The curve C(𝜃) drawn on a smoothly embedded surface 𝜮 in 3-space — where C(𝜃) is defined as the intersection of 𝜮 with a 2-plane perpendicular to 𝜮 at P — leaving the point P at angle 𝜃 will have the well-known formula for curvature:

k(𝜃) = k₁ cos²(𝜃) + k₂ sin²(𝜃)

at P, where k₁ and k₂ are the principal curvatures of 𝜮 at P.

Similarly, we can look at a point P of a smoothly embedded surface 𝜮 in Euclidean n-space, and consider the curves C(𝜃) that are each the intersection of 𝜮 with a hyperplane that is perpendicular to 𝜮 at P and ask what their curvatures are as a function of angle 𝜃.

Does there exist a formula (like k(𝜃) = k₁ cos²(𝜃) + k₂ sin²(𝜃), or different) for the curvatures k(𝜃) of the curves C(𝜃) in this case?

And what about the special case where 𝜮 is a Riemann surface holomorphically embedded in C^m = R^2m?