drawn
- // if O(xo, yo) is a point outside of the ellipse belonging to the tangent at T,
- // then the two following equations holds:
- // (1) a² yt² + b² xt² = a² b² (T belongs to the ellipse)
- // (2) a² yt yo + b² xt xo = a² b² (TP is tangent to the ellipse)
- // using the second equation to eliminate yt from the first equation, we get
- // b² (a² - xt xo)² + a² xt² yo² = a⁴ yo²
- // (3) (a² yo² + b² xo²) xt² - 2 a² b² xo xt + a⁴ (b² - yo²) = 0
- // which can easily be solved for xt
- // To avoid numerical errors, the x and y coordinates in the ellipse plane are normalized using:
- // x' = x / a and y' = y / b
- //
- // This gives:
- // (1) y't² + x't² = 1
- // (2) y't y'o + x't x'o = 1
- //
- // And finally:
- // (3) (x'o² + y'o²) x't² - 2 x't x'o + 1 - y'o² = 0
- //
- // Solving for x't, we get the reduced discriminant:
- // delta' = beta'² - alpha' * gamma'
- //
- // With:
- // beta' = x'o
- // alpha' = x'o² + y'o²
- // gamma' = 1 - y'o²
- //
- // Simplifying to cancel a term of x'o².
- // delta' = y'o² (x'o² + y'o² - 1) = y'o² (alpha' - 1)
- //
- // After solving for xt1, xt2 using (3) the values are substituted into (2) to
- // compute yt1, yt2. Then terms of x'o may be canceled from the expressions for
- // yt1 and yt2. Additionally a point discontinuity is removed at y'o=0 from both
- // yt1 and yt2.
- //
- // y't1 = (y'o - x'o d) / (x'o² + y'o²)
- // y't2 = (x'o y'o + d) / (x + sqrt(delta'))
- //
- // where:
- // d = sign(y'o) sqrt(alpha' - 1)
- // Get the point in ellipse plane frame (2D)
- final Vector2D observer2D = section.toPlane(observer);
- // Normalize and compute intermediary terms
- final double ap = section.getA();
- final double bp = section.getB();
- final double xpo = observer2D.getX() / ap;
- final double ypo = observer2D.getY() / bp;
- final double xpo2 = xpo * xpo;
- final double ypo2 = ypo * ypo;
- final double alphap = ypo2 + xpo2;
- final double gammap = 1. - ypo2;
- // Compute the roots
- // We know there are two solutions as we already checked the point is outside ellipsoid
- final double sqrt = FastMath.sqrt(alphap - 1);
- final double sqrtp = FastMath.abs(ypo) * sqrt;
- final double sqrtSigned = FastMath.copySign(sqrt, ypo);
- // Compute the roots (ordered by value)
- final double xpt1;
- final double xpt2;
- final double ypt1;
- final double ypt2;
- if (xpo > 0) {
- final double s = xpo + sqrtp;
- // xpt1 = (beta' + sqrt(delta')) / alpha' (with beta' = x'o)
- xpt1 = s / alphap;
- // x't2 = gamma' / (beta' + sqrt(delta')) since x't1 * x't2 = gamma' / alpha'
- xpt2 = gammap / s;
- // Get the corresponding values of y't
- ypt1 = (ypo - xpo * sqrtSigned) / alphap;
- ypt2 = (xpo * ypo + sqrtSigned) / s;
- } else {
- final double s = xpo - sqrtp;
- // x't1 and x't2 are reverted compared to previous solution
- xpt1 = gammap / s;
- xpt2 = s / alphap;
- // Get the corresponding values of y't
- ypt2 = (ypo + xpo * sqrtSigned) / alphap;
- ypt1 = (xpo * ypo - sqrtSigned) / s;
- }
- // De-normalize and express the two solutions in 3D
- final Vector3D tp1 = section.toSpace(new Vector2D(ap * xpt1, bp * ypt1));
- final Vector3D tp2 = section.toSpace(new Vector2D(ap * xpt2, bp * ypt2));
- // Return the limb point in the direction of the outside point
- return Vector3D.distance(tp1, outside) <= Vector3D.distance(tp2, outside) ? tp1 : tp2;
- }
- /** Find a point on ellipsoid limb, as seen by an external observer.
