Curves in 3D - 3D Math Primer for Graphics and Game Development - page 646

• Section 13.4 shows how the Bezier form specifies the curve endpoints,

plus interior control points that influence the shape of the curve but
are not interpolated.

• Section 13.5 shows how to subdivide a curve into smaller pieces.

The second half of the chapter covers splines, which are longer curves
created by joining together multiple curves in succession.

• Section 13.6 introduces some basic notation, terminology, and con-

cepts.

• Section 13.7 discusses how to join together Hermite or Bezier curves

into a spline.

• Section 13.8 considers continuity (smoothness) conditions for splines.

• Section 13.9 ends the discussion on splines by considering various

methods for automatically determining the tangents of a spline at
the control points.

13.1 Parametric Polynomial Curves

We focus here almost exclusively on one particular type of curve, the para-
metric polynomial curve. It's important to understand what the two adjec-
tives parametric and polynomial mean, so Section 13.1.1 and Section 13.1.2.
discuss them in detail. Section 13.1.3 reviews some useful alternate nota-
tion. Section 13.1.4 examines the straight line, which is a particularly
instructive example of a parametric polynomial curve. Section 13.1.5 con-
siders the relationship between the endpoints of the curve and polynomial
coe cients. Section 13.1.6 discusses derivatives, such as velocity and ac-
celeration, and shows how they are related to tangent vectors and local
curvature.

13.1.1 Parametric Curves

The word parametric in the phrase “parametric polynomial curve” means
(not altogether surprisingly) that the curve can be described by a function
of an independent parameter, which is often assigned the symbol t. This
curve function is of the form p (t), taking a scalar input (the parameter t)
and returning the point on the curve corresponding to that parameter value
as a vector output. The function p (t) traces out the shape of the curve as