If I understand correctly, what you want to compute is the spline's radius of curvature at each point. Then you want to compare that value to an agent's turning radius to determine if the agent will leave the track there. [1]
To do this, you need to calculateR(s)as you see here: https://en.wikipedia.org/wiki/Curvature (under "Curvature of plane curves" > "Precise definition")
I don't believe Curvy offers second derivativesT'(s)(possibly because some of its spline algorithms do not have continuous second derivatives). So, I'll see if I can cook up an approximation here using Curvy's API: [2]
So, we haveT(s)for free, just using Curvy's CurvySplineBase.GetTangent(float)method.
We can use this call to estimate T'(s). Here I am taking the average of just before and just after, but in practice it may not matter, and it does complicate the endpoint code somewhat.
public bool checkCurvature(float position, CurvySpline spline, Agent agent)
{
// This may be too small a difference for Curvy - try larger values
// if it behaves poorly.
float epsilon = float.epsilon;
// You'll need to enhance this to account for the beginning/end
// of the spline (s = 0.0 or 1.0)
Vector3 tanPrev = spline.GetTangent(position-epsilon);
Vector3 tanCurr = spline.GetTangent(position);
Vector3 tanNext = spline.GetTangent(position+epsilon);
Vector3 prevDiff = tanCurr-tanPrev;
Vector3 nextDiff = tanNext-tanCurr;
// average the two, hopefully a better approximation
Vector3 acceleration = (prevDiff + nextDiff) / (2.0f*epsilon); // dT / ds
float curvature = acceleration.magnitude(); // k(s)
float radius = 1.0f/curvature; // R(s)
// if the agent can turn faster than the curve at this point, return true
if( agent.turnRadius <= radius )
{
return true;
}
return false;
}
Notes:
[1] You may be using a different interpretation of "maximum turning radius" than I am here - perhaps yours is a diameter instead of radius, so you may need to divide by 2. (Or multiplyR(s)by 2)
[2] I do not have Curvy right now so I haven't compiled this or tested it. Sorry. :)
