Parametrised surfaces

To see the effect of such deformations, we apply one or more to a parametrisation of a flat layer. Figure 13 shows the result from using the built-in function ParametricPlot with just the default settings, whereas Fig. 14 shows the same fold using a special adaptation called FoldPlot. This alters various option settings such as adjusting the view-point slightly, changing the ratios of the sides of the box, altering the lighting effects, putting labels on axes, and suppressing the drawing of gridlines on the surface, but putting some onto the sides of the box. The code for FoldPlot, and other functions, is given in the appendix.

ParametricPlot3D[Evaluate[{x,y,0}//axiZY[]],{x,-1,1},{y,-1,1}];
FoldPlot[{x,y,0}//axiZY[],{x,-1,1},{y,-1,1},PlotPoints->{5,45}];

Figure 13: Basic folding, using the default style for a surface plot.

Figure 14: Folding as in Fig. 13, using specially adapted options and styles.

Side remark:The PlotPoints option specifies how many samples to take in each parameter; higher numbers allow a smoother representation of a curving surface. Colour is achieved by simulating the effect of lighting with different hues shining from different directions. Thus, the actual color at a point on the surface is an indication of the orientation and slope of the surface at that point. This allows the 3-dimensional shape to be more easily visualised. Furthermore, the underside is made less reflective than the upper side.

Side remark:Note the use of a ``postfix'' notation, $(x,y,z)//f$, for evaluating a function $f$ at its arguments. This allows a chain of functions to be applied successively $(x,y,z)//f_1^{}//f_2^{}//\,\dots\,$, reading left-to-right.