Five points determine a conic
In Euclidean, non-projective geometry, just as two (distinct) points determine a line (a degree-1 plane curve), five points determine a conic (a degree-2 plane curve). 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. In projective geometry, a line is defined via three points, all families of circles are conics passing through two points of a line at infinity, and coordinates are the basic conic objects of study, and are defined as all conics passing through the line and two points at infinity.
Formally, given any five points in the plane in general linear position, meaning no three collinear, there is a unique conic passing through them, which will be non-degenerate; this is true over both the affine plane and projective plane. Indeed, given any five points there is a conic passing through them, but if three of the points are collinear the conic will be degenerate (reducible, because it contains a line), and may not be unique; see further discussion.
This result can be proven numerous different ways; the dimension counting argument is most direct, and generalizes to higher degree, while other proofs are special to conics.
Intuitively, passing through five points in general linear position specifies five independent linear constraints on the (projective) linear space of conics, and hence specifies a unique conic, though this brief statement ignores subtleties.
More precisely, this is seen as follows:
- conics correspond to points in the five-dimensional projective space
- requiring a conic to pass through a point imposes a linear condition on the coordinates: for a fixed the equation is a linear equation in
- by dimension counting five constraints (that the curve passes through five points) are necessary to specify a conic, as each constraint cuts the dimension of possibilities by 1, and one starts with 5 dimensions;
- in 5 dimensions, the intersection of 5 (independent) hyperplanes is a single point (formally, by Bézout's theorem);
- general linear position of the points means that the constraints are independent, and thus do specify a unique conic;
- the resulting conic is non-degenerate because it is a curve (since it has more than 1 point), and does not contain a line (else it would split as two lines, at least one of which must contain 3 of the 5 points, by the pigeonhole principle), so it is irreducible.
The two subtleties in the above analysis are that the resulting point is a quadratic equation (not a linear equation), and that the constraints are independent. The first is simple: if A, B, and C all vanish, then the equation defines a line, and any 3 points on this (indeed any number of points) lie on a line – thus general linear position ensures a conic. The second, that the constraints are independent, is significantly subtler: it corresponds to the fact that given five points in general linear position in the plane, their images in under the Veronese map are in general linear position, which is true because the Veronese map is biregular: i.e., if the image of five points satisfy a relation, then the relation can be pulled back and the original points must also satisfy a relation. The Veronese map has coordinates and the target is dual to the of conics. The Veronese map corresponds to "evaluation of a conic at a point", and the statement about independence of constraints is exactly a geometric statement about this map.
That five points determine a conic can be proven by synthetic geometry—i.e., in terms of lines and points in the plane—in addition to the analytic (algebraic) proof given above. Such a proof can be given using a theorem of Jakob Steiner, which states:
- Given a projective transformation f, between the pencil of lines passing through a point X and the pencil of lines passing through a point Y, the set C of intersection points between a line x and its image forms a conic.
- Note that X and Y are on this conic by considering the preimage and image of the line XY (which is respectively a line through X and a line through Y).
This can be shown by taking the points X and Y to the standard points and by a projective transformation, in which case the pencils of lines correspond to the horizontal and vertical lines in the plane, and the intersections of corresponding lines to the graph of a function, which (must be shown) is a hyperbola, hence a conic, hence the original curve C is a conic.
Now given five points X, Y, A, B, C, the three lines can be taken to the three lines by a unique projective transform, since projective transforms are simply 3-transitive on lines (they are simply 3-transitive on points, hence by projective duality they are 3-transitive on lines). Under this map A maps to B, since these are the unique intersection points of these lines, and thus satisfy the hypothesis of Steiner’s theorem. The resulting conic thus contains all five points, and is the unique such conic, as desired.
Given five points, one can construct the conic containing them in various ways.
Analytically, the equation for the conic can be found by linear algebra, by writing and solving the five equations in the coefficients: five equations, six unknowns, but homogeneous so scaling removes one dimension; concretely, setting one of the coefficients to 1 accomplishes this.
Synthetically, the conic can be constructed by the Braikenridge–Maclaurin construction, by applying the Braikenridge–Maclaurin theorem, which is the converse of Pascal's theorem. Pascal's theorem states that given 6 points on a conic (a hexagon), the lines defined by opposite sides intersect in three collinear points. This can be reversed to construct the possible locations for a 6th point, given 5 existing ones.
The natural generalization is to ask for what value of k a configuration of k points (in general position) in n-space determines a variety of degree d and dimension m, which is a fundamental question in enumerative geometry.
This is via the analogous analysis of the Veronese map: k points in general position impose k independent linear conditions on a variety (because the Veronese map is biregular), and the number of monomials of degree d in variables (n-dimensional projective space has homogeneous coordinates) is from which 1 is subtracted because of projectivization: multiplying a polynomial by a constant does not change its zeros.
In the above formula, the number of points k is a polynomial in d of degree n, with leading coefficient
In the case of plane curves, where the formula becomes:
whose values for are – there are no curves of degree 0 (a single point is determines a point, which is codimension 2), 2 points determine a line, 5 points determine a conic, 9 points determine a cubic, 14 points determine a quartic, and so forth.
While five points determine a conic, sets of six or more points on a conic are in special position, as demonstrated in Pascal's theorem.
Conversely, four points do not determine a conic, but rather a pencil, the 1-dimensional linear system of conics which all pass through the four points (formally, have the four points as base locus). Similarly, three points determine a 2-dimensional linear system (net), two points determine a 3-dimensional linear system (web), one point determines a 4-dimensional linear system, and zero points place no constraints on the 5-dimensional linear system of all conics.
Similarly, while nine points determine a cubic, if the nine points lie on more than one cubic—i.e., they are the intersection of two cubics—then they are not in general position, and indeed satisfy an addition constraint, as stated in the Cayley–Bacharach theorem.
Instead of passing through points, a different condition on a curve is being tangent to a given line. Being tangent to five given lines also determines a conic, by projective duality, but from the algebraic point of view tangency to a line is a quadratic constraint, so naive dimension counting yields 25 = 32 conics tangent to five given lines, of which 31 must be ascribed to degenerate conics, as described in fudge factors in enumerative geometry; formalizing this intuition requires significant further development to justify.
Another classic problem in enumerative geometry, of similar vintage to conics, is the Problem of Apollonius: a circle that is tangent to three circles in general determines eight circles, as each of these is a quadratic condition and 23 = 8. As a question in real geometry, a full analysis involves many special cases, and the actual number of circles may be any number between 0 and 8, except for 7.
- Cramer's theorem (algebraic curves), for a generalization to n-th degree planar curves
- Interactive Course on Projective Geometry, Chapter Five: The Projective Geometry of Conics: Section Four: Conics on the real projective plane, by J.C. Álvarez Paiva; proof follows Exercise 4.6
- (Coxeter 1961, pp. 252–254)
- The Animated Pascal, Sandra Lach Arlinghaus
- Weisstein, Eric W. "Braikenridge-Maclaurin Construction." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Braikenridge-MaclaurinConstruction.html
- The GNU 3DLDF Conic Sections Page: Pascal's Theorem and the Braikenridge-Maclaurin Construction, Laurence D. Finston
- Coxeter, H. S. M. (1961), Introduction to Geometry, Washington, DC
- Coxeter, H. S. M.; Greitzer, S. L. (1967), Geometry Revisited, Washington, DC: Mathematical Association of America, p. 76
- Dixon, A. C. (March 1908), "The Conic through Five Given Points", The Mathematical Gazette, The Mathematical Association, 4 (70): 228–230, JSTOR 3605147
- Five Points Determine a Conic Section, Wolfram interactive demonstration