Studies the intersection of a normal form for the conic umbilic catastrophe of general codimension K with planes in control space having all but two control variables set to zero, and display in a set of figures and a table the geometry in which strata of the bifurcation set intersect these planes, and the singularity types occurring on them. Five distinct curve forms arise. The authors compare these sections with those of the cuspoids and discuss their occurrence as optical caustics.