Curves in 3D - 3D Math Primer for Graphics and Game Development - page 666

endpoints. So describing a cubic curve in Hermite form boils down to the
following four pieces of information:

• The start position at t = 0,

• The first derivative (initial velocity) at t = 0,

• The end position at t = 1,

• The first derivative (final velocity) at t = 1.

Let's call the desired start and end positions p 0 and p 1 and the start
and end velocities v 0 and v 1 . Figure 13.8 shows some examples of cubic
Hermite curves. Please note that the velocity vectors v 0 and v 1 have been
drawn at one-third their actual length. One reason for doing this is to save
space, and another will make sense later once we learn about Bezier curves
in Section 13.4.
Determining the monomial coe cients from the Hermite values is a rel-
atively straightforward algebraic process of combining equations previously
discussed in this chapter. The four Hermite values can be translated into

Figure 13.8. Some cubic Hermite curves