In mathematics, the matrix representation of conic sections is one way of studying a conic section, its axis, vertices, foci, tangents, and the relative position of a given point. We can also study conic sections whose axes aren't parallel to our coordinate system. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... The axis of rotation of a rotating body is a line such that the distance between any point on the line and any point of the body remains constant under the rotation. ... In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ... In geometry, the focus (pl. ... This article is about the mathematical concept of tangent. For other meanings, see tangent (disambiguation). ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...

Conic sections have the form of a second-degree polynomial: In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...

That can be written as:

Where is the vector: In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ...

And A_Q a matrix: For the square matrix section, see square matrix. ...

Classification

Regular and degenerated conic sections can be distinguished based on the determinant of A_Q. In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

If , the conic is degenerate.

If Q isn't degenerate, we can see what type of conic section it is by computing the subdeterminant resulting from removing the first row and the first column of A_Q (ie the minor A₁₁). In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. ...

• Iff | A₁₁ | < 0, it is a hyperbola.
• Iff | A₁₁ | = 0, it is a parabola.
• Iff | A₁₁ | > 0, it is an ellipse.

In the case of an ellipse, we can make a further distinction between an ellipse and a circle by comparing the last two diagonal elements corresponding to x² and y². A graph of a hyperbola, where h = k = 0 and a = b = 2. ... Wikisource has an original article from the 1911 Encyclopædia Britannica about: Parabola A parabola The parabola (from the Greek: παραβολή) is a conic section generated by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. ... The ellipse and some of its mathematical properties. ...

• If a₁₁ = a₂₂, it is a circle.

If the conic section is degenerate ( | A_Q | = 0), | A₁₁ | still allows us to distinguish its form:

• Iff | A₁₁ | < 0, it is two intersecting lines.
• Iff | A₁₁ | = 0, it is two (possibly coincident) parallel straight lines.
• Iff | A₁₁ | > 0, it is empty.

Center

We can calculate the center by taking the last two rows of the associated matrix, set them equal to 0 and solve the system.

Axes

The major and minor axes are two lines determined by the center of the conic as a point and eigenvectors of the associated matrix as vectors of direction. In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

So we can write a canonical equation:

Because a 2x2 matrix has 2 eigenvectors, we obtain 2 axes.

Vertices

For a general conic we can determine its vertices by calculating the intersection of the conic and its axes — in other words, by solving the system:

Tangents

Through a given point, P, there are generally two lines tangent to a conic. Expressing P as a column vector, p, the two points of tangency are the intersections of the conic with the line whose equation is

When P is on the conic, the line is the tangent there. When P is inside an ellipse, the line is the set of all points whose own associated line passes through P. This line is called the polar of the pole P with respect to the conic.

Just as P uniquely determines its polar line (with respect to a given conic), so each line determines a unique P. This is thus an expression of geometric duality between points and lines in the plane. Duality in the projective plane refers to the interchangeability between points and lines which preserves incidence properties. ...

As special cases, the center of a conic is the pole of the line at infinity, and each asymptote of a hyperbola is a polar (a tangent) to one of its points at infinity.

Using the theory of poles and polars, the problem of finding the four mutual tangents of two conics reduces to finding the intersection of two conics.

Reduced equation

The reduced equation of a conic section is the equation of a conic section translated and rotated so that its center lies in the center of the coordinate system and its axes are parallel to the coordinate axes. This is equivalent to saying that the coordinates are moved to satisfy these properties. See the figure.

Image File history File links Conic_ref_syst. ...

If λ₁ and λ₂ are the eigenvalues of the matrix A, the reduced equation can be written as: In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

Dividing by we obtain a reduced canonical equation. For example, for an ellipse:

From here we get 'a' and 'b'.

The transformation of coordinates is given by: In mathematics, a transformation in elementary terms is any of a variety of different operations from geometry, such as rotations, reflections and translations. ...