# Research Interests

Last Updated: May 15, 2013

## Past Research Interest Statements

My general area of interest is in surfaces and their use in modeling.
The following give a bit more detail about my work in various surface
topics.

## Multivariate Polynomial Data Interpolation

I have been working with Kirk Haller on using polynomials for
multivariate data interpolation. In particular, we are looking
at the polynomial Least of de Boor and Ron, and investigating why
it usually produces much better interpolants than other methods,
why it sometimes produces poor results, and we have been looking
at how to improve it.
## Approximate Continuity

I am interested in a variation of continuity that I
call *approximate continuity*. More formally, two
patches are said to join with approximate continous with
tolerance *e* if the angle between the surface normals of the
two patches at any point along the common boundary is less than *e*.
In my work the the cubic interpolant, I worked backwards to produce
approximately continuous surfaces: I would

- sample a known function,
- fit cubic interpolant patches to the data,
- test the boundaries to see if the continuity condition was met.

If not met, refine the sampling and repeat.

I verified the condition numerically by sampling the
boundary at multiple locations and seeing if the condition was met at
these sample points.
Part of my research involves developing a better test for approximate
continuity. For cubic patches, we can determine the maximum discontinuity
by finding the roots of a degree 18 polynomial. What I would like is to
find a less expensive technique that is accurate to a prescribed tolerance.

With a slightly different thrust, I am trying to develop a construction
of approximately continuous surace patches. That is, given that we
want our patches to interpolate certain data and meet with approximate
continuity, how do we build patches that meet these constraints?

## NC Machining

A variety of surface problems arise in Numerically Control (NC) Machining.
## Geometric Algebra

I have an interest in seeing how to use Geometric Algebra in splines and computer graphics.
I have co-authored a book on Geometric Algebra.
## Parametric Scattered Data Fitting

I have worked on functional and parametric scattered data fitting problems.
While still a topic of interest to me, I am less active in this area.
## Surface Pasting

Surface pasting is a hierarchical technique for adding local detail to
tensor product surfaces. Currently, I am winding down this project.
However, additional details can be found at the
surface pasting project page.