- * @param observer observer position in ellipsoid frame
- * @param outside point outside ellipsoid in ellipsoid frame, defining the phase around limb
- * @return point on ellipsoid limb
- * @exception MathRuntimeException if ellipsoid center, observer and outside
- * points are aligned
- * @param <T> the type of the field elements
- * @since 12.0
- */
- public <T extends CalculusFieldElement<T>> FieldVector3D<T> pointOnLimb(final FieldVector3D<T> observer, final FieldVector3D<T> outside)
- throws MathRuntimeException {
- // There is no limb if we are inside the ellipsoid
- if (isInside(observer)) {
- throw new OrekitException(OrekitMessages.POINT_INSIDE_ELLIPSOID);
- }
- // Cut the ellipsoid, to find an elliptical plane section
- final FieldVector3D<T> normal = FieldVector3D.crossProduct(observer, outside);
- final FieldEllipse<T> section = getPlaneSection(FieldVector3D.getZero(observer.getX().getField()), normal);
- // the point on limb is tangential to the ellipse
- // if T(xt, yt) is an ellipse point at which the tangent is drawn
- // if O(xo, yo) is a point outside of the ellipse belonging to the tangent at T,
- // then the two following equations holds:
- // (1) a² yt² + b² xt² = a² b² (T belongs to the ellipse)
- // (2) a² yt yo + b² xt xo = a² b² (TP is tangent to the ellipse)
- // using the second equation to eliminate yt from the first equation, we get
- // b² (a² - xt xo)² + a² xt² yo² = a⁴ yo²
- // (3) (a² yo² + b² xo²) xt² - 2 a² b² xo xt + a⁴ (b² - yo²) = 0
- // which can easily be solved for xt
- // To avoid numerical errors, the x and y coordinates in the ellipse plane are normalized using:
- // x' = x / a and y' = y / b
- //
- // This gives:
- // (1) y't² + x't² = 1
- // (2) y't y'o + x't x'o = 1
- //
- // And finally:
- // (3) (x'o² + y'o²) x't² - 2 x't x'o + 1 - y'o² = 0
- //
- // Solving for x't, we get the reduced discriminant:
- // delta' = beta'² - alpha' * gamma'
- //
- // With:
- // beta' = x'o
- // alpha' = x'o² + y'o²
- // gamma' = 1 - y'o²
- //
- // Simplifying to cancel a term of x'o².
- // delta' = y'o² (x'o² + y'o² - 1) = y'o² (alpha' - 1)
- //
- // After solving for xt1, xt2 using (3) the values are substituted into (2) to
- // compute yt1, yt2. Then terms of x'o may be canceled from the expressions for
- // yt1 and yt2. Additionally a point discontinuity is removed at y'o=0 from both
- // yt1 and yt2.
- //
- // y't1 = (y'o - x'o d) / (x'o² + y'o²)
- // y't2 = (x'o y'o + d) / (x + sqrt(delta'))
- //
- // where:
- // d = sign(y'o) sqrt(alpha' - 1)
- // Get the point in ellipse plane frame (2D)
- final FieldVector2D<T> observer2D = section.toPlane(observer);
- // Normalize and compute intermediary terms
- final T ap = section.getA();
- final T bp = section.getB();
- final T xpo = observer2D.getX().divide(ap);
- final T ypo = observer2D.getY().divide(bp);
- final T xpo2 = xpo.multiply(xpo);
- final T ypo2 = ypo.multiply(ypo);
- final T alphap = ypo2.add(xpo2);
- final T gammap = ypo2.negate().add(1);
- // Compute the roots
- // We know there are two solutions as we already checked the point is outside ellipsoid
- final T sqrt = FastMath.sqrt(alphap.subtract(1));
- final T sqrtp = FastMath.abs(ypo).multiply(sqrt);
- final T sqrtSigned = FastMath.copySign(sqrt, ypo);
- // Compute the roots (ordered by value)
- final T xpt1;
- final T xpt2;
- final T ypt1;
- final T ypt2;
- if (xpo.getReal() > 0) {
- final T s = xpo.add(sqrtp);
- // xpt1 = (beta' + sqrt(delta')) / alpha' (with beta' = x'o)
- xpt1 = s.divide(alphap);
- // x't2 = gamma' / (beta' + sqrt(delta')) since x't1 * x't2 = gamma' / alpha'
- xpt2 = gammap.divide(s);
- // Get the corresponding values of y't
- ypt1 = ypo.subtract(xpo.multiply(sqrtSigned)).divide(alphap);
- ypt2 = xpo.multiply(ypo).add(sqrtSigned).divide(s);
- } else {
- final T s = xpo.subtract(sqrtp);
- // x't1 and x't2 are reverted compared to previous solution
- xpt1 = gammap.divide(s);
- xpt2 = s.divide(alphap);
- // Get the corresponding values of y't
- ypt2 = ypo.add(xpo.multiply(sqrtSigned)).divide(alphap);
- ypt1 = xpo.multiply(ypo).subtract(sqrtSigned).divide(s);
- }
- // De-normalize and express the two solutions in 3D
- final FieldVector3D<T> tp1 = section.toSpace(new FieldVector2D<>(ap.multiply(xpt1), bp.multiply(ypt1)));
- final FieldVector3D<T> tp2 = section.toSpace(new FieldVector2D<>(ap.multiply(xpt2), bp.multiply(ypt2)));
- // Return the limb point in the direction of the outside point
- return FieldVector3D.distance(tp1, outside).subtract(FieldVector3D.distance(tp2, outside)).getReal() <= 0 ? tp1 : tp2;
- }
- }