I've been reading up on curves, polynomials, splines, knots, etc., and I could definitely use some help. (I'm writing open source code, if that makes a difference.)
Given two end points and any number of control points (including e.g. $0$ and $100$), I need to calculate many different points for curve. This curve must pass through all points, including the end points.
I'm not sure if this means that there is even a difference between the end points and the control points or not; I guess the difference would be that the end points don't have any points on the "outside", and thus they are different in that regard.
I have tried and succeeded with the "De Casteljau's algorithm" method, but the curve it generates doesn't (necessarily) pass through the control points (unless on a straight line or something).
I have also looked into solving for the curve's equation using a generic polynomial curve equation, e.g.:
$y = a + b x + c x ^ 2 + \dots + j x ^ 9$
plugging points into it, and then solving systematic equations. The problem with this approach is that it solves for a function, so then the curve wouldn't be able to go "backwards" any, right (unless it's a "multivalued" function maybe)?
Based on browsing through Wikipedia, I think what I might want is to calculate a spline curve, but even though I know some Calculus I'm having trouble understanding the math behind it.
I asked this question in the Mathematics section because I'm expecting a mathematical answer, but if the solution is easier to explain with pseudocode or something then I'll take that too. :)
Update: I'm looking to curve fit using Spline (low-degree) polynomial interpolation given some points. Order matters (as marty cohen explained it), and I want each polynomial to be continuous in position, tangent, and curvature. I also want minimalized wiggles and to avoid high degree polynomials if possible. :D