1 /* Copyright 2002-2016 CS Systèmes d'Information
2 * Licensed to CS Systèmes d'Information (CS) under one or more
3 * contributor license agreements. See the NOTICE file distributed with
4 * this work for additional information regarding copyright ownership.
5 * CS licenses this file to You under the Apache License, Version 2.0
6 * (the "License"); you may not use this file except in compliance with
7 * the License. You may obtain a copy of the License at
8 *
9 * http://www.apache.org/licenses/LICENSE-2.0
10 *
11 * Unless required by applicable law or agreed to in writing, software
12 * distributed under the License is distributed on an "AS IS" BASIS,
13 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14 * See the License for the specific language governing permissions and
15 * limitations under the License.
16 */
17 package org.orekit.bodies;
18
19 import java.io.Serializable;
20
21 import org.apache.commons.math3.exception.MathArithmeticException;
22 import org.apache.commons.math3.geometry.euclidean.threed.Vector3D;
23 import org.apache.commons.math3.geometry.euclidean.twod.Vector2D;
24 import org.apache.commons.math3.util.FastMath;
25 import org.apache.commons.math3.util.MathArrays;
26 import org.apache.commons.math3.util.Precision;
27 import org.orekit.errors.OrekitException;
28 import org.orekit.errors.OrekitMessages;
29 import org.orekit.frames.Frame;
30
31 /**
32 * Modeling of a general three-axes ellipsoid. <p>
33 * @since 7.0
34 * @author Luc Maisonobe
35 */
36 public class Ellipsoid implements Serializable {
37
38 /** Serializable UID. */
39 private static final long serialVersionUID = 20140924L;
40
41 /** Frame at the ellipsoid center, aligned with principal axes. */
42 private final Frame frame;
43
44 /** First semi-axis length. */
45 private final double a;
46
47 /** Second semi-axis length. */
48 private final double b;
49
50 /** Third semi-axis length. */
51 private final double c;
52
53 /** Simple constructor.
54 * @param frame at the ellipsoid center, aligned with principal axes
55 * @param a first semi-axis length
56 * @param b second semi-axis length
57 * @param c third semi-axis length
58 */
59 public Ellipsoid(final Frame frame, final double a, final double b, final double c) {
60 this.frame = frame;
61 this.a = a;
62 this.b = b;
63 this.c = c;
64 }
65
66 /** Get the length of the first semi-axis.
67 * @return length of the first semi-axis (m)
68 */
69 public double getA() {
70 return a;
71 }
72
73 /** Get the length of the second semi-axis.
74 * @return length of the second semi-axis (m)
75 */
76 public double getB() {
77 return b;
78 }
79
80 /** Get the length of the third semi-axis.
81 * @return length of the third semi-axis (m)
82 */
83 public double getC() {
84 return c;
85 }
86
87 /** Get the ellipsoid central frame.
88 * @return ellipsoid central frame
89 */
90 public Frame getFrame() {
91 return frame;
92 }
93
94 /** Check if a point is inside the ellipsoid.
95 * @param point point to check, in the ellipsoid frame
96 * @return true if the point is inside the ellipsoid
97 * (or exactly on ellipsoid surface)
98 * @since 7.1
99 */
100 public boolean isInside(final Vector3D point) {
101 final double scaledX = point.getX() / a;
102 final double scaledY = point.getY() / b;
103 final double scaledZ = point.getZ() / c;
104 return scaledX * scaledX + scaledY * scaledY + scaledZ * scaledZ <= 1.0;
105 }
106
107 /** Compute the 2D ellipse at the intersection of the 3D ellipsoid and a plane.
