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.
This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise statements thereof, and when writing computer programs (see generic complexity).
General linear position
A set of points in a d-dimensionalaffine space (d-dimensional Euclidean space is a common example) is in general linear position (or just general position) if no k of them lie in a (k− 2)-dimensionalflat for k = 2, 3, ..., d + 1. These conditions contain considerable redundancy since, if the condition holds for some value k0 then it also must hold for all k with 2 ≤ k ≤ k0. Thus, for a set containing at least d + 1 points in d-dimensional affine space to be in general position, it suffices that no hyperplane contains more than d points – i.e. the points do not satisfy any more linear relations than they must.[1]
A set of at most d + 1 points in general linear position is also said to be affinely independent (this is the affine analog of linear independence of vectors, or more precisely of maximal rank), and d + 1 points in general linear position in affine d-space are an affine basis. See affine transformation for more.
Similarly, n vectors in an n-dimensional vector space are linearly independent if and only if the points they define in projective space (of dimension n− 1) are in general linear position.
If a set of points is not in general linear position, it is called a degenerate case or degenerate configuration, which implies that they satisfy a linear relation that need not always hold.
A fundamental application is that, in the plane, five points determine a conic, as long as the points are in general linear position (no three are collinear).
More generally
This definition can be generalized further: one may speak of points in general position with respect to a fixed class of algebraic relations (e.g. conic sections). In algebraic geometry this kind of condition is frequently encountered, in that points should impose independent conditions on curves passing through them.
For example, five points determine a conic, but in general six points do not lie on a conic, so being in general position with respect to conics requires that no six points lie on a conic.
General position is preserved under biregular maps – if image points satisfy a relation, then under a biregular map this relation may be pulled back to the original points. Significantly, the Veronese map is biregular; as points under the Veronese map corresponds to evaluating a degree d polynomial at that point, this formalizes the notion that points in general position impose independent linear conditions on varieties passing through them.
The basic condition for general position is that points do not fall on subvarieties of lower degree than necessary; in the plane two points should not be coincident, three points should not fall on a line, six points should not fall on a conic, ten points should not fall on a cubic, and likewise for higher degree.
This is not sufficient, however. While nine points determine a cubic, there are configurations of nine points that are special with respect to cubics, namely the intersection of two cubics. The intersection of two cubics, which is points (by Bézout's theorem), is special in that nine points in general position are contained in a unique cubic, while if they are contained in two cubics they in fact are contained in a pencil (1-parameter linear system) of cubics, whose equations are the projective linear combinations of the equations for the two cubics. Thus such sets of points impose one fewer condition on cubics containing them than expected, and accordingly satisfy an additional constraint, namely the Cayley–Bacharach theorem that any cubic that contains eight of the points necessarily contains the ninth. Analogous statements hold for higher degree.
For points in the plane or on an algebraic curve, the notion of general position is made algebraically precise by the notion of a regular divisor, and is measured by the vanishing of the higher sheaf cohomology groups of the associated line bundle (formally, invertible sheaf). As the terminology reflects, this is significantly more technical than the intuitive geometric picture, similar to how a formal definition of intersection number requires sophisticated algebra. This definition generalizes in higher dimensions to hypersurfaces (codimension 1 subvarieties), rather than to sets of points, and regular divisors are contrasted with superabundant divisors, as discussed in the Riemann–Roch theorem for surfaces.
Note that not all points in general position are projectively equivalent, which is a much stronger condition; for example, any k distinct points in the line are in general position, but projective transformations are only 3-transitive, with the invariant of 4 points being the cross ratio.
Different geometries
Different geometries allow different notions of geometric constraints. For example, a circle is a concept that makes sense in Euclidean geometry, but not in affine linear geometry or projective geometry, where circles cannot be distinguished from ellipses, since one may squeeze a circle to an ellipse. Similarly, a parabola is a concept in affine geometry but not in projective geometry, where a parabola is simply a kind of conic. The geometry that is overwhelmingly used in algebraic geometry is projective geometry, with affine geometry finding significant but far less use.
