Can Singular Points Be Smoothly Transformed to Achieve Differentiability?

[SW VandeCarr]

Is there a correct mathematical procedure to remove singular points so as to create a smooth continuum, differentiable everywhere? For example,for cusp singularities, is some kind of acceptable "cutting and joining" procedure available at the limit? I asked a similar question in the topology forum some time ago but never got an answer.

If we allow topological transformations, it seems to me that (for example) an inscribed equilateral triangle could be smoothly transformed to a circle without cutting and joining such that the points at the tips of the triangle become differentiable points on the circle.

http://mathworld.wolfram.com/SingularPoint.html

EDIT: I know "at the limit" can be problematical, but I'm trying to avoid arbitrary choices.

 

[CRGreathouse]

Is there a correct mathematical procedure to remove singular points so as to create a smooth continuum, differentiable everywhere?

If all you need is (once-)differentiable* everywhere, you could choose some epsilon around every singularity and rejoin the pieces with Bézier curves. If you want infinite differentiability... nothing immediately comes to mind.

* Or indeed, if all you need is a fixed number of derivatives, just choose a sufficiently large degree.

 

[SW VandeCarr]

If all you need is (once-)differentiable* everywhere, you could choose some epsilon around every singularity and rejoin the pieces with Bézier curves. If you want infinite differentiability... nothing immediately comes to mind.

* Or indeed, if all you need is a fixed number of derivatives, just choose a sufficiently large degree.

Thanks GR. Bezier curves seem to be what I was looking for. I was trying a constructive method by defining an isosceles triangle with two points on the limbs of the cusp and the third point at the singular point. I would then bisect the apical angle with a line through the singular point and then take the perpendicular to the bisector at the singular point and call that the first estimate of the tangent. I could refine the estimate by shortening the limbs of the triangle. If I understand Bezier's method correctly, this constructive method seems to follow the same line of reasoning.

I don't really need anything specific in terms of the number of derivatives. As you suggest, I should, in principle, get any finite number of derivatives just by refining the estimate.