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Shape Statistics: Procrustes Superimpositions and Tangent Spaces

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  • F. James Rohlf

Abstract

The shape of a set of labeled points corresponds to those attributes of the configuration that are invariant to the effects of translation, rotation, and scale. Procrustes distance may be used to compare different shapes and also serve as a metric that may be used to define multidimensional shape spaces. This paper demonstrates that the preshape space of planar triangles Procrustes aligned to a reference triangle corresponds to a unit hemisphere. An overview of methods used as linear approximations of D. G. Kendall's non-Euclidean shape space is given, and the equivalence of several methods based on orthogonal projections is shown. Some problems with approximations based on stereo graphic projections are also discussed. A simple example using artificial data is included. Copyright Springer-Verlag New York Inc. 1999

Suggested Citation

  • F. James Rohlf, 1999. "Shape Statistics: Procrustes Superimpositions and Tangent Spaces," Journal of Classification, Springer;The Classification Society, vol. 16(2), pages 197-223, July.
    • Handle: RePEc:spr:jclass:v:16:y:1999:i:2:p:197-223
    DOI: 10.1007/s003579900054
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    Citations

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    Cited by:

    1. Brandon P Hedrick & Peter Dodson, 2013. "Lujiatun Psittacosaurids: Understanding Individual and Taphonomic Variation Using 3D Geometric Morphometrics," PLOS ONE, Public Library of Science, vol. 8(8), pages 1-13, August.
    2. Robert G Wallace, 2002. "The Shape of Space: Applying Geometric Morphometrics to Geographic Data," Environment and Planning A, , vol. 34(1), pages 119-144, January.
    3. Sean T. Osis & Blayne A. Hettinga & Shari L. Macdonald & Reed Ferber, 2015. "A novel method to evaluate error in anatomical marker placement using a modified generalized Procrustes analysis," Computer Methods in Biomechanics and Biomedical Engineering, Taylor & Francis Journals, vol. 18(10), pages 1108-1116, July.
    4. Yousuke Kaifu & Iwan Kurniawan & Soichiro Mizushima & Junmei Sawada & Michael Lague & Ruly Setiawan & Indra Sutisna & Unggul P. Wibowo & Gen Suwa & Reiko T. Kono & Tomohiko Sasaki & Adam Brumm & Gerri, 2024. "Early evolution of small body size in Homo floresiensis," Nature Communications, Nature, vol. 15(1), pages 1-13, December.
    5. Irene Epifanio & María Victoria Ibáñez & Amelia Simó, 2018. "Archetypal shapes based on landmarks and extension to handle missing data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 12(3), pages 705-735, September.
    6. Katherine L Bell & Christopher A Hamm & Arthur M Shapiro & Chris C Nice, 2017. "Sympatric, temporally isolated populations of the pine white butterfly Neophasia menapia, are morphologically and genetically differentiated," PLOS ONE, Public Library of Science, vol. 12(5), pages 1-17, May.
    7. Manuel Dehon & Denis Michez & André Nel & Michael S Engel & Thibaut De Meulemeester, 2014. "Wing Shape of Four New Bee Fossils (Hymenoptera: Anthophila) Provides Insights to Bee Evolution," PLOS ONE, Public Library of Science, vol. 9(10), pages 1-16, October.
    8. Maria Carla de Francesco & Anna Loy, 2016. "Intra- and Interspecific Interactions as Proximate Determinants of Sexual Dimorphism and Allometric Trajectories in the Bottlenose Dolphin Tursiops truncatus (Cetacea, Odontoceti, Delphinidae)," PLOS ONE, Public Library of Science, vol. 11(10), pages 1-19, October.

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