In algebraic geometry, Cramer's theorem on algebraic curves gives the necessary and sufficient number of points in the real plane falling on an algebraic curve to uniquely determine the curve in non-degenerate cases.
The theorem is due to Gabriel Cramer, who published it in 1750.
Likewise, a non-degenerate conic (polynomial equation in x and y with the sum of their powers in any term not exceeding 2, hence with degree 2) is uniquely determined by 5 points in general position (no three of which are on a straight line).
The intuition of the conic case is this: Suppose the given points fall on, specifically, an ellipse.
Then five pieces of information are necessary and sufficient to identify the ellipse—the horizontal location of the ellipse's center, the vertical location of the center, the major axis (the length of the longest chord), the minor axis (the length of the shortest chord through the center, perpendicular to the major axis), and the ellipse's rotational orientation (the extent to which the major axis departs from the horizontal).
The number of distinct terms (including those with a zero coefficient) in an n-th degree equation in two variables is (n + 1)(n + 2) / 2.
numbering n + 1 in total; the (n − 1) degree terms are
numbering n in total; and so on through the first degree terms
numbering 2 in total, and the single zero degree term (the constant).
For example, a fourth degree equation has the general form with 4(4+3)/2 = 14 coefficients.
Determining an algebraic curve through a set of points consists of determining values for these coefficients in the algebraic equation such that each of the points satisfies the equation.
Given n(n + 3) / 2 points (xi, yi), each of these points can be used to create a separate equation by substituting it into the general polynomial equation of degree n, giving n(n + 3) / 2 equations linear in the n(n + 3) / 2 unknown coefficients.
More than this number of points would be redundant, and fewer would be insufficient to solve the system of equations uniquely for the coefficients.
An example of a degenerate case, in which n(n + 3) / 2 points on the curve are not sufficient to determine the curve uniquely, was provided by Cramer as part of Cramer's paradox.
Thus these points do not determine a unique cubic, even though there are n(n + 3) / 2 = 9 of them.
More generally, there are infinitely many cubics that pass through the nine intersection points of two cubics (Bézout's theorem implies that two cubics have, in general, nine intersection points) Likewise, for the conic case of n = 2, if three of five given points all fall on the same straight line, they may not uniquely determine the curve.
If the curve is required to be in a particular sub-category of n-th degree polynomial equations, then fewer than n(n + 3) / 2 points may be necessary and sufficient to determine a unique curve.
where the center is located at (a, b) and the radius is r. Equivalently, by expanding the squared terms, the generic equation is
Two restrictions have been imposed here compared to the general conic case of n = 2: the coefficient of the term in xy is restricted to equal 0, and the coefficient of y2 is restricted to equal the coefficient of x2.