Ellipse.java
/* Copyright 2002-2016 CS Systèmes d'Information
* Licensed to CS Systèmes d'Information (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.apache.commons.math3.geometry.euclidean.threed.Vector3D;
import org.apache.commons.math3.geometry.euclidean.twod.Vector2D;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathArrays;
import org.orekit.frames.Frame;
import org.orekit.utils.TimeStampedPVCoordinates;
/**
* Model of a 2D ellipse in 3D space.
* <p>
* These ellipses are mainly created as plane sections of general 3D ellipsoids,
* but can be used for other purposes.
* </p>
* <p>
* Instances of this class are guaranteed to be immutable.
* </p>
* @see Ellipsoid#getPlaneSection(Vector3D, Vector3D)
* @since 7.0
* @author Luc Maisonobe
*/
public class Ellipse implements Serializable {
/** Serializable UID. */
private static final long serialVersionUID = 20140925L;
/** Convergence limit. */
private static final double ANGULAR_THRESHOLD = 1.0e-12;
/** Center of the 2D ellipse. */
private final Vector3D center;
/** Unit vector along the major axis. */
private final Vector3D u;
/** Unit vector along the minor axis. */
private final Vector3D v;
/** Semi major axis. */
private final double a;
/** Semi minor axis. */
private final double b;
/** Frame in which the ellipse is defined. */
private final Frame frame;
/** Semi major axis radius power 2. */
private final double a2;
/** Semi minor axis power 2. */
private final double b2;
/** Eccentricity power 2. */
private final double e2;
/** 1 minus flatness. */
private final double g;
/** g * g. */
private final double g2;
/** Evolute factor along major axis. */
private final double evoluteFactorX;
/** Evolute factor along minor axis. */
private final double evoluteFactorY;
/** Simple constructor.
* @param center center of the 2D ellipse
* @param u unit vector along the major axis
* @param v unit vector along the minor axis
* @param a semi major axis
* @param b semi minor axis
* @param frame frame in which the ellipse is defined
*/
public Ellipse(final Vector3D center, final Vector3D u,
final Vector3D v, final double a, final double b,
final Frame frame) {
this.center = center;
this.u = u;
this.v = v;
this.a = a;
this.b = b;
this.frame = frame;
this.a2 = a * a;
this.g = b / a;
this.g2 = g * g;
this.e2 = 1 - g2;
this.b2 = b * b;
this.evoluteFactorX = (a2 - b2) / (a2 * a2);
this.evoluteFactorY = (b2 - a2) / (b2 * b2);
}
/** Get the center of the 2D ellipse.
* @return center of the 2D ellipse
*/
public Vector3D getCenter() {
return center;
}
/** Get the unit vector along the major axis.
* @return unit vector along the major axis
*/
public Vector3D getU() {
return u;
}
/** Get the unit vector along the minor axis.
* @return unit vector along the minor axis
*/
public Vector3D getV() {
return v;
}
/** Get the semi major axis.
* @return semi major axis
*/
public double getA() {
return a;
}
/** Get the semi minor axis.
* @return semi minor axis
*/
public double getB() {
return b;
}
/** Get the defining frame.
* @return defining frame
*/
public Frame getFrame() {
return frame;
}
/** Get a point of the 2D ellipse.
* @param theta angular parameter on the ellipse (really the eccentric anomaly)
* @return ellipse point at theta, in underlying ellipsoid frame
*/
public Vector3D pointAt(final double theta) {
return toSpace(new Vector2D(a * FastMath.cos(theta), b * FastMath.sin(theta)));
}
/** Create a point from its ellipse-relative coordinates.
* @param p point defined with respect to ellipse
* @return point defined with respect to 3D frame
* @see #toPlane(Vector3D)
*/
public Vector3D toSpace(final Vector2D p) {
return new Vector3D(1, center, p.getX(), u, p.getY(), v);
}
/** Project a point to the ellipse plane.
