This demo shows the graph of x(u,v),y(u,v),z(u,v).
In the graph labeled u,v,x(u,v), there are two pink cones through a
point (u0,v0,x(u0,v0)),
one with a slope ε greater than that of the tangent plane and one with
a
slope ε less than that of the tangent plane, while in the graph labeled
u,v,y(u,v), there are two cyan cones through a point (u0,v0,y(u0,v0)),
one with a slope ε greater than that of the tangent plane and one with
a
slope ε less than that of the tangent plane, and finally in the graph
labeled u,v,z(u,v), there are two magenta cones through a point (u0,v0,y(u0,v0)),
one with a slope ε greater than that of the tangent plane and one with
a
slope ε less than that of the tangent plane. If the curve is
differentiable at (u0,v0), you should always be
able to
choose a δ small enough that the portion of
the surfaces u,v,x(u,v), u,v,t(u,v), and u,v,z(u,v) in a disc of radius
δ centered at (u0,v0)
lies inside the pink, cyan, and magenta cones respectively.