A Mathematica animation in physical 3D was one of the works on display in the Art Gallery at last month’s SIGGRAPH 2000 (the 27th International Conference on Computer Graphics and Interactive Techniques). Simply titled 3-D Zoetrope, the subject of the animation was the homotopy, or metamorphosis, of a ring torus (a donut-shaped object) into a Costa minimal surface.
A zoetrope, according to Webster’s Dictionary, is “an optical toy, in which figures made to revolve on the inside of a cylinder, and viewed through slits in its circumference, appear like a single figure passing through a series of natural motions as if animated or mechanically moved.”
In 3-D Zoetrope, 60 phases of the torus-Costa surface transformation are attached to the edge of a wheel. The rotation of the wheel is then “frozen” using a stroboscopic light optically synchronized to the “spokes” of the wheel, where the objects project from its edge.
The computer-rendered proposal for the zoetrope had previously won first prize at the 1999 First International Digital Sculpture Competition. Stewart Dickson, the artist, is a pioneer in using computer-aided, rapid mechanical prototyping technologies to visualize mathematical objects in three physical dimensions.
Dickson, a technical director at Walt Disney Feature Animation and a long-time Mathematica user, found Mathematica to be uniquely suited for this project since the Costa surface depends upon the WeierstrassP function, which is not available in most computer math packages. In Mathematica, it is a built-in function that renders the code extremely compact. Dickson added, “The fact that Mathematica makes the language on the page both dynamic and concrete (in regard to 3D CAD/CAM compatibility), I find extremely powerful.”
|A detailed description of the construction of the zoetrope, including several QuickTime animations, is included in the original proposal. For more information on Mr. Dickson and his work, visit the MathArt web site.||