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Volume 9, Issue 3

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

Minimum Functions

The geometric definitions can be used to define minimum functions. For example, the distance from node p to node q, |pq|, is zero and minimized when and the area, defined by the three nodes p, q, r, is zero and minimized when is on the line . The R-function formulation is similarly dependent on minimum functions [7].

Define a function of the variable point that is strictly zero along the line qr:

The equation applies in any dimension.

Figure 1. Contours that are minimum at a point, along a line, and along a line segment.

Linearity

The perimeter function can be modified so that the contour lines are linear along a broken line segment. Four nodal points are required: say, , , , . Construct a function that is zero along the path and linear along the paths and , with angles and . Again define the variable point :

The functions , , and are defined in terms of and :

Here we combine all the pieces:

Figure 2. Contours of a function that are minimized along a line segment (in blue) and linear along two adjacent broken line segments (in red).



     
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