Architectural Applications of Several 3D Geometric Transformations

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This Demonstration considers the following surfaces: ellipsoid, hyperboloid of one sheet, elliptic paraboloid, hyperbolic paraboloid, helicoid, and Möbius strip, which can be represented by parametric equations of the general form . These surfaces can undergo further transformations, including rotation, translation, helical motion, and ruling.

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To illustrate the application of these methods in architecture, sketches of real-world examples are displayed, as well as maps showing their locations.

This Demonstration shows a set of six structures: their animated transformation in 3D, one or two matching sketches of a building, and a map for each of the six 3D surfaces. Short descriptions or legends for each graphic are given.

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Contributed by: Guenther Gsaller (February 2016)
Open content licensed under CC BY-NC-SA


Snapshots


Details

According to Pottmann [1], traditional surface classes are rotational, translational, ruled, helical, and piped. They are the source for these methods.

The six surfaces considered in this Demonstration are the ellipsoid, hyperboloid of one sheet, elliptic paraboloid, hyperbolic paraboloid, helicoid, and Möbius strip.

To find information about them (e.g. the parametric equation), the Related Links give a list of six links to mathworld.wolfram.com (Wolfram MathWorld).

For the half ellipsoid, the parametric equation is:

with running from to and from to . A rotation is implemented through the additional parameter , with running from to , from to , and from 8 to 14.

From ten buildings or installations around the world sketches were prepared, fitting the chosen surface example. In Mathematica they were imported as PNG files.

Via www.openstreetmap.org, an SVG file of the location for each of the ten buildings above was exported. They were the sources for the minimized sketches, fitting the chosen building example. In Mathematica they were again imported as PNG files.

References

[1] H. Pottmann, A. Asperl, M. Hofer, and A. Kilian, Architectural Geometry, Exton: Bentley Institute Press, 2007.

[2] OpenStreetMap Foundation Wiki. (Jan 22, 2016) www.openstreetmap.org.



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