My name is Ricky and I run a site called Race Optimal.
http://www.raceoptimal.com/about/physics
I'm working on converting the 2D physics model to 3D. I have an algorithm to calculate a continuous piecewise polynomial for the x, y, and z coordinates. As the z coordinate changes, more and less normal force is experienced depending on the acceleration relative to the z-axis. Typically this would be solved by calculating the radius of curvature. However, since the path is 3D, I'm not sure how to calculate curvature relative to the z axis. Any suggestions?
Wikipedia to the rescue!
https://en.wikipedia.org/wiki/Curvature#Curvature_of_space_curves
Basically, when the 3D curve is in parameterized form, the first derivative with respect to t gives the tangent vector to the path. The second derivative gives the normal vector pointing in the direction of acceleration due to curvature. Radius of curvature is then given by
$$ R=\frac{(x'^{2}+y'^{2}+z'^{2})^{3/2}}{\sqrt{(z''y'-y''z')^{2}+(x''z'-z''x')^{2}+(y''x'-x''y')^{2}}} $$
The radius of curvature relative to the vertical can be found by dividing this total radius by the z component of the unit normal. The curvature relative to the x and y directions, i.e. for cornering, can be found by dividing by the magnitude of the x and y components combined. Thus, the effective radius of curvature in for cornering or vertical acceleration with be equal to or larger than the 3D radius of curvature.