In
Euclidean and
projective geometry
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 (''p ...
, five points determine a conic (a degree-2 plane curve), just as two (distinct)
points
A point is a small dot or the sharp tip of something. Point or points may refer to:
Mathematics
* Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...
determine a
line (a degree-1
plane curve
In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane c ...
). There are additional subtleties for
conic
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, thou ...
s that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines.
Formally, given any five points in the plane in
general linear position, meaning no three
collinear
In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
, there is a unique conic passing through them, which will be non-
degenerate; this is true over both the
Euclidean plane
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
and any
pappian 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.
Proofs
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.
Dimension counting
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
In algebraic geometry, Bézout's theorem is a statement concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the de ...
);
* 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
In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, of three gloves, at least two must be right-handed or at least two must be l ...
), 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