A Chmutov surface of degree algebraic surface
with affine equation
(1)
where Chebyshev polynomial of the
first kind and polynomial defined by
(2)
where the matrices have dimensions et al. 2008, p. 49).
The projective closures of these surfaces lie
in ordinary double points as singularities . The first few surfaces
are given by
The surface has
(6)
singular points (Chmutov 1992; Breske et al. 2008, p. 50), giving the sequence 0, 1, 3,
14, 28, 57, 93, 154, 216, 321, 425, 576, 732, 949, 1155, ... (OEIS A057870 )
for surfaces give explicit lower bounds
for the maximum number of ordinary double points
on a surface of degree
Based on Chmutov's equations, Banchoff (1991) defined the simpler set of surfaces
(7)
where even number and Chebyshev
polynomial of the first kind . This is a separate family whose projective closures
are nonsingular. For example, the surfaces illustrated
above have orders 2, 4, and 6 and are given by the equations
See also Goursat's Surface ,
Ordinary
Double Point ,
Superellipse ,
Tanglecube
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References Banchoff, T. F. "Computer Graphics Tools for Rendering Algebraic Surfaces and for Geometry of Order." In Geometric
Analysis and Computer Graphics: Proceedings of a Workshop Held May 23-25, 1988 Breske, S.; Labs, O.; and van Straten, D. "Real
Line Arrangements and Surfaces with Many Real Nodes." In Geometric Modeling
and Algebraic Geometry (Ed. B. Jüttler and R. Piene). Berlin,
Germany: Springer, pp. 47-54, 2008. https://doi.org/10.1007/978-3-540-72185-7_3 . Chmutov,
S. V. "Examples of Projective Surfaces with Many Singularities." J.
Algebraic Geom. 1 , 191-196, 1992. Hirzebruch, F. "Singularities
of Algebraic Surfaces and Characteristic Numbers." In The
Lefschetz Centennial Conference, Part I: Proceedings of the Conference on Algebraic
Geometry, Algebraic Topology, and Differential Equations, Held in Mexico City, December
10-14, 1984 Sloane, N. J. A. Sequence A057870
in "The On-Line Encyclopedia of Integer Sequences." Trott,
M. Graphica
1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael
Trott. Trott,
M. The
Mathematica GuideBook for Graphics. https://www.mathematicaguidebooks.org/ . Referenced
on Wolfram|Alpha Chmutov Surface
Cite this as:
Weisstein, Eric W. "Chmutov Surface."
From MathWorld https://mathworld.wolfram.com/ChmutovSurface.html
Subject classifications
is an algebraic surface
with affine equation
![P_d(x_1,x_2)+1/2[T_d(x_3)+1]=0,](/images/equations/ChmutovSurface/NumberedEquation1.svg)
is a Chebyshev polynomial of the
first kind and 
is a polynomial defined by
(Chmutov 1992; Breske et al. 2008, p. 49).
The projective closures of these surfaces lie
in 
and have only ordinary double points as singularities. The first few surfaces
are given by
th
order such surface has
,
2, .... These surfaces give explicit lower bounds
for the maximum number of ordinary double points
on a surface of degree 
, with 
as 
.
is a positive even number and 
is again a Chebyshev
polynomial of the first kind. This is a separate family whose projective closures
are nonsingular. For example, the surfaces illustrated
above have orders 2, 4, and 6 and are given by the equations
![2[x^2(3-4x^2)^2+y^2(3-4y^2)^2+z^2(3-4z^2)^2]=3.](/images/equations/ChmutovSurface/Inline18.svg)