* @param p point defined with respect to 3D frame
* @return point defined with respect to ellipse
* @see #toSpace(Vector2D)
*/
public Vector2D toPlane(final Vector3D p) {
final Vector3D delta = p.subtract(center);
return new Vector2D(Vector3D.dotProduct(delta, u), Vector3D.dotProduct(delta, v));
}
/** Find the closest ellipse point.
* @param p point in the ellipse plane to project on the ellipse itself
* @return closest point belonging to 2D meridian ellipse
*/
public Vector2D projectToEllipse(final Vector2D p) {
final double x = FastMath.abs(p.getX());
final double y = p.getY();
if (x <= ANGULAR_THRESHOLD * FastMath.abs(y)) {
// the point is almost on the minor axis, approximate the ellipse with
// the osculating circle whose center is at evolute cusp along minor axis
final double osculatingRadius = a2 / b;
final double evoluteCuspZ = FastMath.copySign(a * e2 / g, -y);
final double deltaZ = y - evoluteCuspZ;
final double ratio = osculatingRadius / FastMath.hypot(deltaZ, x);
return new Vector2D(FastMath.copySign(ratio * x, p.getX()),
evoluteCuspZ + ratio * deltaZ);
}
if (FastMath.abs(y) <= ANGULAR_THRESHOLD * x) {
// the point is almost on the major axis
final double osculatingRadius = b2 / a;
final double evoluteCuspR = a * e2;
final double deltaR = x - evoluteCuspR;
if (deltaR >= 0) {
// the point is outside of the ellipse evolute, approximate the ellipse
// with the osculating circle whose center is at evolute cusp along major axis
final double ratio = osculatingRadius / FastMath.hypot(y, deltaR);
return new Vector2D(FastMath.copySign(evoluteCuspR + ratio * deltaR, p.getX()),
ratio * y);
}
// the point is on the part of the major axis within ellipse evolute
// we can compute the closest ellipse point analytically
final double rEllipse = x / e2;
return new Vector2D(FastMath.copySign(rEllipse, p.getX()),
FastMath.copySign(g * FastMath.sqrt(a2 - rEllipse * rEllipse), y));
} else {
final double k = FastMath.hypot(x / a, y / b);
double projectedX = x / k;
double projectedY = y / k;
double deltaX = Double.POSITIVE_INFINITY;
double deltaY = Double.POSITIVE_INFINITY;
int count = 0;
final double threshold = ANGULAR_THRESHOLD * ANGULAR_THRESHOLD * a2;
while ((deltaX * deltaX + deltaY * deltaY) > threshold && count++ < 100) { // this loop usually converges in 3 iterations
final double omegaX = evoluteFactorX * projectedX * projectedX * projectedX;
final double omegaY = evoluteFactorY * projectedY * projectedY * projectedY;
final double dx = x - omegaX;
final double dy = y - omegaY;
final double alpha = b2 * dx * dx + a2 * dy * dy;
final double beta = b2 * omegaX * dx + a2 * omegaY * dy;
final double gamma = b2 * omegaX * omegaX + a2 * omegaY * omegaY - a2 * b2;
final double deltaPrime = MathArrays.linearCombination(beta, beta, -alpha, gamma);
final double ratio = (beta <= 0) ?
(FastMath.sqrt(deltaPrime) - beta) / alpha :
-gamma / (FastMath.sqrt(deltaPrime) + beta);
final double previousX = projectedX;
final double previousY = projectedY;
projectedX = omegaX + ratio * dx;
projectedY = omegaY + ratio * dy;
deltaX = projectedX - previousX;
deltaY = projectedY - previousY;
}
return new Vector2D(FastMath.copySign(projectedX, p.getX()), projectedY);
}
}
/** Project position-velocity-acceleration on an ellipse.
