Whitney Umbrella
Whitney Umbrella is a ruled surface, defined by an implicit equation, immersed with singularities.
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aperiodic def-implicit implemented ruled singular tradition-classical
About
The model singularity for a surface mapped into space: a pinch point where the sheet closes up to a single point, with a line of self-intersection running out of it -- the handle of the umbrella. Whitney showed that this and ordinary crossings are the only singularities a generic map of a surface into three-space can have, which is why it turns up so often.
Formula
Properties
- Family
- ruled
- Given by
- an implicit equation
- Curvature
- no condition imposed
- Periodicity
- not periodic
- Ends
- 0
- Embedding
- immersed with singularities
- Fidelity
- exact
- Exactness
- elementary
Sources
Closed form reproduced numerically against the shipped implementation (math_art/surfaces/algebraic.py) over 240 sample points: matches oracle x 1 over 240 points (worst 0).
- H. Whitney, 'The general type of singularity of a set of 2n-1 smooth functions of n variables', Duke Math. J. 10 (1943).