Ellipsoid.java
/* Copyright 2002-2023 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,
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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;
}
}