To draw a torus

Aug 2025

talk gallery art

Summary:

I spend a lot of time drawing pictures of surfaces, so I often wonder how to do so correctly. Shadows, shading, and reflections are all determined by the normal vector of the surface, and its orientation relative to the light and the viewer. The classical differential geometry of a surface is also entirely controlled by the normal vector, through the “Gauss map”. The curvature of the surface dictates how shadows fall, the shape of specular highlights, and even the number of highlights. We will reflect on how it is we draw a tours.

Presented at:

Illustrating Mathematics Reunion/Expansion at ICERM, Summer 2025

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Gallery

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To draw a torus

When I draw a torus, all the homunculi in my head get constructing. Some are planning, others applying shading, some airbrushing on ambient reflections, and one gingerly applying highlights at the end. Every single one of them is doing differential geometry.

Hand with reflecting torus

A riff on MC escher’s famous self portrait. I used Escher’s original portrait to map out what a mirror would see if pointed in any direction. Then, I used this map with the normals of a torus to color each point on the surface.

Torus

A nicely rendered torus.

Lines of curvature on a little bird

A timelapse of me measuring the curvature of a little porcelain bird, using reflections. I have a circular light behind my head, so the reflection is shaped like a small ellipse. I am marking the semimajor and semiminor axes of the ellipse. These axes are oriented in the directions of the lines of curvature. If the curvature is small, then the highlight is large and I draw a long line. If it is large, then the highlight and my line is small. Repeating everywhere on the surface, I can map out the direction and magnitude of curvature.