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Mapping Constraints
Surfaces with more that four boundary edges may be mapped. Mapping first finds four
sequences of boundary edges forming four logical.5 The constraint formulated for each mapping
surface is that the sum of the intervals on opposing sets of edges must be equal.1112
Submapping Constraints
To facilitate the constraint formulation for submappable surfaces, the surface is placed in an
abstract i-j coordinate system. Starting at an arbitrary vertex on the surface and proceeding around
the surface in a counter-clockwise direction, each edge is classified as being parallel to the [i] or [j]
axis, and directed in the positive or negative direction, i.e., [+i], [-i], [+j], or [-j].513
The constraints for the submappable and mappable surfaces are similar. That is, the sum of
the intervals on the [+i] edges must be equal to the sum of the intervals on the [-ij edges, and
similarly for the [+j] and [-j] edges, or
E ti= ti (3)
E I[+j] = E I[ j] (4)
There are additional constraints which enforce that the surface does not self-overlap in the i-j space.
These constraints are added to the linear program dynamically, as they are violated. These
constraints are similar to the sweep volume constraints.
Other Constraints
Other meshing schemes can also supply constraints to the edges on a surface. For example,
paving,14 an advancing-front meshing algorithm, requires that the sum of the intervals on all the
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BENZLEY, STEVEN; MITCHELL, SCOTT A. & SHEPHERD, JASON F. Interval Assignment for Volumes with Holes, article, September 7, 1999; Albuquerque, New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc627482/m1/9/: accessed August 15, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.