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Algebraic Construction of Smooth Interpolants on Polygonal Domains

# 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).