Nutshell Abstract

Salient assemblage representation models recursive parametric systems.

Abstract

A multidimensional curvilinear coordinate net allows an asymptotic C^infinity salient that depends on local geometric properties, like principal directions. Metric tensor computations compensate for parameter stretching, coordinate curve nonlinearity, and obliquity. A series of orthogonal functions approximates a given salient shape. Successive salients are assembled recursively with rules on shape, size, alignment, and dihedral. These rules determine invariant attributes during assemblage deformation. The resulting patch can be folded, yet remains homeomorphic to one parameter space. That simplifies geodesic calculation. Representation algebra is complex, but allows efficient computation because parameter arguments are effective predictors of negligible terms.

Prototypical Salient Assemblage

Assemblage constructed from 3 salient units.

Requirements

• Add, remove, reposition, deform salient units.
• Asymptotic salient blending.
• Topologically invariant, homeomorphic with one parameter space.
• Local control of salient direction, shape, size, and volume, at least approximately.
• Recursive attachment rules, like alignment with principal directions.
• Multidimensional.

Characteristics

As a consequence of the above requirements, a salient assemblage has the following characteristics:

• Constructive formulation, finite skeleton substructure formed by salient semiaxes.
• Concise data storage.
• One patch, thus no patch boundary, avoid geodesic cusp.
• Parameters usually have physical significance.
• Nowhere flat.
• Complicated algebraic expressions require computer.

More Information

For more information, see slide presentation Salient Assemblage Representation of Multidimensional, Recursive, Deforming Geometry.

For an example application of a salient assemblage manifold see Optimization Using A Geodesic.