Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 452

When a curve is constructed from a set of points and the curve passes through the points, it is said to
interpolate the points. However, if the points are used to control the general shape of the curve, with the
curve not necessarily passing through them, then the curve is said to approximate the points . Interpo-
lation is also used generally to refer to all approaches for constructing a curve from a set of points. For a
given interpolation technique, if the resulting curve is guaranteed to lie within the convex hull of the set
of points, then it is said to have the convex hull property .

B.5.2 Simple linear interpolation: geometric and algebraic forms
Simple linear interpolation is given by Equation B.64 and shown in Figure B.33 . Notice that the inter-
polants, 1

u and u , sum to one. This property ensures that the interpolating curve (in this case a
straight line) falls within the convex hull of the geometric entities being interpolated (in this simple
case the convex hull is the straight line itself).

PðÞ¼ ð 1 u ÞP 0 þ uP 1 (B.64)

Using more general notation, one can rewrite Equation 64 as in Equation B.65 . Here, F 0 and F 1 are
called blending functions. This is referred to as the geometric form because the geometric information,
in this case P 0 and P 1 , is explicit in the equation.
PðÞ¼F 0 ðÞP 0 þ F 1 ðÞP 1 (B.65)

The linear interpolation equation can also be rewritten as in Equation B.66 . This form is typical of
polynomial equations in which the terms are collected according to coefficients of the variable raised to
a power. It is more generally written as Equation B.67 . In this case there are only linear terms. This way
of expressing the equation is referred to as the algebraic form.

PðÞ¼P 1 P 0 ð Þu þ P 0 (B.66)

PðÞ¼a 1 u þ a 0 (B.67)

Alternatively, both of these forms can be put in a matrix representation . The geometric form
becomes Equation B.68 and the algebraic form becomes Equation B.69 . The geometric form is useful
in situations in which the geometric information (the points defining the curve) needs to be frequently
updated or replaced. The algebraic form is useful for repeated evaluation of a single curve for different
values of the parameter. The fully expanded form is shown in Equation B.70 . The curves discussed next
can all be written in this form. Of course, depending on the actual curve type, the U (variable), M
(coefficient), and B (geometric information) matrices will contain different values.

P (1)

P ( u )

P 1

P (0)

P 0

FIGURE B.33
Linear interpolation.