108 * @param planePoint point belonging to the plane, in the ellipsoid frame
109 * @param planeNormal normal of the plane, in the ellipsoid frame
110 * @return plane section or null if there are no intersections
111 */
112 public Ellipse getPlaneSection(final Vector3D planePoint, final Vector3D planeNormal) {
113
114 // we define the points Q in the plane using two free variables τ and υ as:
115 // Q = P + τ u + υ v
116 // where u and v are two unit vectors belonging to the plane
117 // Q belongs to the 3D ellipsoid so:
118 // (xQ / a)² + (yQ / b)² + (zQ / c)² = 1
119 // combining both equations, we get:
120 // (xP² + 2 xP (τ xU + υ xV) + (τ xU + υ xV)²) / a²
121 // + (yP² + 2 yP (τ yU + υ yV) + (τ yU + υ yV)²) / b²
122 // + (zP² + 2 zP (τ zU + υ zV) + (τ zU + υ zV)²) / c²
123 // = 1
124 // which can be rewritten:
125 // α τ² + β υ² + 2 γ τυ + 2 δ τ + 2 ε υ + ζ = 0
126 // with
127 // α = xU² / a² + yU² / b² + zU² / c² > 0
128 // β = xV² / a² + yV² / b² + zV² / c² > 0
129 // γ = xU xV / a² + yU yV / b² + zU zV / c²
130 // δ = xP xU / a² + yP yU / b² + zP zU / c²
131 // ε = xP xV / a² + yP yV / b² + zP zV / c²
132 // ζ = xP² / a² + yP² / b² + zP² / c² - 1
133 // this is the equation of a conic (here an ellipse)
134 // Of course, we note that if the point P belongs to the ellipsoid
135 // then ζ = 0 and the equation holds at point P since τ = 0 and υ = 0
136 final Vector3D u = planeNormal.orthogonal();
137 final Vector3D v = Vector3D.crossProduct(planeNormal, u).normalize();
138 final double xUOa = u.getX() / a;
139 final double yUOb = u.getY() / b;
140 final double zUOc = u.getZ() / c;
141 final double xVOa = v.getX() / a;
142 final double yVOb = v.getY() / b;
143 final double zVOc = v.getZ() / c;
144 final double xPOa = planePoint.getX() / a;
145 final double yPOb = planePoint.getY() / b;
146 final double zPOc = planePoint.getZ() / c;
147 final double alpha = xUOa * xUOa + yUOb * yUOb + zUOc * zUOc;
148 final double beta = xVOa * xVOa + yVOb * yVOb + zVOc * zVOc;
149 final double gamma = MathArrays.linearCombination(xUOa, xVOa, yUOb, yVOb, zUOc, zVOc);
150 final double delta = MathArrays.linearCombination(xPOa, xUOa, yPOb, yUOb, zPOc, zUOc);
151 final double epsilon = MathArrays.linearCombination(xPOa, xVOa, yPOb, yVOb, zPOc, zVOc);
152 final double zeta = MathArrays.linearCombination(xPOa, xPOa, yPOb, yPOb, zPOc, zPOc, 1, -1);
153
154 // reduce the general equation α τ² + β υ² + 2 γ τυ + 2 δ τ + 2 ε υ + ζ = 0
155 // to canonical form (λ/l)² + (μ/m)² = 1
156 // using a coordinates change
157 // τ = τC + λ cosθ - μ sinθ
158 // υ = υC + λ sinθ + μ cosθ
159 // or equivalently
160 // λ = (τ - τC) cosθ + (υ - υC) sinθ
161 // μ = - (τ - τC) sinθ + (υ - υC) cosθ
162 // τC and υC are the coordinates of the 2D ellipse center with respect to P
163 // 2l and 2m and are the axes lengths (major or minor depending on which one is greatest)
164 // θ is the angle of the 2D ellipse axis corresponding to axis with length 2l
165
166 // choose θ in order to cancel the coupling term in λμ
167 // expanding the general equation, we get: A λ² + B μ² + C λμ + D λ + E μ + F = 0
168 // with C = 2[(β - α) cosθ sinθ + γ (cos²θ - sin²θ)]
169 // hence the term is cancelled when θ = arctan(t), with γ t² + (α - β) t - γ = 0
170 // As the solutions of the quadratic equation obey t₁t₂ = -1, they correspond to
171 // angles θ in quadrature to each other. Selecting one solution or the other simply
172 // exchanges the principal axes. As we don't care about which axis we want as the
173 // first one, we select an arbitrary solution
174 final double tanTheta;
175 if (FastMath.abs(gamma) < Precision.SAFE_MIN) {
176 tanTheta = 0.0;
177 } else {
178 final double bMA = beta - alpha;
179 tanTheta = (bMA >= 0) ?
180 (-2 * gamma / (bMA + FastMath.sqrt(bMA * bMA + 4 * gamma * gamma))) :
181 (-2 * gamma / (bMA - FastMath.sqrt(bMA * bMA + 4 * gamma * gamma)));
182 }
183 final double tan2 = tanTheta * tanTheta;
184 final double cos2 = 1 / (1 + tan2);
185 final double sin2 = tan2 * cos2;
186 final double cosSin = tanTheta * cos2;
187 final double cos = FastMath.sqrt(cos2);
188 final double sin = tanTheta * cos;
189
190 // choose τC and υC in order to cancel the linear terms in λ and μ
191 // expanding the general equation, we get: A λ² + B μ² + C λμ + D λ + E μ + F = 0
192 // with D = 2[ (α τC + γ υC + δ) cosθ + (γ τC + β υC + ε) sinθ]
193 // E = 2[-(α τC + γ υC + δ) sinθ + (γ τC + β υC + ε) cosθ]
194 // θ can be eliminated by combining the equations
