General position in the context of Computational geometry

⭐ Core Definition: General position

In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings.

For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). One also says "two generic lines intersect in a point", which is formalized by the notion of a generic point. Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case.

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⭐ Core Definition: General position
General position in the context of Degree (algebraic geometry)
General position in the context of Weddle surface

General position in the context of Degree (algebraic geometry)

In mathematics, the degree of an affine or projective variety of dimension $n$ is the number of intersection points of the varietywith $n$ hyperplanes in general position. For an algebraic set, the intersection points must be counted with their intersection multiplicity, because of the possibility of multiple components. For (irreducible) varieties, if one takes into account the multiplicities and, in the affine case, the points at infinity, the hypothesis of general position may be replaced by the much weaker condition that the intersection of the variety has the dimension zero (that is, consists of a finite number of points). This is a generalization of Bézout's theorem. (For a proof, see Hilbert series and Hilbert polynomial § Degree of a projective variety and Bézout's theorem.)

The degree is not an intrinsic property of the variety, as it depends on a specific embedding of the variety in an affine or projective space.

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General position in the context of Weddle surface

In algebraic geometry, a Weddle surface, introduced by Thomas Weddle (1850, footnote on page 69), is a quartic surface in 3-dimensional projective space, given by the locus of vertices of the family of cones passing through 6 points in general position.

Weddle surfaces have 6 nodes and are birational to Kummer surfaces.

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General position in the context of Computational geometry

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⭐ Core Definition: General position

General position in the context of Degree (algebraic geometry)

General position in the context of Weddle surface