Curves in 3D - 3D Math Primer for Graphics and Game Development - page 701

Figure 13.22
Continuity conditions for
cubic B ezier splines.

In Figure 13.19 the curve is not C 1 continuous at k 4 because the tan-
gents are not equal. However, the curve is G 1 continuous at this location.
The hint, of course, is that the tangents are parallel at the knot. If the
tangents at a knot are not parallel, then there's no way to move along the
curve in a smooth way. However, if the tangents are parallel, then the dis-
continuity is purely a change in speed, not a change in direction. We could
remove this discontinuity by carefully introducing an offsetting discontinu-
ity in the time-to-parameter function s(t) that exactly “undoes” the jump
in speed.
Higher-order geometric continuity extends this idea, although it is a
bit more di cult to visualize. We say that a curve is G 2 continuous if its
curvature changes continuously.

13.8.3 How Smooth Can a Curve Be?
We end our discussion on continuity by asking an important question:
what's the highest level of continuity we can expect from a polynomial
spline? We said earlier that any particular curve segment has C

conti-
nuity, because we can differentiate it as many times as we want and the
result is always a continuous function. Can we achieve this same level of
smoothness with a spline?
Consider two adjacent cubic Bezier segments. Let's fix the first segment
and consider what happens to the second segment as we demand higher and
higher levels of continuity at the knot. When we demand C 0 continuity, we
lock in the first Bezier control point. Clearly, the first endpoint must match
the last endpoint of the first segment for the spline to be C 0 continuous.

∞