Apparatus For Removing Hidden Lines from Bezier Surfaces

Jack Dikian

November 2000

This paper describes the work carried out by the author in implementing an apparatus for removing hidden lines from Bezier surfaces on the Tektronix storage tube technology of the early 80’s.

Bézier surface

A Bézier surface is formed as the cartesian product of the blending functions of two orthogonal Bézier curves. Bézier surfaces, first described in the early 60’s by the French engineer Pierre Bézier and used in automobile body design.

Hidden lines

When rendering a three dimensional surface on a two dimensional plane such as a computer screen, lines which should otherwise not seen by the viewer must be removed. The shape of the surface, if opaque, should not be cluttered by overlapping lines. Importantly, our real world experience does not allow for us to “see” through what is a solid surface or object.

In order to remove these lines, hidden line algorithms are applied in the surface rendering software to create a wire-frame which contains only visible lines and hides the lines covered by the surface.

There are a number of algorithms used to remove hidden lines. Arthur Appel’s work at IBM in the late 1960’s for example works by propagating the visibility from a segment with a known visibility to a segment whose visibility is yet to be determined. By a comparison of the two following images, the line removal algorithm can be seen at work as the wireframe representation of the surface shaded object removes the lines which are not in view.

Whilst much of the initial work in hidden line removal was done by Arthur Appel, the field is still growing as there are exceptions when his algorithm is not effective. There is a variety of other algorithms which are implemented in computer-assisted design such as the object-precision algorithms of Weiss and Galimberti/Montenari and the image-precision algorithms Encarnacao (priority-edge intersection test and scan grid – point/surface test), Warnock, and Watkins.

The Approach