Thus, in Euclidean geometry three non-collinear points determine a circle (as the circumcircle of the triangle they define), but four points in general do not (they do so only for cyclic quadrilaterals), so the notion of "general position with respect to circles", namely "no four points lie on a circle" makes sense. In projective geometry, by contrast, circles are not distinct from conics, and five points determine a conic, so there is no projective notion of "general position with respect to circles".
General position is a property of configurations of points, or more generally other subvarieties (lines in general position, so no three concurrent, and the like). General position is an extrinsic notion, which depends on an embedding as a subvariety. Informally, subvarieties are in general position if they cannot be described more simply than others. An intrinsic analog of general position is general type, and corresponds to a variety which cannot be described by simpler polynomial equations than others. This is formalized by the notion of Kodaira dimension of a variety, and by this measure projective spaces are the most special varieties, though there are other equally special ones, meaning having negative Kodaira dimension. For algebraic curves, the resulting classification is: projective line, torus, higher genus surfaces (), and similar classifications occur in higher dimensions, notably the Enriques–Kodaira classification of algebraic surfaces.
Other contexts
In intersection theory, both in algebraic geometry and in geometric topology, the analogous notion of transversality is used: subvarieties in general intersect transversally, meaning with multiplicity 1, rather than being tangent or other, higher order intersections.
General position for Delaunay triangulations in the plane
When discussing Voronoi tessellations and Delaunay triangulations in the plane, a set of points in the plane is said to be in general position only if no four of them lie on the same circle and no three of them are collinear. The usual lifting transform that relates the Delaunay triangulation to the bottom half of a convex hull (i.e., giving each point p an extra coordinate equal to |p|2) shows the connection to the planar view: Four points lie on a circle or three of them are collinear exactly when their lifted counterparts are not in general linear position.
Abstractly: configuration spaces
In very abstract terms, general position is a discussion of generic properties of a configuration space; in this context one means properties that hold on the generic point of a configuration space, or equivalently on a Zariski-open set.
This notion coincides with the measure theoretic notion of generic, meaning almost everywhere on the configuration space, or equivalently that points chosen at random will almost surely (with probability 1) be in general position.
Algebraic geometry is a branch of mathematics which classically studies zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines.
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points to Euclidean points, and vice-versa.
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In geometry, a degenerate conic is a conic that fails to be an irreducible curve. This means that the defining equation is factorable over the complex numbers as the product of two linear polynomials.
In projective geometry, Pascal's theorem states that if six arbitrary points are chosen on a conic and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon meet at three points which lie on a straight line, called the Pascal line of the hexagon. It is named after Blaise Pascal.
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist embedded in two, three, or higher dimensional spaces. The word line may also refer to a line segment in everyday life that has two points to denote its ends (endpoints). A line can be referred to by two points that lie on it or by a single letter.
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family.
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An incidence structure is what is obtained when all other concepts are removed and all that remains is the data about which points lie on which lines. Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure. Such fundamental results remain valid when additional concepts are added to form a richer geometry. It sometimes happens that authors blur the distinction between a study and the objects of that study, so it is not surprising to find that some authors refer to incidence structures as incidence geometries.
In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane.
In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves in the projective plane P2. The original form states:
In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.
In algebraic geometry, the conic sections in the projective plane form a linear system of dimension five, as one sees by counting the constants in the degree two equations. The condition to pass through a given point P imposes a single linear condition, so that conics C through P form a linear system of dimension 4. Other types of condition that are of interest include tangency to a given line L.
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.
In Euclidean and projective geometry, five points determine a conic, just as two (distinct) points determine a line. There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines.
The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction of the general methods, initiated by David Hilbert and the Italian school of algebraic geometry in the beginning of the century, and later formalized by André Weil, Jean-Pierre Serre and Alexander Grothendieck. Much of the classical terminology, mainly based on case study, was simply abandoned, with the result that books and papers written before this time can be hard to read. This article lists some of this classical terminology, and describes some of the changes in conventions.
In enumerative geometry, Steiner's conic problem is the problem of finding the number of smooth conics tangent to five given conics in the plane in general position. If the problem is considered in the complex projective plane CP2, the correct solution is 3264. The problem is named after Jakob Steiner who first posed it and who gave an incorrect solution in 1848.
References
Yale, Paul B. (1968), Geometry and Symmetry, Holden-Day
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