Getting pebble spline points after a certain distance?

Question

If I have a Catmull-Rom spline of a certain length, how can I calculate its position at a certain distance? As a rule, to calculate the point in the splice with cat rum, you enter a value from 0 to 1 to get your position through the proportions, how can I do this for distances? For example, if my spline is 30 units long, how can I get its position at a distance of 8?

The reason I'm asking for is because it seems that using catmull rom splines giving points in the [0,1] domain does not guarantee that it will give you a point at that distance in the spline, for example, if I enter 0.5 in a catmull romspline of length 30, this does not mean that I will get a position at a distance of 15 splines if the spline itself is not essentially straight.

Answer 1

The usual way is to save the length of each segment, and then to determine the partial length of the segment that you increase t by epsilon and calculate the linear distance between the two points until you click on your answer. Obviously, the smaller your epsilon, the better the result you get, but it gives surprisingly good results. I used this method to move at a constant speed along camouflage, and you cannot see how it accelerates and decelerates ... it moves at a constant speed. Obviously, depending on how much your segments will have an epsilon value, you will need to change it, but overall you can choose a "good enough" epsilon and everything will be fine.

Finding an answer is not iteratively INCREDIBLY expensive (I saw the output of a back, and it wasn’t very;)). You will need to have a tiny epsilon value to degrade performance ...

Answer 2

Another link: Adaptive subdivision of Bezier curves in the Anti-Grain Geometry library
mainly due to the different problem of drawing Bezier curves on a pixel grid with a wide brush, but see the very end.
(Added :) Antigrain also has great examples of /bspline.cpp where you can move nodes and change the number of intermediate points.

Answer 3

Goz's answer is correct - here is a related discussion of the length of Bezier curves . The summary of the posters was that fewer calculations (and much easier) to make an approximation than to calculate the exact answer. This is applicable because you can change the base of parametric splines so that you can convert the Catmull-Rom curve to Bezier segments.

For approximation, you basically break it down into primitives with a simple analytic length, and then add up all the simple lengths. While most people use line segments, you tend to shrink. You can minimize the error with small segments, but your approximation will always be less than the true length for non-linear curves.

If you need better accuracy, a jgt paper discusses how to use circles as approximation primitives, which seems to be faster / bigger but not much harder to implement. These include an implementation of Example C.