The transform command generates a procedure that can be used to change plot data structures by applying the procedure f to all the points in the plot structure.

The procedure f represents a mapping

where m and n can take values 2 or 3. The procedure must take as input m arguments and output a list of n components.

Examples of the use of this command include embedding 2-D plots into 3-D space, changing coordinate systems or transforming curves in the complex plane into the Riemann Sphere.

The result of a call to transform is a procedure that takes a 2-D or 3-D plot data structure as parameter. You can assign the procedure to a variable, and apply the procedure to various plots.  For more information about plot data structures, see plot/structure.

The transform command is usually used with [m, n] equal to [2, 2], [2, 3], or [3, 3].  It returns a transformation procedure in the case where m is 3 and n is 2.  However, this procedure works only on those 3-D plot objects that have 2-D counterparts.

Several commands in the plottools package can transform plots. For a list, see the plottools help page.  The plots[changecoords] and plots[display] commands can also be used to transform plots.

If the transform command is applied to a filled shape, such as an arrow or rectangle, the polygon that is generated might not resemble the result one would expect from a transformation of the entire shape. That is because the transform command works only on the points of the plot data structure, which are typically the vertices and not interior points.

$\mathrm{with}\left(\mathrm{plottools}\right)\:$

$\mathrm{with}\left(\mathrm{plots}\right)\:$

$p≔\mathrm{plot3d}\left(-\frac{5x}{x^2+y^2+1}\,x=-3..3\,y=-3..3\,\mathrm{style}=\mathrm{contour}\,\mathrm{contours}=8\right)\:$

$q≔\mathrm{contourplot}\left(-\frac{5x}{x^2+y^2+1}\,x=-3..3\,y=-3..3\,\mathrm{filled}=\mathrm{true}\right)\:$

$f≔\mathrm{transform}\left(\left(x\,y\right)\mapsto \left[x\,y\,\mathrm{-3}\right]\right)\:$

$\mathrm{display}\left(\left\{p\,f\left(q\right)\right\}\,\mathrm{orientation}=\left[50\,40\right]\right)$

$p≔\mathrm{plot3d}\left(\left[{1.3}^x\sin \left(y\right)\,x\,y\right]\,x=-1..2\mathrm{\pi }\,y=0..\mathrm{\pi }\right)\:$

$f≔\mathrm{transform}\left(\left(r\,\mathrm{th}\,\mathrm{ph}\right)\mapsto \left[r\cdot \sin \left(\mathrm{ph}\right)\cdot \cos \left(\mathrm{th}\right)\,r\cdot \sin \left(\mathrm{ph}\right)\cdot \sin \left(\mathrm{th}\right)\,r\cdot \cos \left(\mathrm{ph}\right)\right]\right)\:$

$\mathrm{display}\left(\left\{p\,f\left(p\right)\right\}\,\mathrm{orientation}=\left[145\,20\right]\,\mathrm{scaling}=\mathrm{constrained}\right)$