Robotics
Circuit Design
Medicine
Glass Melting
Regular rectangular Heptagon: c=0
Equations of the configuration space of a robot (in SINGULAR ):

We fix   P7 = (0,0,0) , P1 = (1,0,0), P6 = (0,1,0)   and consider the coordinates (xi,yi,zi) of Pi with respect to Pi-1. It is clear that   x2 = 0 , y6 = 0.  We obtain 12 equations in 13 variables.
ring S=0,(x,y,z,e,f,t,u,v,w,a,b,c,d,h),dp;
ideal J=
  f2+w2-1,
  x2+t2+a2-1,
  y2+u2+b2-1,
  z2+v2+c2-1,
  e2+d2-1,
  ft+wa,
  xy+tu+ab,
  yz+uv+bc,
  ze+cd,
  x+y+z+e+1,
  f+t+u+v-1,
  w+a+b+c+d;
The equations describe a curve in R13. The projections to the different planes should be computed. The projection to the (e,f)-plane can be obtained as follows:
J=homog(J,h);
ideal L=std(J);
intvec hi=hilb(L,1);
ideal K=eliminate(J,xyztuvwabcd,hi);
K=subst(K,h,1);
K[1];
Plot the computed projection to the (e,f)-plane:
 
LIB "surf.lib";
plot(K[1]);
Image of the curve in the (e,f)-plane

KL, 06/03 http://www.singular.uni-kl.de