2 * Equation for a Moebius strip
11 #include "parameter.hh"
13 Parameter<double> stiff("stiff","stiffness of tangent specs.",0.4,0.1,0.0,1.0);
15 Point MoebiusStrip::edgepoint(double t) {
16 double theta= (t-0.5)*4.0*PI;
17 double r= 1.0 - halfgap*(1.0 - sin(theta/2.0));
18 return Point(r*sin(theta),
20 halfbreadth*cos(theta/2.0));
23 Point MoebiusStrip::middlepoint(double t, double u) {
24 return edgepoint(t*0.5)*u + edgepoint((t+1)*0.5)*(1.0-u);
30 Bezier(double x0, double x1, double dx0, double dx1);
31 double operator()(double t) { return h + t*(g + t*(f + t*e)); }
35 Bezier::Bezier(double x0, double x1, double dx0, double dx1) {
36 // e= 2*x0 - 2*x1 + dx0 + dx1;
37 // f= x1 - dx0 - x0 - e;
40 e= g + 2*h + dx1 - 2*x1;
44 void Bezier::debug() {
45 fprintf(stderr,"bz e %7.4f f %7.4f g %7.4f h %7.4f\n",e,f,g,h);
48 // The first end is at [sin(theta/2),-cos(theta/2),0]
49 // The second end is at [-theta/pi,0,sin(theta)]
50 // The first end is oriented towards [0,cos(theta),sin(theta)]
51 // The second end is oriented towards [0,-1,0]
53 Point MoebiusEnfoldment::middlepoint(double t, double u) {
54 if (t > bottomportion) {
56 t /= (1.0 - bottomportion);
57 double sizehere= sqrt(1-t*t);
58 return Point((u*2.0-1.0) * sizehere,
60 sizehere * thickness * sin(u*2.0*PI));
63 double theta= (.5-u)*2*PI;
64 Bezier bx(sin(theta*.5), -theta/PI, 0, 0);
65 double ypushiness= (1-cos(theta))*2.5+1;
66 // double fudge= (PI*sin(theta*.5)*cos(theta*.5))*(.5-u)*(.5-u)*4;
67 double fudge= (.5-u)*(.5-u)*4*cos(theta*.5);
68 Bezier by(-cos(theta*.5), 0,
69 cos(theta)*ypushiness + fudge*ypushiness,
72 Bezier bz(0, sin(theta), sin(theta), 0);
73 return Point( bx(t), by(t), thickness * bz(t) );
77 double smoothinterp(double t, const double values[], const double rates[]) {
79 int n1= (int)floor(t*ncells);
81 double r= t*ncells-n1;
82 double v1= values[n1];
83 double v2= values[n2];
84 double r1= rates[n1]/ncells;
85 double r2= rates[n2]/ncells;
88 double b= 3*dv-2*r1-r2;
90 double vr= a*r*r + b*r*r + c*r;
92 // printf("smoothinterp %7.6f %7.6f %g(%g)..%g(%g) %g\n",t,r,v1,r1,v2,r2,v);
96 Point MoebiusNewEnfoldment::middlepoint(double t, double u) {
97 const double agammas[]= {
98 /* angle at rim for surface near edge, for values of t:
99 * 0.0 .25 0.5 0.75 1.0, not usu. used */
100 PI, 0.75*PI, 0.5*PI, -0.25*PI, -PI, -PI
102 const double agammarates[]= {
103 /* rates of change of gamma with t */
104 -PI, -PI, -PI, -3*PI, -3*PI, -PI
106 const double alambdas[]= { 0.0, 0.0, 0.0, 0.0, 0.0 };
107 const double alambdarates[]= { 0.0, 0.0, 0.0, 0.0, 0.0 };
108 const double bgammas[]= {
109 /* angle at middle (u=0.5) in direction of decreasing u, for values of t:
110 * 0.0 .25 0.5 0.75 1.0, not usu. used */
111 0, 0.5*PI, 0.5*PI, 0.5*PI, PI, PI
113 const double bgammarates[]= {
114 /* rates of change of gamma with t */
115 2*PI, 0, 0, 0, 2*PI, 2*PI
117 const double blambdas[]= {
118 /* shear along middle (u=0.5) for values of t:
119 * 0.0 .25 0.5 0.75 1.0, not usu. used */
120 0.0, 0.5, 0.5, 0.5, 0, 0
122 const double blambdarates[]= { 0.0, 0.0, 0.0, 0.0, 0.0 };
124 Point p= middlepoint(1.0-t,1.0-u);
125 return Point(-p[0],p[1],-p[2]);
128 double asine= sin(aangle);
129 double acosine= cos(aangle);
130 double agamma= smoothinterp(t,agammas,agammarates);
131 double arotnl= smoothinterp(t,alambdas,alambdarates);
132 double aradial= cos(agamma);
133 double aaxial= sin(agamma);
134 Point a(asine, acosine, 0);
135 Point da(arotnl*acosine+aradial*asine,aradial*acosine-arotnl*asine,-aaxial);
136 double bangle= 2*PI*t;
137 double bsine= sin(bangle);
138 double bcosine= cos(bangle);
139 double bgamma= smoothinterp(t,bgammas,bgammarates);
140 double brotnl= smoothinterp(t,blambdas,blambdarates);
141 double bradial= cos(bgamma);
142 double baxial= sin(bgamma);
143 Point b(0, bcosine-1.0, -bsine);
144 Point db(baxial,bradial*bcosine-brotnl*bsine,brotnl*bcosine+bradial*bsine);
146 double ab= (a-b).magnitude();
147 double adscale= ab*stiff/da.magnitude();
148 double bdscale= ab*stiff/db.magnitude();
149 double facta= (1-u)*(1-u)*(1-u);
151 double factan= 3*(1-u)*(1-u)*u;
152 double factbn= 3*(1-u)*u*u;
153 Point dan= a+da*adscale;
154 Point dbn= b+db*bdscale;
155 Point result= a*facta + dan*factan + dbn*factbn + b*factb;
156 // printf("t=%7.6f u=%7.6f a=%-30s dan=%-30s dbn=%-30s b=%-30s\n",
157 // t,u, a.printing(), dan.printing(), dbn.printing(), b.printing());