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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