This page serves as storage unit for old ideas and projects that I've either retired or never finished. Some of these were formulated as early as high school, but I leave them up to remind me of my roots.
Given some space curve f(x,y), we loosely define a 'ribbon curve' as any of the curves that result from the tangent, normal, and binormal vectors for every (x,y,f(x,y)). It may be interesting to see how these ribbon curves and the space curve itself all interact. It may also be useful to represent this ribbon 'system' (dare I say) in a physical sense and see how the system and its curve components are affected by certain fields. I don't know man, I just think it's cool and I recognize the world of ideas that are birthed from it.
UPDATE: I have successfully visualized this project in Mathematica. Check my LinkedIn for more information.
In a typical surface z=f(x,y), the gradient vector (notated grad f) is a vector that lives in the domain (xy plane) of the surface and points in the direction of maximum positive change in z at some point on the surface. In other words, the steepest direction from some point on a surface. The gradient vector has the partial of f with respect to x in its x component and the partial of f with respect to y in the y component.
But what if we have a parametrized surface r(u,v) = (x, y, z) where each component of r is a function of u,v? This means our domain is some defined boundary in the u,v plane. What does "grad r" even mean, and how does one compute it? (I believe it would be some vector in the domain with the components (grad x, grad y, grad z))
When we apply the gradient operation to a parametrized surface of order 2 (such as r(u, v) = (x(u, v), y(u, v), z(u, v))) we get the Jacobian matrix. The question is still the same; what does this matrix mean? Can we represent it visually? I suspect it can be represented as some vector in the domain which points in the direction of maximum change on our surface.
The question isn't quite answered yet, but here's another to see where we're going with this! Suppose we have a parametrized surface h(q, r, s) = (x, y, z). What does a gradient mean in this situation, and how is it represented mathematically? What about when the parametrization is of an order higher than the dimensionality of the surface, such as g(k, l, m, n) = (x, y, z)?
We have a system of four things which produce a fifth thing. Imagine, for a moment, that we have the following:
Some kind of space to operate in
A subset of that space, for example a surface
Another subset of that space, for example a solid
Some kind of light source
These are the four components of a shadow, the projection or mapping of a region onto another with respect to the properties of the light source. Each component itself has several properties and contributes to the system in various ways. With this idea established, we can begin to ask a million questions.
Can we write a generalized equation for this shadow?
This equation would have to account for all components and properties.
What are the properties of a space?
What are the properties of a surface?
What are the properties of a solid?
What are the properties of a "light source?"
Then, of course, we can continue to propose ideas such as "what if there are two lights?" or "what if there is a black hole between the light and the solid, or between the solid and the surface?" Shadow Math is an interesting topic for sure.