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Croissant Surface

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croissant surface with its singular point marked in orange

The name croissant surface is used for a pinched quartic surface in Hauser's gallery. The RSME-IMAGINARY exhibition catalog gives the following equation for it:

(x^2+y^2+z^2+(7sqrt(5)-11)/2)^2-[(1+sqrt(5))x-7+3sqrt(5)]^2-(1+sqrt(5))^2y^2=0

(Real Sociedad Matemática Española 2011, p. 43). This equation has exactly one real singular point,

((1+sqrt(5))/2,0,0),

which is an ordinary double point.

Although its pinched shape resembles a Dupin cyclide, this quartic is not a member of that family. Comparing its coefficients with the canonical equations listed by Alcázar et al. (2018) gives incompatible parameter conditions.

See also

Quartic Surface

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References

Alcázar, J. G.; Dahl, H. E. I.; and Muntingh, G. "Symmetries of Canal Surfaces and Dupin Cyclides." Comput. Aided Geom. Design 59, 68-85, 2018. https://doi.org/10.1016/j.cagd.2017.10.001.Hauser, H. "Gallery of Singular Algebraic Surfaces: Croissant." https://homepage.univie.ac.at/herwig.hauser/gallery.html.Real Sociedad Matemática Española. RSME-IMAGINARY: Una mirada matemática. Madrid, Spain: Real Sociedad Matemática Española, p. 43, 2011. https://fme.upc.edu/ca/exposicio-imaginary-una-simfonia-matematica-fme-2021/material-docemt/rsmeimaginary_catalogue_es.pdf.

Cite this as:

Weisstein, Eric W. "Croissant Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CroissantSurface.html

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