I have been studying interpolation methods because I find then fascinating and they give me a greater sense of freedom for mathematically defining curves that have a given shape, but when I began studying Calculus 3, or Multivariable Calculus in the beginning of last year I gradually came to notice that there is not a very big number of examples of smooth Jordan curves ( that is smooth closed curves with no interceptions ) that I can describe mathematically to use to demonstrate the validity of Stokes theorem for example, and it doesn't help that textbook examples (or even my professor's examples) usually use circles and ellipses to demonstrate the validity of these Theorems, raising questions like what if we have some curve that is not convex ? would the Stokes theorem still work ? (of course they will !!!, but isn't it quite sad to know this and not be able to actually make the calculations for even one special case ??? ) , that is, the theorems of Multivariable Calculus express very strong and powerful results, nevertheless I always find myself questioning what is the use of having such strong results if I am only able to use these theorems in a handful of cases where the objects being studied are "simple enough" to be described (these objects being the aforementioned curves, or their parametrization, or a surface that has the given curve as a boundary, or its parametrization, etc).

I do understand that the justifications and demonstrations of the theorems of Multivariable Calculus (Green's theorem, Stokes theorem, Gauss's theorem and the Gradient theorem) do not rely on any given person being able to describe the objects that they make reference to, and that such fact is exactly why they are so important, for if in every given case one could make the calculations and come to the results of these theorems there would be no need to learn them, one would be fine by just learning how to do these calculations. So of course I know that there exists probably way more objects than those we can mathematically describe in some closed-form solution (that is, there is probably a non-enumerable number of such objects).

Notwithstanding, it would at least comfort me to be able to describe a greater multitude of such objects than those that I am currently able to mathematically describe, and to me the natural progression of this "quest" of mine is to learn interpolation methods that could be used to arbitrarily approximate any one given type of those objects, that is smooth closed curves, smooth surfaces, smooth tridimensional paths or even continuously varying vector fields.

Considering that the only one of those tha i was able to find any thing about was the first, that is smooth closed curves, when i searched for polar interpolation and found the following site: Calculus VII - Peanut allergy and Polar interpolation, that does some interesting things on some given polar interpolation problems but that doesn't in fact give a general method to solve such problems and kind of guesses some pre-processing functions until finding the final result, i would very much appreciate if there was any other methods that are more meticulous than what was used there. Other site where i found something relevant was in A Primer on Bézier Curves, more specifically on chapter 32 - Forming poly-Bézier curves.

If anyone has any recommendations on books, articles, sites or other materials that might help me in my quest to learn how to increase the number of smooth closed curves, smooth surfaces, smooth tridimensional paths or continuously varying vector fields that i can represent mathematically in some closed-form, be it by interpolation methods or some other ways, I politely ask that you share it with me so I can satiate a little of my desire to learn new mathematical modeling techniques, if I may call them so.


Sounds like what you need is some examples of curves and surfaces that are a bit more interesting than circles and ellipses.

One approach is the one you mention -- you could use Bezier curves, splines, or some other interpolation/approximation scheme, and this would allow you to construct examples that are as complex as you like.

But there is a very rich assortment of curves and surfaces with fairly simple equations that you can construct without interpolation/approximation techniques.

For curves, you could start with "A Book of Curves" by Lockwood. It describes dozens of interesting curves.

For other curves and surfaces, try looking in textbooks on classical differential geometry, such as those by DoCarmo, Alfred Gray, Struik, Darboux, and others. If you search for "differential geometry of curves and surfaces", you'll find lots of stuff.

Finally, there is this gallery of curves, and this gallery of surfaces.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.