ParametricPlottingTools Package

DirectedParametricPlot[{x[t],y[t]}, {t,tmin,tmax}] is essentially ParametricPlot, but it produces a color curve which is green at the beginning (green for go - when t is near tmin) and red at the end (red for stop - when t is near tmax). DirectedParametricPlot accepts the same options as ParametricPlot with the single exception that PlotStyle settings will have no effect. Specific options for DirectedParametricPlot include ColorSegments (set to 8 by default) which controls the number of different shades along the curve from start to finish; and HueRange (set to {.3,1} by default) which controls the range of Hues along the curve.

ParametricPlotMovie[{x[t],y[t]},{t,tmin,tmax}] uses the same syntax as the built-in command ParametricPlot, but produces a movie instead of a static plot (double click any frame to animate). The movie illustrates how the curve is traced out in the plane as t moves linearly from tmin to tmax. The option Frames controls the number of frames in the movie. It is set to 20 by default, but may be set to any natural number. The option setting LeftFrame->False will supress the left image on each frame. The option setting ShowVector->True will replace the point on each right image with a vector eminating from the origin. The option PointStyle may be set to any graphics directive (or list thereof); it controls the appearance of the moving point. ParametricPlotMovie will also accept any options accepted by ParametricPlot (such as AspectRatio->Automatic).

HomotopyMovie[{x[s,t],y[s,t]},{s,t}] generates a movie illustrating the homotopy {x[s,t],y[s,t]} mapping the unit square [0,1]^2 into the plane. i.e. the homotopy is between {x[0,t],y[0,t]} and {x[1,t],y[1,t]}. The option Frames controls the number of frames in the movie. It is set to 15 by default, but may be set to any natural number. HomotopyMovie will also accept any options accepted by ParametricPlot; it is a good idea to use the option setting PlotRange->{{xmin,xmax},{ymin,ymax}} so that the image on the right will cover the same portion of the plane in each frame. You may use the syntax HomotopyMovie[{x[s,t],y[s,t],{s},{t,tmin,tmax}] to make a movie of a homotopy between functions defined on the interval [tmin,tmax].