There is no need to cast rays when you have divided the space into convex polygons. If you know all the vertices of a convex polygon and two of them form a visible wall, you can determine the texture column and height of the screen space wall column by means of interpolation. This is what the Doom engine does.

The PC version of Wolfenstein 3D used a more straight forward ray casting approach, but I think the SNES port was Id's first use of BSP.