I think your question is not complete, are you projecting the surface S on the XY plane? are you projecting something on the surface S? projection is meaningless unless you say what on what...

Anyway,

The X+Y+Z=1 in the first octant looks like the face of a pyramid,

when projected on the XY plane it will look like a filled triangle.

The R looks like a quarter of a ball, so they aren't the same.

To solve the (r,t) surface of the triangle is not so simple, so I won't do it unless you say this is what you want.

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