How do I calculate the function of a curve with the known arc length and ending point coordinates?

I'm trying to plot a curve, and I know the length of the curve, the angle of the curve (pi-2*pi), and the ending point coordinates.

My thought is to use the Arc Length formula in reverse, since the curve will always be an arc, and derive each side. Here is a link to an example finding the Arc Length.

https://www.mathworks.com/help/matlab/math/integration-to-find-arc-length.html

But I'm not sure how to handle the Arc Length (L) derivation.

E.g., my arc length is 6, going from pi to 2*pi, and points are [4,10] and [0,16]; If I say L=6 and derive both sides to cancel the integral, then I end up getting imaginary numbers, and I know something went wrong. Since L is really an equation (e.g., cos(x)+3 from pi to 2*pi). I think I may have 2 equations, but I'm generally confused on how to move forward, or if it's even possible.

Can anyone tell me if this is solvable, and if so, how?

Thanks a lot for your time!

[Rik]

Can't you use the formula for a circle and use trigonometry to find the angles? (x=r*cos(theta), y=r*sin(theta)) Or do you still need some sort of offset? Can you add an image as an example (even if you make it with MS paint)?

[Fred]

I tried what you are saying, and I don't think Matlab can handle the calculations. Because the center of the theoretical circle is so far away or so large relative to the plot dimensions, Matlab cannot consider enough significant digits to calculate the differences.

I'm trying to write a script that can plot the curve below from 18 to 23. I know the length of the curve, and the coordinates of each point.

Is there a way to use the curvature to plot the line between the two points?

[David Goodmanson]

Hi Fred, could you provide some more information on exactly how this problem is stated, and how pi to 2*pi gets into it? If your end points are [4,10] and [0,16], the straight-line distance between them is 7.2. An arc length of 6 is not possible, but I probably don't understand the problem.

[John D'Errico]

I need some additional information. From your comments, you say that you know "the angle of the curve". What does this mean?

If the question is simply to find a circular arc that passes between the two points drawn,, with a known arc length, then the computation does not require going through the arc length integral at all.

I'll guess you are saying (from the picture you drew) that IF you took lines that are normal to the curve at each endpoint, computed the angles of those normal lines, and subtracted those angles, that would yield a known value? Or you could do the same thing using tangent lines along the curve, subtracting the associated angles there.

Is this what you mean?

If so, then a circular arc is too constraining. Any other arc that you draw will result in an arc length computation that is no longer computable analytically.

So in order to provide an intelligent answer, I need to fully understand your question. What do you mean by "angle of the curve"? I would also need to understand any other implicit requirements, such as must the curve be monotonic in some way? On the picture that you drew, there are a lot of numbers and extra lines that do not seem to have meaning. So you need to be more precise to get a good answer.

I'll look back in on this in a few hours today.

Answer 1

[John D'Errico]

This is not really a MATLAB question, since you are trying to solve a problem that is poorly posed, with insufficient information to give an answer.

Sorry but you cannot "reverse" the arclength formula as you wish to generate an analytical solution, nor even easily find an approximation solution.

There is not sufficient information to recover the equation of the curve, just from that integral, and the endpoint information. This will be a variation of what is called an integral equation. At best, you would probably need to build the curve as a spline, then optimize the spline parameters, such that the end slopes are given, and the arclength meets your goal. Since the length of the curve is only one piece of information, coupled with the end conditions, it just won't be sufficient information to define a spline.

Integral equations of this sort are rarely easy to solve.