Computer Vision: From Surfaces to 3D Objects (C.W. Tyler. Ed): Corrections to Ch. 0, Fig, 0.4

Several corrections are necessary for the caption to Fig. 0.4 of Computer Vision: From Surfaces to 3D Objects (C. W. Tyler, Ed), which erroneously characterized the images as depicting a group of minimal surfaces.  They were incorrectly attributed to Shoichi Fujimori, who has visualized numerous minimal surfaces, but none of the surfaces actually reproduced in the figure.

Of the four images in Fig. 0.4, only the one at top left is a minimal surface in the sense of having zero mean curvature. It is a view of the Costa minimal surface (Costa, 1984). Such surfaces form the mathematical model of a soap film stretched on a frame or other non-uniform starting conditions. The other three are CMC surfaces of constant (non-zero) mean curvature, which form a mathematical model for a soap bubble that minimizes the surface area enclosing a fixed volume.  Two of these (upper right and lower left) are embedded in Euclidian 3-space, while the lower right image is a form of the Smyth (1993) surface, which is embedded in three-dimensional sphere.

The upper-left image was realized by the GRAPE consortium, which has been disbanded and can no longer be contacted. The other three were made by Nick Schmitt, an impressive selection of whose visualizations of this genus can be found at his website.

I am very grateful to Shoichi Fujimori for pointing out the errors in this figure caption and to Nick Schmitt for post-facto permission to reproduce his figures.

I propose to change the figure in future editions of the book to substitute Nick Schmitt’s Costa surface for the one I used. Thus, the caption to Fig. 0.4 will be revised to read:

Fig. 0.4. Depictions of four mathematical surfaces illustrating the principle of constant mean curvature (see text).  Upper left: a surface of zero mean curvature. The other three surfaces have various forms of constant (non-zero) mean curvature.   (Reproduced from Nick Schmitt, with gracious permission.)


Costa, A. (1984) “Examples of a Complete Minimal Immersion in R3 of Genus One and Three Embedded Ends.” Bil. Soc. Bras. Mat. 15, 47-54.

Smyth, B. (1993) A generalization of a theorem of Delaunay on constant mean curvature surfaces, statistical thermodynamics and differential geometry of microstructured materials (Minneapolis, MN, 1991), IMA Vol. Math. Appl., Springer, New York, 123-13