Curves in 3D - 3D Math Primer for Graphics and Game Development - page 695

Remember that a Hermite curve segment is defined by its starting and
ending positions and velocities. When we were focused on a single segment,
we denoted the positions by p 0 and p 1 , and the velocities by v 0 and v 1 . In
the context of a spline, we use a notation organized around a knot rather
than a segment. For positions, we don't use the p s because, as we've said
earlier, the knot k i , which is the starting position of the segment q i (0),
also serves as the ending position of the previous segment at q i−1 (1). For
velocities, the notation v ou i refers to the outgoing velocity at knot i and
defines the starting velocity for the segment q i . Likewise, the incoming
velocity from the left side of k i is denoted v i i and defines the ending velocity
of the previous segment q i−1 . We also refer to these velocity vectors as
tangents.
Figure 13.19 shows a spline with five Hermite segments. All of the
knots, segments, and tangents are labeled according to the notation just
described.
Be warned that the tangents in Figure 13.19—and all the figures of
Hermite curves in this chapter—are drawn at one-third scale. O cially
we'd like to tell you that this was done so that the diagrams would be
smaller and this topic would consume less of the Earth's natural resources.
A more accurate reason is that we draw the tangents at one-third length
so the tangents will be the same as the edges of the Bezier control polygon.
Matching the Bezier control polygon has some educational benefits, but,
more importantly, it facilitates laziness on the part of the authors: the
tools we used to create the curves in the diagrams are based on Bezier
splines.
The splines in the diagrams in this topic were created in Adobe Photo-
shop by making a path and then “stroking” the path. The arrows for the
tangent vectors were drawn by putting one end at a knot and the other end

Figure 13.19. Our notation for splines with segments in Hermite form