CG Modeling 2: NURBS - An Introduction to Computer Graphics for Artists - page 249

14.2.3 Tangent Curves

Tangent curves are curves created based on tangency with other curves. The difference
between this type of curve and a tangent arc is that a tangent arc will always have
curvature characteristic of a circle, but a tangent curve can be calculated based
on the tangent vector of two curves at any given selection location. The new curve
will blend between these tangents in the middle, and fl ow smoothly into both source
curves on either end. Depending on their tangents, the new curve may not be a
perfect arc.

14.2.4 Periodic Curves

A periodic curve is a curve without beginning or end. This isn't literally true
because your application has to pick a part of the curve as the origin, but conceptu-
ally, that is what it is. To defi ne a periodic curve, the endpoints must cross with at
least two overlapping CVs on either side of the join, for a total of fi ve CVs. These
must all have the same tangent. If these conditions are satisfi ed, and the curve is
identifi ed to your software as “periodic,” then it will be periodic. The reason it has
to be identifi ed as periodic is that otherwise the endpoints will be free to move
away from each other. As a periodic curve, they cannot be pulled apart. Most pro-
grams have only two ways to make a curve periodic; make it that way to begin with
by using a periodic curve creation tool or invoke a command that will make an
open curve periodic.
Tip: Sometimes a tool looks like it has made an open curve periodic by closing
it, but instead it has only made the endpoints tangent. If the endpoints can be moved
independently, it is not periodic.

14.2.5 Curve on Surface

A curve on surface is a curve that has been projected onto a surface. The easiest way
to visualize this is to imagine the curve extruded straight through the surface it is
projected onto. The points of intersection are used as the basis for a new curve, and
their tangents adjusted to ensure that they lie perfectly within the target surface. By
itself, a curve on surface doesn't accomplish a great deal, but when combined with
curve duplication, a curve on surface can be used to create a shape that would have
been very diffi cult to make without the aid of a reference surface. Combined with a
trimming tool, unwanted sections of a NURBS surface may be cut away into almost
any shape you like (Fig. 14.9 ).