CG Modeling 2: NURBS - An Introduction to Computer Graphics for Artists - page 247

Fig. 14.6 The number of control vertices per span −1 is equal to the curve degree

Fig. 14.7 Sections of a NURBS curve are built from tangent arcs

A higher degree curve, or higher-level curve, gives its user greater control over
its shape, but is primarily used to achieve either tangency or curvature continuity .

14.2.1.3 Tangency

The simple defi nition of a tangent is a line that touches but does not intersect a curve
or surface (Fig. 14.7 ). In CG, it is a little different. In CG, a tangent can intersect the
curve it touches, but not at its origin, and only if the curve is not a conic section , or
circle. Also, in CG, a tangent line begins at the point it fi rst touches a curve, and then
extends in the same direction as the curve a certain length. This is unlike a tangent
in geometry that is a line extending indefi nitely in both directions at the point of
contact. The length of the tangent helps defi ne the strength of the control point that
marks its origin.
Tangency in a NURBS curve is when a curve's out tangent is equal to another
curve's in tangent at a common point (Fig. 14.8 ). If you have a three-degree curve
connected to another three-degree curve at a common point, then aligning the last
two points on the fi rst curve with the fi rst two points of the second curve will cause
the two curves to be tangent, because any pair of control points that starts or ends at
an open end of a curve will defi ne the tangent at that location. By aligning them, you
ensure they are the same for both curves.