a=ListInterpolation[{-1,-1,0,0,-1},{{0,1}}];
c=ListInterpolation[{0,1,1,1,1},{{0,1}}];
p=ListInterpolation[{1,0,0,0,0},{{0,1}}];
r=ListInterpolation[{0,1,1,0,0},{{0,1}}];

Animate[{ContourPlot3D[{x^2-y^2-z^2 == 0, a[t](x-p[t])+c[t](z-r[t])== 0},{x,-5,5},{y,-5,5},{z,-5,5},PerformanceGoal-> "Quality",Mesh->None],ContourPlot[{(c[t]*y+p[t])^2-x^2-(r[t]-a[t]*y)^2 == 0},{x,-5,5},{y,-5,5},PerformanceGoal-> "Quality"]},{t,0,1},AnimationRate->0.1]