A surface which can be interpreted as a self-intersecting rectangle in three dimensions. The Whitney umbrella is the only stable singularity of mappings from to . It is given by the parametric equations
(1)
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(2)
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(3)
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for . The center of the "plus" shape which is the end of the line of self-intersection is a pinch point. The coefficients of the first fundamental form are
(4)
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(5)
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(6)
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and the second fundamental form are
(7)
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(8)
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(9)
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giving area element
(10)
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and Gaussian curvature and mean curvature
(11)
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(12)
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Note that the ruled cubic surface given by the equation:
(13)
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is the union of Whitney umbrella and the ray , , called the handle of the Whitney umbrella.