* @param pv position-velocity-acceleration to project, in the reference frame
* @return projected position-velocity-acceleration
*/
public TimeStampedPVCoordinates projectToEllipse(final TimeStampedPVCoordinates pv) {
// find the closest point in the meridian plane
final Vector2D p2D = toPlane(pv.getPosition());
final Vector2D e2D = projectToEllipse(p2D);
// tangent to the ellipse
final double fx = -a2 * e2D.getY();
final double fy = b2 * e2D.getX();
final double f2 = fx * fx + fy * fy;
final double f = FastMath.sqrt(f2);
final Vector2D tangent = new Vector2D(fx / f, fy / f);
// normal to the ellipse (towards interior)
final Vector2D normal = new Vector2D(-tangent.getY(), tangent.getX());
// center of curvature
final double x2 = e2D.getX() * e2D.getX();
final double y2 = e2D.getY() * e2D.getY();
final double eX = evoluteFactorX * x2;
final double eY = evoluteFactorY * y2;
final double omegaX = eX * e2D.getX();
final double omegaY = eY * e2D.getY();
// velocity projection ratio
final double rho = FastMath.hypot(e2D.getX() - omegaX, e2D.getY() - omegaY);
final double d = FastMath.hypot(p2D.getX() - omegaX, p2D.getY() - omegaY);
final double projectionRatio = rho / d;
// tangential velocity
final Vector2D pDot2D = new Vector2D(Vector3D.dotProduct(pv.getVelocity(), u),
Vector3D.dotProduct(pv.getVelocity(), v));
final double pDotTangent = pDot2D.dotProduct(tangent);
final double pDotNormal = pDot2D.dotProduct(normal);
final double eDotTangent = projectionRatio * pDotTangent;
final Vector2D eDot2D = new Vector2D(eDotTangent, tangent);
final Vector2D tangentDot = new Vector2D(a2 * b2 * (e2D.getX() * eDot2D.getY() - e2D.getY() * eDot2D.getX()) / f2,
normal);
// velocity of the center of curvature in the meridian plane
final double omegaXDot = 3 * eX * eDotTangent * tangent.getX();
final double omegaYDot = 3 * eY * eDotTangent * tangent.getY();
// derivative of the projection ratio
final double voz = omegaXDot * tangent.getY() - omegaYDot * tangent.getX();
final double vsz = -pDotNormal;
final double projectionRatioDot = ((rho - d) * voz - rho * vsz) / (d * d);
// acceleration
final Vector2D pDotDot2D = new Vector2D(Vector3D.dotProduct(pv.getAcceleration(), u),
Vector3D.dotProduct(pv.getAcceleration(), v));
final double pDotDotTangent = pDotDot2D.dotProduct(tangent);
final double pDotTangentDot = pDot2D.dotProduct(tangentDot);
final double eDotDotTangent = projectionRatio * (pDotDotTangent + pDotTangentDot) +
projectionRatioDot * pDotTangent;
final Vector2D eDotDot2D = new Vector2D(eDotDotTangent, tangent, eDotTangent, tangentDot);
// back to 3D
final Vector3D e3D = toSpace(e2D);
final Vector3D eDot3D = new Vector3D(eDot2D.getX(), u, eDot2D.getY(), v);
final Vector3D eDotDot3D = new Vector3D(eDotDot2D.getX(), u, eDotDot2D.getY(), v);
return new TimeStampedPVCoordinates(pv.getDate(), e3D, eDot3D, eDotDot3D);
}
/** Find the center of curvature (point on the evolute) at the nadir of a point.
* @param point point in the ellipse plane
* @return center of curvature of the ellipse directly at point nadir
* @since 7.1
*/
public Vector2D getCenterOfCurvature(final Vector2D point) {
final Vector2D projected = projectToEllipse(point);
return new Vector2D(evoluteFactorX * projected.getX() * projected.getX() * projected.getX(),
evoluteFactorY * projected.getY() * projected.getY() * projected.getY());
}
}