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This paper introduces a new 2D representation of the orientation distribution function for an arbitrary material texture. The approach is based on the isometric square torus mapping of the Clifford torus, which allows for points on the unit quaternion hypersphere (each corresponding to a 3D orientation) to be represented in a periodic 2D square map. The combination of three such orthogonal mappings into a single RGB (red–green–blue) image provides a compact periodic representation of any set of orientations. Square torus representations of five different orientation sampling methods are compared and analyzed in terms of the Rieszsenergies that quantify the uniformity of the samplings. The effect of crystallographic symmetry on the square torus map is analyzed in terms of the Rodrigues fundamental zones for the rotational symmetry groups. The paper concludes with example representations of important texture components in cubic and hexagonal materials. The new RGB representation provides a convenient and compact way of generating training data for the automated analysis of material textures by means of neural networks.
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How periodic surfaces bend without stretching
Many compliant shell mechanisms are periodically corrugated or creased. Being thin, their preferred deformation modes are inextensional, i.e., isometric. Here, we report on a recent characterization of the isometric deformations of periodic surfaces. In a way reminiscent of Gauss theorem, the result builds a constraint that relates the ways in which the periodic surface stretches, effectively but isometrically, to the ways in which it bends and twists. Several examples and use cases are presented.
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- Award ID(s):
- 2045881
- PAR ID:
- 10580073
- Publisher / Repository:
- International Association for Shell and Spatial Structures
- Date Published:
- Format(s):
- Medium: X
- Location:
- Zurich
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Conference Proceeding:
- The DOI is not currently available.
