Michael L. V. Pitteway

Bresenham's algorithm with Grey scale

The control parameter in Bresenhams algorithm, when interpreted as a measure of distance from a straight line boundary, gives a weighted average of the intensities on each side of the line. This can be used to produce an aesthetically pleasing visual effect with modern display devices.

The Ellipse and the Five-Centred Arch

There has been a long history in the approximation of ellipses by circular arcs in order to simplify their construction and manipulation. Such approximation was of use for a wide variety of applications, in fields such as mathematics (generating figures), astronomy (analysing orbits), art (marking out large oval frames for ceiling painting), architecture (building masonry arches, floor plans, etc), and, more recently, the conversion of fonts from a general conic specification to circular arcs. Documented evidence goes as far back as the Italian Renaissance when various schemes were published by the architect Sebastiano Serlio in the sixteenth century. More contentiously, it has been argued that fifteen centuries previously the Romans used such approximations when designing and building their amphitheatres.

Drawing conics on a hexagonal grid

An algorithm is presented which can be used to outline ellipses, circles, or any of the other conic sections on a hexagonal lattice. The basic algorithm requires just one test and three add operations in the inner loop, though an additional test is required to detect a change in the overall direction between two adjacent sectants.

Hardware aspects of algorithm generation

Bezier splines can be used to define the outlines of attractive fonts for high resolution laser printers. Pratt (1985) has shown how Pitteway’s conic drawing algorithm can be constrained to work with integers only, with the control of start and end points necessary to avoid bleeding from the seedfill of the outlines. Modern display devices cannot match the resolution of the printers, but many offer a grey-scale capability which can be used to improve the appearance of the edges bewteen lit and unlit regions. The conic drawing algorithm can be adapted for this purpose, while retaining integer control.

On modelling the Bezier cubic in algebraic form

Splines based on four-point Bezier cubic arcs show some disadvantages compared to general conic arcs. Efficient rendering and data fitting algorithms require that the cubic be expressed in algebraic form, but there are problems in the conversion from the parametric form because of a singularity associated with a special case, the parabola. In conversion from the general conic to the Bezier cubic format, on the other hand, the authors note that the arcs of an ellipse can be well approximated, but not sharply turning hyperbolic arcs.