4901.sagews

Strophoid

Determine the locus of the foci of the ellipse obtained when the plane cutting the cylinder rotates about the tangent at its vertex.

R.<t,b,u,v>=PolynomialRing(QQ, order='lex')

A=(-b,0); O=(0,0); F1=(u,v); F2=(-u,2*t-v); M=(0,t)

h1=(M[0]-F1[0])^2+(M[1]-F1[1])^2-t^2 
h2=(F1[1]-A[1])*(M[0]-A[0])-(F1[0]-A[0])*(M[1]-A[1])

I=[h1,h2]*R

I.groebner_basis()

[t*b + t*u - b*v, t*v - 1/2*u^2 - 1/2*v^2, b*u^2 - b*v^2 + u^3 + u*v^2]

x,y=var('x,y')
C=I.groebner_basis()[2].substitute(b=5,u=x,v=y)
implicit_plot(C,(x,-5,5),(y,-5,5),figsize=5,aspect_ratio=1)