What curve is it? [closed]

Problem: Need to find geodesic curvature of a curve on a surface and i have curve like this: $w(x,y)=const$ $$$$ I know some formulas for geodesic curvature, but they use curves like $r=r(x(t),y(t))$ $$$$So my question is: How can i understand w-curve? Can i remake it into r-curve someway?

Solution 1:

Curves in the plane can be given in a few different ways:

• Parameterized curves: $r(t) = (x(t),y(t)$
• Graphs: $y=f(x)$ or sometimes $x=g(y)$
• Implicit curves: $w(x,y) = c$

You do need to know how to convert from one format to another. You may have learned how to do this in calculus.

For example: to convert a graph $y=f(x)$ into a parameterized curve, use the input variable as the parameter: $x(t)=t$, $y(t)=f(t)$

For another example: to convert an implicit curve $w(x,y)=c$ into a graph, solve for $y$ or perhaps for $x$ (this is not always possible globally, but it is almost always possible locally; I presume that since you have used the differential-geometry tag, you know the Implicit Function Theorem).

You can put these together: to convert an implicit curve into a parametric curve, first convert it into a graph, then convert that into a parametric curve.