I'm reading Rourke and Sanderson's intro to PL topology and I'm having a hard time. There's a counterexample as to why projection maps aren't necessarily PL. It says that the projection of an arc on to another has a graph that is part of a hyperbola. In the book we have established that a map between polyhedra is pl if and only if the graph is itself a polyhedron. I have tried to find explicit map formulas for such a projection but they all appear to be straight lines. Can you help me see it?
2026-04-06 18:15:55.1775499355
Projection Maps aren't always PL
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Your mistake is to consider the orthogonal projection. Instead, you should consider the central projection. For instance, the formula for the central projection with the center $(0,0)$ to the line $x+y=1$ is given by the formula $$ p(x,y)= \left(\frac{x}{x+y}, \frac{y}{x+y}\right). $$ I will leave it to you to verify that the restriction of this map to generic vertical (or horizontal) lines is not PL.