195 // D cosθ - E sinθ = 2[α τC + γ υC + δ]
196 // E cosθ + D sinθ = 2[γ τC + β υC + ε]
197 // hence the terms D and E are both cancelled (regardless of θ) when
198 // τC = (β δ - γ ε) / (γ² - α β)
199 // υC = (α ε - γ δ) / (γ² - α β)
200 final double denom = MathArrays.linearCombination(gamma, gamma, -alpha, beta);
201 final double tauC = MathArrays.linearCombination(beta, delta, -gamma, epsilon) / denom;
202 final double nuC = MathArrays.linearCombination(alpha, epsilon, -gamma, delta) / denom;
203
204 // compute l and m
205 // expanding the general equation, we get: A λ² + B μ² + C λμ + D λ + E μ + F = 0
206 // with A = α cos²θ + β sin²θ + 2 γ cosθ sinθ
207 // B = α sin²θ + β cos²θ - 2 γ cosθ sinθ
208 // F = α τC² + β υC² + 2 γ τC υC + 2 δ τC + 2 ε υC + ζ
209 // hence we compute directly l = √(-F/A) and m = √(-F/B)
210 final double twogcs = 2 * gamma * cosSin;
211 final double bigA = alpha * cos2 + beta * sin2 + twogcs;
212 final double bigB = alpha * sin2 + beta * cos2 - twogcs;
213 final double bigF = (alpha * tauC + 2 * (gamma * nuC + delta)) * tauC +
214 (beta * nuC + 2 * epsilon) * nuC + zeta;
215 final double l = FastMath.sqrt(-bigF / bigA);
216 final double m = FastMath.sqrt(-bigF / bigB);
217 if (Double.isNaN(l + m)) {
218 // the plane does not intersect the ellipsoid
219 return null;
220 }
221
222 if (l > m) {
223 return new Ellipse(new Vector3D(1, planePoint, tauC, u, nuC, v),
224 new Vector3D( cos, u, sin, v),
225 new Vector3D(-sin, u, cos, v),
226 l, m, frame);
227 } else {
228 return new Ellipse(new Vector3D(1, planePoint, tauC, u, nuC, v),
229 new Vector3D(sin, u, -cos, v),
230 new Vector3D(cos, u, sin, v),
231 m, l, frame);
232 }
233
234 }
235
236 /** Find a point on ellipsoid limb, as seen by an external observer.
237 * @param observer observer position in ellipsoid frame
238 * @param outside point outside ellipsoid in ellipsoid frame, defining the phase around limb
239 * @return point on ellipsoid limb
240 * @exception OrekitException if the observer is inside the ellipsoid
241 * @exception MathArithmeticException if ellipsoid center, observer and outside
242 * points are aligned
243 * @since 7.1
244 */
245 public Vector3D pointOnLimb(final Vector3D observer, final Vector3D outside)
246 throws OrekitException, MathArithmeticException {
247
248 // there is no limb if we are inside the ellipsoid
249 if (isInside(observer)) {
250 throw new OrekitException(OrekitMessages.POINT_INSIDE_ELLIPSOID);
251 }
252 // cut the ellipsoid, to find an elliptical plane section
253 final Vector3D normal = Vector3D.crossProduct(observer, outside);
254 final Ellipse section = getPlaneSection(Vector3D.ZERO, normal);
255 final double a2 = section.getA() * section.getA();
256 final double b2 = section.getB() * section.getB();
257
258 // the point on limb is tangential to the ellipse
259 // if T(xt, yt) is an ellipse point at which the tangent is drawn
260 // if O(xo, yo) is a point outside of the ellipse belonging to the tangent at T,
261 // then the two following equations holds:
262 // a² yt² + b² xt² = a² b² (T belongs to the ellipse)
263 // a² yt yo + b² xt xo = a² b² (TP is tangent to the ellipse)
264 // using the second equation to eliminate yt from the first equation, we get
265 // b² (a² - xt xo)² + a² xt² yo² = a⁴ yo²
266 // (a² yo² + b² xo²) xt² - 2 a² b² xo xt + a⁴ (b² - yo²) = 0
267 // which can easily be solved for xt
268 final Vector2D observer2D = section.toPlane(observer);
269 final double xo = observer2D.getX();
270 final double yo = observer2D.getY();
271 final double xo2 = xo * xo;
272 final double yo2 = yo * yo;
273 final double alpha = a2 * yo2 + b2 * xo2;
274 final double beta = a2 * b2 * xo;
275 final double gamma = a2 * a2 * (b2 - yo2);
276 // we know there are two solutions as we already checked the point is outside ellipsoid
277 final double sqrt = FastMath.sqrt(beta * beta - alpha * gamma);
278 final double xt1;
279 final double xt2;
280 if (beta > 0) {
281 final double s = beta + sqrt;
282 xt1 = s / alpha;
283 xt2 = gamma / s;
284 } else {
285 final double s = beta - sqrt;
286 xt1 = gamma / s;
287 xt2 = s / alpha;
288 }
289
290 // we return the limb point in the direction of the outside point
291 final Vector3D t1 = section.toSpace(new Vector2D(xt1, b2 * (a2 - xt1 * xo) / (a2 * yo)));
292 final Vector3D t2 = section.toSpace(new Vector2D(xt2, b2 * (a2 - xt2 * xo) / (a2 * yo)));
293 return Vector3D.distance(t1, outside) <= Vector3D.distance(t2, outside) ? t1 : t2;
294
295 }
296
297 }