Fitting a transition + circle + transition curve to a set of measured points

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I am dealing with a reverse-engineering problem regarding road geometry and estimation of design conditions.

Suppose you have a set of points obtained from the measurement of positions of a road. This road has straight sections as well as curve sections. Straight sections are, of course, represented by lines, and curves are represented by circles of unknown center and radius. There are, as well, transition sections, which may be clothoids / Euler spirals or any other usual track transition curve. A representation of the track may look like this: Track measured positions We know in advance that the road / track was designed taking this transition + circle + transition principle into account for every curve, yet we only have the measurement points, and the goal is to find the parameters describing every curve on the track, this is, the transition parameters as well as the circle's center and radius.

I have written some code using a nonlinear optimization algorithm, where a user can select start and end points and fit a circle that to the arc section between them, as it shows in next figure: Fitted circle in curve However, I don't find a suitable way to take the transition into account. After giving it some thought I came to think that this s because, given a set of discrete points -with their measurement error- representing a full curve, it is not entirely clear where to consider it "begins" and where it "ends" and, moreover, it is less clear where to consider the transition, the proper circle and the exit transition "begin" and "end".

Is there any work on this subject which I may have missed? is there a proper way to fit the whole transition + curve + transition structure into the set of points?

numerical-methods curve-fitting computational-geometry geometry
2 Answers

As far as I know, there's no method to fit a sequence clothoid1-circle-clothoid2 into a given set of points.

Basic facts are that two points define a straight, and three points define a unique circle.

The clothoid is far more complex, because you need: The parameter A, the final radius Rf, an initial point px,py, the radius Ri at that point, and the tangent T (angle with X-axis) at that point.
These are 5 data you may use to find the solution.

Due to clothoid coords are calculated by expanded Fresnel integrals (see https://math.stackexchange.com/a/3359006/688039 a little explanation), and then apply a translation & rotation, there's no an easy way to fit this spiral into a set of given points.

When I've had to deal with your issue, what I've done is:

  1. Calculate the radius for triplets of consecutive points: p1p2p3, p2p3p4, p3p4p5, etc
  2. Observe the sequence of radius. Similar values mean a circle, increasing/decreasing values mean a clothoid; Big values would mean a straight.
  3. For each basic element (line, circle) find the most probably characteristics (angles, vertices, radius) by hand or by some regression method. Many times the common sense is the best.
  4. For a spiral you may start with aproximated values, taken from the adjacent elements. These values may very well be the initial angle and point, and the initial and final radius. Then you need to iterate, playing with Fresnel and 'space change' until you find a "good" parameter A. Then repeat with small differences in the other values, those you took from adjacents.
  5. Make the changes you consider as good. For example, many values (A, radius) use to be integers, without decimals, just because it was easier for the designer to type.

If you can make a small applet to do these steps then it's enough. Using a typical roads software helps, but doesn't avoid you the iteration process.

user_38710280

If the points are dense compared to the effective radii of curvature, estimate the local curvature by least square fitting of a circle on a small number of points, taking into account that the curvature is most of the time zero.

You will obtain a plot with constant values and ramps that connect them. You can use an estimate of the slope at the inflection points to figure out the transition points.

user_11965490
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