Abstract
The creation of smooth interpolating curves and surfaces is an important
aspect of computer graphics. Trigonometric interpolation in the form of the
Fourier transform has been a popular technique. For computer graphics, simpler
curves and surfaces like the B´ezier curve and B-spline curve have been
more popular due to the computational efficiency. Fitting B-spline or B´ezier
curves or surfaces to unorganised data points has been more challenging since
these curves are not naturally interpolating. Normally a system of equations
needs to be solved to obtain the curves or surfaces with the added problem of
identifying data points to form piecewise continuous surfaces. We solve the
problem of periodic interpolating curves and surfaces using harmonic interpolation
[73]. We extend harmonic interpolation to handle an even number
of data points. We then show how harmonic interpolation can be applied
using geometry images [29] to create smooth interpolating surfaces. We introduce
algorithms to manipulate the amount of interpolated points, and the
location of the interpolated points. Finally, we show how a smooth interpolating
surface created by harmonic interpolation can be converted to a series
of B´ezier surfaces. The combination of techniques allows us to quickly create
a smooth interpolating surface from a set of unorganised points that have a
known spherical structure.
Keywords: Interpolation, harmonic interpolation,
trigonometric interpolation, B´ezier curves
surface fitting, tensor product surfaces.
Prof. W.F. Steeb