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A Normalized Local Area Coordinate System

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 .


xshen@
Sat Dec 3 17:51:03 EST 1994