There are two main definitions of "cone":
1. (geometry) A surface of revolution formed by rotating a segment of a line around another line that intersects the first line.
2. (geometry) A solid of revolution formed by rotating a triangle around one of its altitudes.
I think neither meaning of "cone" reflects {konju}. I think {konju} describes a *segment* of a line (possibly infinite thus identical to a line in such case) revolving around another linear segment thus forming a *surface*. As for konju1 it describes an object (possibly immaterial) with *a property* of such cone-ness.
This explains why only vertex is specified. but konju2 seems to be superflous then since it is not always necessary.