Digital Artform

Here's the texture UV-mapped in the usual way onto two NURBS patches, a torus and a plane.

The results are as you would expect.

By connecting the UV channels of the object to the surface shader red and green colors, we can visualize the natural UV coordinates of the NURBS surfaces.

Suppose instead of accepting the natural "default" UV values that come "baked" into the NURBS surfaces, we supply our own. Since the objects have been conveniently modeled small and near the world origin they happen (because I planned it this way) to sit in the cubical region of world space with corners at (0,0,0) and (1,1,1)

Instead of letting the file texture node get the usual UV values, lets pass world space X to U and world space Y to V.

We have just mathematically modelled an XY planar projection without using a Maya projection node.

If you know the math, you can make all kinds of projectors and other fun textural things for yourself.

Caveat

If you try to do this any Maya 4.5 and later, you may need to implement a workaround: if you try to REPLACE 2dplacement uv info with your own exotic math, Maya detects this as a missing placement node and ignores user-supplied uv info, going into a default behavior instead. If however you KEEP the placement node, but multiply its data by zero, you can SUM it with your own exotic math, thereby "tricking" Maya into accepting some very slick shader math right in the hypershade.

I could do some very exotic "slit-scan" effects on animated file textures if I could map a unique frame number or frame offset to every pixel, but Maya seems to allow the frame number to be updated once per frame, not once per pixel.