The integral considered in Chapters 2 and 3 is
where is a flat triangle
in the parametric
space with an area of
.
The most convenient way to evaluate the integral
in the parametric
space is to transform coordinates to a local
system of area
coordinates (Chapter 8 [49]) within triangle
. The
triangle
is divided into three regions of area
,
, and
which are constrained to satisfy
shown in Figure A.1.
We define the nomalized area coordinates as
which, because of the area constraint, must satisfy
Only two of the normalized area coordinates can be considered as independent variables. Each
point within triangle
can be represented in terms of
,
, and
and its vertices as
where ,
,
, and
are points in
vector
form in the
space. They are denoted by
,
,
, and
as shown in Figure
A.1.
Substituting these into (
) and using (
) to eliminate
, we obtain
Differentiating in (
) with respect to
and
, respectively, gives
where and
are the unit vectors of
and
, respectively, shown in Figure A.2.
The Jacobian for the local area coordinates is given by
so that, it can easily be shown that the surface integral over in
(
)
transforms as follows:
where and
, given by
(
), are linear functions of
and
.