Parameterization by differential equations

[rabbed]

Hi

Is it possible to solve something like this (and are there any errors in the math)?

A given curve with implicit function f(x,y) = 0 (for example r^2-x^2-y^2 = 0), has a
normal (df/dx, df/dy) and a tangent with direction according to (-df/dy, df/dx).

A parameterization of the implicit function curve, p(t) = (g(t), h(g(t))), has
velocity p'(t) = (g'(t), g'(t)*h'(g(t))) where g(t) and h(g(t)) are to be found.

The velocity of the parameterization should be orthogonal to the tangent of
the implicit function curve and the speed of the parameterization should be 1,
giving the differential equation system:

p'(t)·(-df/dy,df/dx) = 0
p'(t)·p'(t) = 1

or

g'(t) * -df( g(t),h(g(t)) )/dy + g'(t)*h'(g(t)) * df( g(t),h(g(t)) )/dx = 0
g'(t) * g'(t) + g'(t)*h'(g(t)) * g'(t)*h'(g(t)) = 1

 

[gtoguy97]

Yes, it is possible to solve this problem. The two equations form a system of two equations with two unknowns, which can be solved using techniques such as substitution or elimination. There do not appear to be any errors in the math.