Curves in 3D - 3D Math Primer for Graphics and Game Development - page 691

′ ′ −1

(Note that the computation of b

j will “blend” the nonexistent point b

with a weight of zero.)
For Hermite curves, we usually are interested only in odd values for n,
so that we have the same number of derivatives at each endpoint.
A higher degree polynomial has the ability to describe a curve with
more “wiggles,” but, unfortunately, in general it suffers from several short-
comings:

• The curve has global support. Each control point exerts some nonzero

weight on every point along the curve, with the exception of the
endpoints.

• The curve has extraneous “wiggles” that sometimes show up in places

we don't want, oscillating back and forth between the control points.
This is known as the Runge 17 phenomenon.

• Somewhat related to the extra wiggles is the fact that higher degree

curves are very sensitive. Due to the curve's global support, a change
to any one of the control points will result in a change over the entire
curve; due to the high sensitivity, this response can be very large.

• Having ruled out polynomial interpolation as a viable curve design

tool, we cannot directly specify a point that we want the curve to
interpolate, other than the endpoints.

The basic problem is that we are asking too much from a single polynomial.
Splines do not have these shortcomings.
Here's what's in store. First, to facilitate the discussion, we must expand
our notation and introduce a level of indirection between the local and
global parameterization, which we do in Sections 13.6.1 and 13.6.2. Then,
in Section 13.7, we talk about Hermite and Bezier splines, which are used
in many software packages, such as Adobe Photoshop and Autodesk 3DS
Max. From there, our focus naturally gravitates towards deciding what to
do at the “seams.” The first hurdle is to define the criteria that must be
met so that the curve is smooth at these junction points. Such continuity
conditions are the subject of Section 13.8. Once we understand these issues,
we will have finally reached our goal set at the start of this chapter, a spline
system that provides an intuitive means to define a curved shape.
Having developed a flexible design tool where the user can specify the
position and tangent at each control point, Section 13.9 then investigates
methods by which the designer need specify only the positions of the control
points, and the tangents are computed automatically based on a set of
intuitive user controls.

17 Pronounced “RUN-guh.”