Hi Zooen, if the control points on either side of a corner point are tangent continuous the curve will be smooth at that spot and it is allowed for a single curve segment to contain such "potential corner points". Your offset case is an example of this.

If you manipulate the control points so that a tangent discontinuity (a sharp point) is formed, that is not allowed within a single segment and the segment will be divided at that discontinuity point.

re:
> However, if, for example, you extrude the curve, you don’t see any edges on the extruded surface.

It's the segmentation of the curve that controls this. Edges on the extruded surface will occur between curve segments.

- Michael