Interpolation requires that the fitting curve passes all the data points. Fitting with parametric curves begins with assigning parameter values to the data points. One approach to assigning values is known as the chord length parametrization method, described here:
- Given a set of discrete data points
, let
where
is the chord length between two consecutive points. - The overall chord length d is
- The parametric value associated with point
is
which is normalized such that
with 
and 
corresponding to 
and 
.
There are several ways to solve this problem. You may need to solve a linear system. You may use functions from Numerical Recipes in C, such as Gaussian elimination with pivoting (available on Athena) to solve this problem.
- athena% add recipes
- athena% cd /mit/recipes/src/recipes_c-ansi/recipes
The source code of the linear solver is ``gaussj.c''. Reference [35] in the course notes explain the detail.
Hand in the screen output of your program and the plots of three representations of the fitting curve. Your program for chord length parametrization should be able to handle any value n.
and 
, where 
and
- Convert these curves to Bézier form. You may choose to either use matrix
operations or the end conditions (position and
derivative) to determine the control polygon of the curves.
- Plot the control polygons of the two curves in OpenGL with the profiles.
- Plot the convex hulls of the two curves in OpenGL with the profiles.
Hand in your plots and submit your code electronically.
