Game Engine Architecture

562 11. Animation Systems

bx

LERP

Bottom Left

Bottom Right

LERP

Top Left

Top Right

LERP Output Pose

by

Figure 11.49. A simple 2D LERP blend, implemented as cascaded binary blends.

11.10.3.3. Simple Two-Dimensional LERP Blend

In Section 11.6.3.2, we saw how a two-dimensional LERP blend can be realized

by simply cascading the results of two binary LERP blends. Given a desired

two-dimensional blend point b = [ bx by ], Figure 11.49 shows how this kind of

blend can be represented in tree form.

11.10.3.4. Triangular LERP Blend

Section 11.6.3.3 introduced us to triangular LERP blending, using the barycen-

tric coordinates α, β, and γ = (1 – α – β) as the blend weights. To represent this

kind of blend in tree form, we need a ternary (three-input) expression tree

node, as shown in Figure 11.50.

Triangular

LERP Output Pose

Clip A

Clip B

Clip C (γ = 1 − α–β)

αβ

Figure 11.50. A triangular 2D LERP blend, represented as a ternary expression tree.

11.10.3.5. Generalized Triangular LERP Blend

In Section 11.6.3.4, we saw that a generalized two-dimensional LERP blend

can be specifi ed by placing clips at arbitrary locations on a plane. A desired

output pose is specifi ed by a point b = [ bx by ] on the plane. This kind of

blend can be represented as a tree node with an arbitrary number of inputs,

as shown in Figure 11.51.

A generalized triangular LERP blend can always be transformed into a

ternary tree by using Delaunay triangulation to identify the triangle that sur-

rounds the point b. The point is then converted into barycentric coordinates

α, β, and γ = (1 – α – β), and these coordinates are used as the blend weights