Curves in 3D - 3D Math Primer for Graphics and Game Development - page 650

The matrix C must be as “tall” as the number of dimensions the data
have; for example, three if we have 3D data. However, we don't need to
refer to specific x, y, or z coordinates much in this chapter because most
of the ideas work the same in 3D or 2D (or 1D!). We can just leave each
coe cient c i in vector form and assume that it is a vector of the appropriate
dimension, so that each c i corresponds to a single column of C :

2
4 1

3
5

2
4 | | | |

3 2

4 | | | |

3

Coe cients as column
vectors

t
t 2
t 3

5 5

C = c 0 c 1 c 2 c 3

|

, p (t) = Ct = c 0 c 1 c 2 c 3

|

.

| | | | | |

When dealing with a higher degree polynomial, the matrix C is wider
and the power vector t is taller, since we have more coe cients and more
powers of t. This not only makes sense, it's the law: for the product Ct
to be legal according to linear algebra rules, the number of columns in C
must match the number of rows in t .

13.1.4 Two Trivial Types of Curves
Although you're reading this section because you want to learn how to draw
a curve, allow a brief digression to mention two trivial types of “curves”: a
straight line segment and a point.
We showed how to represent a line segment parametrically in Section 9.2
when we discussed rays. Consider a ray from the point p 0 to the point p 1 . If
we let d be the delta vector p 1 − p 0 , then the ray is expressed parametrically
as

p (t) = p 0 + d t. (13.3)

Parametric line segment

Observe that this is a polynomial of the type we've been considering, where
c 0 = p 0 , c 1 = d , and the other coe cients are zero. In other words, this
linear curve is a polynomial curve of degree 1.
As boring as lines are, there's an even less interesting shape that can be
represented in parametric polynomial form: the point. Lowering the degree
of the polynomial from 1 to 0 results in a so-called constant curve. In this
case, the function p (t) = c 0 always returns the same value, resulting in a
“curve” that is a single stationary point.

13.1.5 Endpoints in Monomial Form
Clearly, one of the most basic properties of a curve that we want to control
are the locations of its start and end, p (0) and p (1), respectively. Let's