Animated Bezier Surface Generator
- Malcolm Harper
Animated Bezier Surface Generator. A fascinating display of changing shapes for your entertainment. This is an animated version of a previously uploaded model for creating bezier surfaces. Set the sliders alive and enjoy. A challenge - animate the colours, change the points and upload. ------------------------------------------------------------------------------------------------- Animated Bezier Surface Generator - 4 x 6 grid - Spreadsheet entry. Note that a reasonably fast computer is required to run this model. Developed on HP i7, 8GB RAM. Hiding "Surface" during input of data may overcome slowness. By selecting appropriately from a grid of 4 rows of 6 points this model creates a system of bezier curves from which is generated a surface having a boundary that may be rectangular, triangular, waist-like or bulbous. For each of the 4 rows of the grid, from 1 to 6 points may be chosen to be included in shaping the surface. Point selection proceeds in sequence from point A1 (B1,C1, D1) through to A6 (B6, C6, D6) and is achieved by operating sliders . The value of the slider corresponds to the degree of the bezier applied to a row: viz 0 (a point), 1 (straight line), 2 (quadratic bezier) and 3 (cubic bezier), etc. Having the "degree" (or power) of each of the four lateral beziers alterable leads to the ability to create irregular shaped boundaries. The x, y, z values of each point are introduced via the Spreadsheet. Although not implemented here, the spreadsheet could be used to transform polynomial parametric functions of up to power 5 into Bezier Control Points corresponding to that function, enabling the modelling of recognised curves. Points not included in the generation of the surface are blanked out on the screen. Bezier curves have been limited to lie between their two end points. Accuracy falls away when the curve continues beyond these limits. Never- the-less Slider limits are alterable. Non-rational Beziers are at the foundation of this model although Rational Beziers could be implemented, having the advantage of being capable of describing a greater range of curve types (a circular arc is one such curve). There are four levels to implementing this model and it was found necessary to describe the surface by using only functions, functions of functions, ........... Although the shorthand mathematical definition of a Bezier Surface is given by a Sum Sum notation, this unfortunately is not understood by GeoGebra. Background. A Bezier Surface is the trace of a point running along a single "longitudinal" bezier curve whose Control Points are themselves points on "lateral" bezier curves. In this model the degree of the longitudinal bezier is always three, requiring four such lateral bezier curves. The trace-point on the longitudinal bezier has its position controlled by parameter "v" while parameter "u" applies to the 4 lateral beziers. The degree of each of the four lateral beziers is individually set by a slider () to: 0 (a point), 1 (straight line), 2 (quadratic bezier) and 3 (cubic bezier) etc. By varying sliders "u" and "v" point S (red) and its associated bezier CPs may be moved over the surface. The edge bezier connecting end points A and M is shown in red. Also shown are the two lateral beziers through A and M . Methodology By entering four rows of six points all alternative boundaries options become possible. Each row is bound to one "lateral" bezier curve. Row 0 comprises points A1, A2, A3, A4, A5, A6; Row 1= B1 ..... ; Row 2= C1 .......; Row 3= D1 ...... For Bezier 0, slider m0 sets its degree and commencing with point A1, selects Control Points from that row. Similarly for the remaining three lateral beziers. Non-relevant points and polylines are hidden. Lateral beziers, associated CPs and the polyline connecting CPs are colour coded. Animation The speed of the Sliders are based on prime numbers.