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A visualization method of four-dimensional polytopes by oval display of parallel hyperplane slices

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Abstract

A method to visualize polytopes in a four-dimensional euclidian space (x, y, z, w) is proposed. A polytope is sliced by multiple hyperplanes that are parallel to each other and separated by uniform intervals. Since the hyperplanes are perpendicular to thew-axis, the resulting multiple slices appear in the three-dimensional (x, y, z) space and they are shown by the standard computer graphics. The polytope is rotated extrinsically in the four-dimensional space by means of a simple input method based on keyboard typings. The multiple slices are placed on a parabola curve in the three-dimensional world coordinates. The slices in a view window form an oval appearance. Both the simple and the double rotations in the four-dimensional space are applied to the polytope. All slices synchronously change their shapes when a rotation is applied to the polytope. The compact display in the oval of many slices with the help of quick rotations facilitate a grasp of the four-dimensional configuration of the polytope.

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Acknowledgments

This work was supported by JSPS KAKENHI Grant No. 20260052. The author thanks a reviewer for pointing out the importance of subsidiary data in the slice visualization. The colored parallel coordinates are added by the reviewer’s suggestion.

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Authors and Affiliations

  1. Department of Computational Science, Kobe University, Kobe, 657-8501, Japan

    Akira Kageyama

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  1. Akira Kageyama

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Correspondence toAkira Kageyama.

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Kageyama, A. A visualization method of four-dimensional polytopes by oval display of parallel hyperplane slices.J Vis19, 417–422 (2016). https://doi.org/10.1007/s12650-015-0319-5

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