# Computational Applied Mathematics

### Contouring algorithms with terrain mapping applications

#### Summary:

When studying geological models below the Earth's surface, or indeed the surface itself, we often wish to concentrate on particular values on the terrain. This is often visualised by means of contour lines. One of the challenges of producing contour lines is the estimation of values between data points - especially if data points are sparse. Another challenge is to produce smooth, 'sensible' contours - contours which do not cross or have sharp corners or loops.

In this thesis we investigate some of the interpolation methods for contouring, highlighting their advantages and limitations, and compare the outputs produced. We propose that our new contouring algorithm described in this thesis will run faster than existing contouring methods, whilst also being able to contour difficult areas such as at discontinuities.

We present the main algorithms used in the new contouring program, Amlin, which utilises the same data structures as TetSim [44] andf builds on them to produce the contour outputs shown in this thesis. This leads to a new method for contouring, which uses an interpolating subdivisions scheme based on the Butterfly scheme, which generates C^1-continuous surfaces from arbitrary meshes.

The modified butterfly scheme used in the Amlin program expands beyond the original domain, whilst still respecting the original nodes, including those at the boundary. This can be seen as either an advantage, or a disadvantage, depending on the results required. The expansion beyond the boundary enables us to estimate the nature of the domain, and hence the contour, beyond the boundary. This can be particularly advantageous if the domain contains missing data and holes, as the expansion into these holes makes it more straightforward to estimate the values of the missing data. The problem with this expansion is that if we wished to retain the original domain we would need to perform some trimming or even retirangulation near the boundary.

We present a new method of butterfly subdivision which is constrained to the original boundary. This is especia;lly important when the domain forms part of a larger data set, where expanding beyond the boundary would cause an overlap of data.

We discuss the extent to which the hypothesis has been proved within this thesis, and for the methods implemented, results and outputs are presented, along with comparisons, suggestions for improvement and further work.

#### Published in:

University of Wales, Bangor, PhD thesis (2010)