Math Behind Computer Graphics: Piecewise Smooth Interpolation

Abstract

Modern computers are able to create complex imagery with only a small set of information. For example, the fonts on your computer are saved as a set of points and the computer is told how to connect them. Many 3D animations start the same way, where the animation starts as a grid before the rest of the shape is systematically filled in. But how does the computer know how to connect the dots into a mesh? Or know how to create the smooth surface so that it doesn’t look blocky? To solve these problems, we implement mathematical algorithms to generate computer graphics. In this talk, we will discuss 1D and 2D interpolation techniques which tell the computer how to algorithmically connect points to follow certain criteria. We can create smooth lines which connect all our points using high order polynomials like Lagrange or Newton forms. We could also define a function between each pair of points so that the final image appears smooth. If we introduce additional points, then we can utilize Bezier curves. This is how your computer creates fonts. We can also combine methods, such as our new “quadrubic” technique which combines quadratic and cubic splines. These methods can then be adapted to create 3D surfaces.