Curvature is a geometric property that describes the amount of bending or deviation from being flat on a surface. It is typically defined in terms of the normal direction, which is the direction perpendicular to the surface at each point. The curvature in the normal direction is called the normal curvature or the principal curvature.
When we talk about the curvature of a surface in a direction that is not normal to the surface, we are essentially considering the curvature in a different direction or along a specific curve on the surface. This type of curvature is known as the geodesic curvature or the curve curvature.
The geodesic curvature of a curve on a surface measures how the curve deviates from being a straight line on the surface. It is influenced by both the intrinsic curvature of the surface and the extrinsic properties of the curve itself. The geodesic curvature can be positive, negative, or zero, depending on whether the curve bends away from, toward, or remains tangent to the surface.
Calculating the geodesic curvature involves considering the surface's geometry and the behavior of the curve on the surface. It is often done using differential geometry techniques, such as the Gauss-Bonnet theorem or the use of curvature tensors.
In summary, while the curvature of a surface is typically defined in terms of the normal direction, it is also possible to calculate the curvature of a surface in a specific direction or along a curve on the surface. This curvature is known as the geodesic curvature and requires additional considerations beyond the normal curvature.