How to Calculate the Curvature of a Level Curve?

[Mapprehension]

I can’t find the notation I need to look up what I need to know.

I have a mapping defined thus:

x = F (a, b, c)
y = G (a, b, c)

Where [a, b, c] is any point from some 3-space surface and [x, y] is in cartesian space. F, G are continuously differentiable (but may contain discontinuities, a fact not relevant here).

I have a notion of a level curve defined by all points whose metric function M yields the same value. M is some function of all first partial derivatives of a, b, c with respect to x and y. I need the curvature of this level curve, where curvature is defined in the usual way: that is, the radius of the circle whose tangent vector changes direction at the same rate as a tangent vector at the point in question on the level curve.

It is easy enough to find the tangent vector of the level curve; it is merely a 90° rotation of the the gradient vector of M at the point in question. I can find the change in that tangent numerically easily enough. However, I do not know the notation for this operation of finding the derivative of the gradient in the direction of the level curve (that is, perpendicular to the gradient). Since I have nothing that I recognize as a parameterization of the level curve, I find myself stymied by the usual descriptions of curvature.

Presumably the curvature is some horrible jumble of first, second, and third partials of a, b, c, with respect to x and y, with the amount of horror dependent on the complexity of M. However, my notational confusion leaves my thinking on the matter confused as well.

Any tips appreciated. Happy new year!
— Mapprehension

 

[jvicens]

Dear Mapprehension,

Thank you for your forum post. It seems like you are looking for the notation to calculate the curvature of a level curve defined by a metric function M. The curvature of a curve is typically defined as the rate of change of the tangent vector along the curve. In your case, the tangent vector is a 90° rotation of the gradient vector of M at the point in question on the level curve.

To calculate the curvature, you will need to take the derivative of the gradient vector in the direction of the level curve. This can be denoted as ∇M, where ∇ is the gradient operator and M is the metric function. This operation will give you the directional derivative of the gradient vector, which can then be used to calculate the curvature.

However, since you do not have a parameterization of the level curve, the calculation of the curvature may involve a complex combination of first, second, and third partial derivatives of a, b, c with respect to x and y. The complexity of this calculation will depend on the complexity of the metric function M.

I suggest consulting a calculus textbook or speaking with a mathematics expert for further assistance with the notation and calculation of the curvature of a level curve. I hope this helps and happy new year to you as well!