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COSTA'S SURFACE
Surface studied by da Costa in 1982.
Celso Jose da Costa (1949-....): Brazilian mathematician. en.wikipedia.org/wiki/Costa%27s_minimal_surface mathworld.wolfram.com/CostaMinimalSurface.html |
Gray
parametrization: where c and |
Until 1982, it was conjectured that the only complete (i.e. without boundary) non-periodic minimal surfaces without self-intersection were: the plane, the catenoid and its associated surfaces. Costa's surface, and other ones afterwards, refuted this conjecture.
It can be obtained by taking
in the Weierstrass parametrization of a minimal
surface:
.
Costa's surface is invariant under the action of a half-turn around the axis x = y, z = 0 (in the above parametrization, y(u,v)=x(v,u) and z(v,u)=-z(u,v)), and under the action of this half-turn, the two faces are swapped. Opposite, two half surfaces. Check that each one is the image by a half-turn of the other one |
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Animation of the surface the two faces of which have
different colors.
Imagine water poured inside the upper funnel; it will come out by the external face of the bottom funnel... |
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Costa's surface is topologically equivalent to a torus
minus 3 points (it is therefore of genus
1).
It is also topologically very close to the cubic
algebraic surface with the very simple Cartesian equation: Note that the horizontal sections of the latter surface are ellipses or hyperbolas. Note also that this surface is asymptotic for large x
and y to Plücker's conoid
|
![]() |
Costa's surface can be generalized to a surface with order n rotational symmetry, see here.
Compare Costa's surface to the finite
Riemann minimal surface, which also has a plane sheet and two flared
sheets, but that intersect with each other.
![]() Costa's surface, by Alain Esculier |
Costa's surface by Patrice Jeener, with his kind authorization |
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© Robert FERRÉOL 2017