Graphics Reference
In-Depth Information
Figure 1.18.
Barycentric coordinates and volume ratios.
v 2
D 1
D 0
w
D 2
v 0
v 1
of another simplex t. Let s = v 0 v 1 ··· v k and t = w 0 w 1 ··· w s . If we express points of s
in terms of the (unique) barycentric coordinates with respect to its vertices, then f
induces a well-defined map
f:st
Æ
defined by
k
k
Ê
Á
ˆ
˜ =
ÂÂ
()
f
a
v
f
v
.
ii
i
i
i
=
0
i
=
0
Definition.
The map |f| is called the map from s to t induced by the vertex map f.
In Chapter 6 we shall see that the map f is a special case of what is called a sim-
plicial map between simplicial complexes and |f| is the induced map on their under-
lying spaces. The main point to note here is that a map f of vertices induces a map
|f| on the whole simplex. (This is very similar to the way a map of basis vectors in a
vector space induces a well-defined linear transformation of the whole vector space.)
This gives us a simple abstract way to define linear maps between simplices, although
a formula for this map in Cartesian coordinates is not that simple. See Exercises 1.7.6
and 1.7.7.
1.8
Principal Axes Theorems
The goal of this section is to state conditions under which a linear transformation can
be diagonalized. We shall be dealing with vector spaces over either the reals or the
complex numbers. We refer to the main theorems of this section as “principal axes
theorems” because they can be interpreted as asserting the existence of certain coor-
dinate systems (coordinate axes) with respect to which the transformation has a par-
ticularly simple description. Such diagonalization theorems are special cases of what
are usually called “spectral theorems” in the literature because they deal with the
eigenvalues (the “spectrum”) of the transformation.
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