Copyright (c) Susan Laflin. August 1999.

Recommended Text Books.

I.D.Faux and M.J.Pratt."Computational Geometry for Design and Manufacture. "Ellis Horwood. 1979. ISBN 0-470-27069-1. section 7.2.

J.D.Foley, A. VanDam, S.K.Feiner, J.F.Hughes and R.L.Phillips. "Introduction to Computer Graphics." Addison Wesley 1994 ISBN 0-201-60921-5. section 9.3

D.F.Rogers and J.A.Adams. "Mathematical Elements for Computer Graphics." 2nd edition. McGraw Hill 1989. ISBN 0-07-100289-8. sections 6.11 to 6.14

J.Rooney and Ph.Steadman. "Principles of Computer-aided Design." Pitman. 1987. ISBN 0-273-02672-0. section 8.2

Other References.

"Piecewise Quadratic Approximations on Triangles". Powell and Sabin. ACM Transactions on Mathematical Software, Vol 3 No 4 Dec.1977 pages 316-325.

"Computing Dirichlet Tessellations in the Plane". Green and Sibson. Computer J. Vol 21 No 2 May 1978 pages 168-173.

"Algorithm for Surface Fitting with Spline Functions". P.Dierckx. IMA J. Num.Anal. Vol 1 No 3 July 1981 pages 267-283.

Surface Representation

To record a surface fully in three dimensions would require [x,y,z] coordinates at every point on the surface. This is obviously impractical in any real situation. The usual compromise is to extend a rectangular grid over the area of interest and record the heights of the surface at the crossing points of the grid or mesh. I shall call this "regular data" and refer to a surface described in this way as a "regular surface". Mathematicians might object to the latter term since it is a property of the data not of the surface, but it is convenient for our purposes so I shall use it in this sense in this document. This data format gives an array of "spot-heights" which can be interpolated in various ways. The detailed discussion of this method may be found in the section on "regular surfaces"

It is possible to choose an equation to represent the whole surface and use some approximation method such as least squares (beware of ill-conditioning) or minimax to calculate the values of the various parameters which provide the best fit to the surface. This works well in the few cases when we can predict the form of equation which will represent the surface.

In most cases, this is not possible and so it is much more usual to cover the surface with a mosaic of "patches" or "tiles", each of which is fitted to a square of the mesh. There are several forms of equation for such tiles, the simplest of which is the "Bi-Linear Surface Patch". Others which are widely used for various applications are Lofted and Coons' Patches or the Bi-Cubic Surface Patches.

If we are trying to design a surface rather than merely represent it, it will probably be most convenient to use a control net. The type of surfaces defined using this approach are Bezier, B-Spline or Beta-spline Surfaces

When the data is not recorded over a rectangular mesh, but is instead supplied as (x,y,z) coordinates dotted across the surface, it is referred to as "irregular data" or "an irregular surface" and the methods in this case are descussed fully in the section on irregular data.