/**
* CurvedSpringHelper.java
*
* Copyright (c) 2013-2016, F(X)yz
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
* * Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* * Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* * Neither the name of F(X)yz, any associated website, nor the
* names of its contributors may be used to endorse or promote products
* derived from this software without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
* WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
* DISCLAIMED. IN NO EVENT SHALL F(X)yz BE LIABLE FOR ANY
* DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
* (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
* ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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* SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
package org.fxyz3d.shapes.primitives.helper;
import java.util.ArrayList;
import java.util.List;
import org.fxyz3d.geometry.Point3D;
import org.fxyz3d.geometry.GaussianQuadrature;
/** Spring base on the curve of helix around a torus
*
* Ecuation: r[t]={Cos[t] (R+r Cos[h t]),(R+r Cos[h t]) Sin[t],r Sin[h t]};
*
* Tube around helix: S[t,u]=r[t]+a·cos(u)·n[t]+a·sin(u)·b[t] according
* Frenet-Serret trihedron
* http://www.usciences.edu/~lvas/math430/Curves.pdf
*
* @author jpereda
*/
public class CurvedSpringHelper {
public static final int tR = 0;
public static final int tN = 1;
public static final int tB = 2;
private final double R, r, h;
private double arc;
private List<Point3D[]> trihedrons;
private int subDivLength;
public CurvedSpringHelper(double R, double r, double h){
this.R=R;
this.r=r;
this.h=h;
}
public void calculateTrihedron(int subDivLength, double arc){
// Create points
trihedrons=new ArrayList<>();
this.subDivLength=subDivLength;
this.arc=arc;
for (int t = 0; t <= subDivLength; t++) { // 0 - length
trihedrons.add(getTrihedron((float) t / subDivLength * arc));
}
}
/*
t [0,<=2Pi]
*/
private Point3D[] getTrihedron(double t){
// r[t]
Point3D vR = new Point3D((float)(Math.cos(t)*(R + r*Math.cos(h*t))),
(float)((R + r*Math.cos(h*t))*Math.sin(t)),
(float)(r*Math.sin(h*t)));
// r'[t]
Point3D dR= new Point3D((float)(-((R + r*Math.cos(h*t))*Math.sin(t))-h*r*Math.cos(t)*Math.sin(h*t)),
(float)(Math.cos(t)*(R + r*Math.cos(h*t)) - h*r*Math.sin(t)*Math.sin(h*t)),
(float)(h*r*Math.cos(h*t)));
float nT=dR.magnitude(); // || r'[t] ||
// r''[t]
Point3D ddR= new Point3D((float)(-(Math.cos(t)*(R + (1 + h*h)*r*Math.cos(h*t))) + 2*h*r*Math.sin(t)*Math.sin(h*t)),
(float)(-((R + (1 + h*h)*r*Math.cos(h*t))*Math.sin(t)) - 2*h*r*Math.cos(t)*Math.sin(h*t)),
(float)(-(h*h*r*Math.sin(h*t))));
// (|| r'[t] ||^2)'[t]
float dn=(float)(-2*h*r*(R + r*Math.cos(h*t))*Math.sin(h*t));
// T'[t]=r''[t]/||r't||-r'[t]*(|| r'[t] ||^2)'[t]/2/|| r'[t] ||^3
Point3D dT=ddR.multiply(1f/nT).substract(dR.multiply(dn/((float)(Math.pow(nT,3d)*2d))));
// T[t]=r'[t]/||r'[t]||
Point3D T=dR.normalize();
// N[t]=T'[t]/||T'[t]||
Point3D N=dT.normalize();
// B[t]=T[t]xN[t]/||T[t]xN[t]||
Point3D B=T.crossProduct(N).normalize();
// R,N,B
return new Point3D[]{vR,N,B};
}
public Point3D getS(int t, float cu, float su){
Point3D[] trihedron = trihedrons.get(t);
// S[t,u]
Point3D p = trihedron[CurvedSpringHelper.tR]
.add(trihedron[CurvedSpringHelper.tN].multiply(cu)
.add(trihedron[CurvedSpringHelper.tB].multiply(su)));
p.f=((float)(t*arc)/(float)subDivLength); // [0-<=2Pi]
return p;
}
public double getLength(double arc){ // [0-<=2Pi]
GaussianQuadrature gauss = new GaussianQuadrature(5,0,arc);
// || r'[t] ||
return gauss.NIntegrate(t->Math.sqrt(r*r+2d*h*h*r*r+2d*R*R+4d*r*R*Math.cos(h*t)+r*r*Math.cos(2d*h*t))/Math.sqrt(2d));
}
public double getKappa(double t){
// r'[t]
Point3D dR= new Point3D((float)(-((R + r*Math.cos(h*t))*Math.sin(t))-h*r*Math.cos(t)*Math.sin(h*t)),
(float)(Math.cos(t)*(R + r*Math.cos(h*t)) - h*r*Math.sin(t)*Math.sin(h*t)),
(float)(h*r*Math.cos(h*t)));
float nT=dR.magnitude(); // || r'[t] ||
// r''[t]
Point3D ddR= new Point3D((float)(-(Math.cos(t)*(R + (1 + h*h)*r*Math.cos(h*t))) + 2*h*r*Math.sin(t)*Math.sin(h*t)),
(float)(-((R + (1 + h*h)*r*Math.cos(h*t))*Math.sin(t)) - 2*h*r*Math.cos(t)*Math.sin(h*t)),
(float)(-(h*h*r*Math.sin(h*t))));
// || r''[t]xr'[t] ||
float nddRxdR=ddR.crossProduct(dR).magnitude();
// kappa[t] = || r''[t]xr'[t] || / || r'[t] ||^3
return nddRxdR/(float)Math.pow(nT,3d);
}
public double getTau(double t){
Point3D dR= new Point3D((float)(-((R + r*Math.cos(h*t))*Math.sin(t))-h*r*Math.cos(t)*Math.sin(h*t)),
(float)(Math.cos(t)*(R + r*Math.cos(h*t)) - h*r*Math.sin(t)*Math.sin(h*t)),
(float)(h*r*Math.cos(h*t)));
// r''[t]
Point3D ddR= new Point3D((float)(-(Math.cos(t)*(R + (1 + h*h)*r*Math.cos(h*t))) + 2*h*r*Math.sin(t)*Math.sin(h*t)),
(float)(-((R + (1 + h*h)*r*Math.cos(h*t))*Math.sin(t)) - 2*h*r*Math.cos(t)*Math.sin(h*t)),
(float)(-(h*h*r*Math.sin(h*t))));
// r'''[t]
Point3D dddR = new Point3D((float)((R + (1 + 3*h*h)*r*Math.cos(h*t))*Math.sin(t) + h*(3 + h*h)*r*Math.cos(t)*Math.sin(h*t)),
(float)(-(Math.cos(t)*(R + (1 + 3*h*h)*r*Math.cos(h*t))) + h*(3 + h*h)*r*Math.sin(t)*Math.sin(h*t)),
(float)(-(h*h*h*r*Math.cos(h*t))));
// r'[t]xr''[t] . r'''[t]
float dRxddRxdddR=dR.crossProduct(ddR).dotProduct(dddR);
// || r''[t]xr'[t] ||
float ndRxddR=dR.crossProduct(ddR).magnitude();
// tau[t] = r'[t]xr''[t].r'''[t] / || r''[t]xr'[t] ||^2
return Math.abs(dRxddRxdddR/(float)Math.pow(ndRxddR,2d));
}
}