Calculating Clothoid between two tangents

curvesfresnel-integralsgeometry

I am trying to connect two points with a Clothoid (Euler-Spiral) https://en.wikipedia.org/wiki/Euler_spiral . It is mandatory to connect the points with the correct start and endHeading of the tangents.

ÜE and ÜA are member of the tangents g and f

The tangents are intersecting in point E(2,0) and intersecting arc is t = 45° which is also needed for the calculation of the clothoid.

Now some basic Clothoid Formulas I have tried so far:
Declaration of Formula

Clothoids have the following parameters:

  1. the shape parameter A
  2. the radius of curvature R at the end point
  3. the arc length L
  4. the tangent angle t
  5. the tangent disengagement deltaR

In order to calculate a clothoid 2 determinants are needed. If a third is added, the clothoid is overdetermined.
In my case the tangent angle with t = 45° exists in any case and I mean the shape parameter A must be determined numerically?
At the moment I am trying to calculate A iteratively by just increasing it and calculating the
Radius with R = A / sqrt(2*t) und L = (A * A)/R. But I never get my Endpoint in that case (4,2) correctly. The Endpoint is calculated by the Solution of the Fresnel Integrals for

X = L * (1-((t * t)/10) * (1-((t * t)/21.6) * (1 – ((t * t)/43.333))));

Y = ((L * t) / 3) * ((1-((t * t)/14) * (1 – ((t * t)/31.429))));

Those formula are correct because I am able to check my solution with a graphic editor software.
Does anyone have a solution for this problem or can help me what am I doing wrong?

Best Answer

So I spent some research and for this problem does not exist a trivial solution which is easily solvable. But there is a nice library which solves my problem. Exists for MATLAB and for C++.

https://github.com/ebertolazzi/G1fitting https://de.mathworks.com/matlabcentral/fileexchange/42113-ebertolazzi-g1fitting

Additional information: https://www.researchgate.net/publication/237062806_Fast_and_accurate_G1_fitting_of_clothoid_curves

Related Question