Companion

The clamped spline on uniform knots: the first and last knot have multiplicity p + 1, so the curve starts at the first control point and ends at the last, in the directions of the first and last edge of the control polygon.

The closed spline on uniform knots: the first p control points are repeated at the end, so that the curve returns to where it started with continuous derivatives up to order p - 1. It doesn't pass through the control points.