Journal of Prime Research in Mathematics

# Quality Surface Construction

Cripps R. J
University of Birmingham, Edgbaston, B15 2TT, England.

$$^{1}$$Corresponding Author: r.cripps@bham.ac.uk

### Abstract

Current surface construction methods in CADCAM use parametric polynomial equations in the form of a NURBS. This representation is ideal for computer-based implementations, allowing efficient interrogation. However, issues exist in constructing and manipulating such surfaces. When constructing a NURBS surface there are difficulties in determining constraints such as parameterisation, tangent magnitudes and twist vectors. Controlling the geometric features like curvature profiles of sectional/longitudinal curves on a NURBS surface is problematical as is joining several such surfaces together. A cause of these difficulties in control is that the control points do not lie on the surface itself. An alternative approach to surface construction is to specify the curvature and construct the surface so that it satisfies the curvature constraints. Since NURBS does not directly allow this, a fundamentally different approach is required. The key is to adopt a point-based approach where the surface is defined by a small number of points lying on the surface. Intermediate points are then constructed using a recursive approach which is defined to ensure that the curvature profile between adjacent points is of a very  high quality. A case study is presented that illustrates the point-based approach.

#### Keywords:

Surface construction, Genralised Corn Spiral, NURBS, parameterisation.