The Mathematica Journal
Volume 9, Issue 3

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Algebraic Construction of Smooth Interpolants on Polygonal Domains
Elisabeth Anna Malsch
Gautam Dasgupta

Geometric Measures and Support Functions

It is useful to construct the interpolant with respect to geometric measures as opposed to nodal locations. The benefit of such an approach is that it is invariant of coordinate system and dimension.

The Distance Function

The distance between two points and with coordinates and is:

The function can be written as the norm of the vector connecting the points, norm[]. The built-in Norm function would be appropriate if the points were represented as complex numbers rather than as pairs of real numbers.

The Signed Area of a Triangle

The area of the triangle with vertices , , with coordinates , , can be defined in terms of the determinant:

The area is signed: .

The Unsigned Area Function

The unsigned area of a triangle can be written in terms of side lengths using Heron's formula:

where

The resulting area is necessarily nonnegative. The area measure has no physical meaning for a node located on the edge of the domain. In such a case, defining the unsigned area to be one when the signed area is zero allows the concave element shape function routines to apply to elements with side nodes.

The Angle Functions

The cosine or sine of an angle defined by three vertices can be determined using line lengths and the unsigned area. Care is taken to choose the proper sign for the cosine and sine functions. For the angle Beta defined by the nodes , , with the angle at :

The angle Beta never has to be evaluated.



     
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