Conics

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with 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 historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.

The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, and some particular line, called a directrix, are in a fixed ratio, called the eccentricity. The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables, which may be written in matrix form. This equation allows deducing and expressing algebraically the geometric properties of conic sections.

In the Euclidean plane, the three types of conic sections appear quite different, but share many properties. By extending the Euclidean plane to include a line at infinity, obtaining a projective plane, the apparent difference vanishes: the branches of a hyperbola meet in two points at infinity, making it a single closed curve; and the two ends of a parabola meet to make it a closed curve tangent to the line at infinity. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically.

Standard forms in Cartesian coordinates

Standard forms of an ellipse

Standard forms of a parabola

Standard forms of a hyperbola

After introducing Cartesian coordinates, the focus-directrix property can be used to produce the equations satisfied by the points of the conic section. By means of a change of coordinates (rotation and translation of axes) these equations can be put into standard forms. For ellipses and hyperbolas a standard form has the x-axis as principal axis and the origin (0,0) as center. The vertices are (±a, 0) and the foci (±c, 0). Define b by the equations c² = a² − b² for an ellipse and c² = a² + b² for a hyperbola. For a circle, c = 0 so a² = b². For the parabola, the standard form has the focus on the x-axis at the point (a, 0) and the directrix the line with equation x = −a. In standard form the parabola will always pass through the origin.

For a rectangular or equilateral hyperbola, one whose asymptotes are perpendicular, there is an alternative standard form in which the asymptotes are the coordinate axes and the line x = y is the principal axis. The foci then have coordinates (c, c) and (−c, −c).

• Circle: $x^{2}+y^{2}=a^{2}$
• Ellipse: ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$
• Parabola: $y^{2}=4ax$ with $a>0$
• Hyperbola: ${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1$
• Rectangular hyperbola: $xy={\frac {c^{2}}{2}}$

The first four of these forms are symmetric about both the x-axis and y-axis (for the circle, ellipse and hyperbola), or about the x-axis only (for the parabola). The rectangular hyperbola, however, is instead symmetric about the lines {{{1}}} and {{{1}}}.

These standard forms can be written parametrically as,

• Circle: (a cos θ, a sin θ),
• Ellipse: (a cos θ, b sin θ),
• Parabola: (at², 2at),
• Hyperbola: (a sec θ, b tan θ) or (±a cosh u, b sinh u),
• Rectangular hyperbola: $(d\,t,{\frac {d}{t}})$ where $d={\frac {c}{\sqrt {2}}}.$

General Cartesian form

In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section (though it may be degenerate), and all conic sections arise in this way. The most general equation is of the form

$Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0,$

with all coefficients real numbers and A, B, C not all zero.

Circle

The circle is the simplest and best known conic section. As a conic section, the circle is the intersection of a plane perpendicular to the cone's axis.

The geometric definition of a circle is the locus of all points a constant distance $r$ from a point $(h,k)$ and forming the circumference (C). The distance $r$ is the radius (R) of the circle, and the point $O=(h,k)$ is the circle's center also spelled as centre. The diameter (D) is twice the length of the radius.

Circle illustration with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta.

Equations

Standard Form

The equation for a circle with center $(h,k)$ and radius $r$ is

$(x-h)^{2}+(y-k)^{2}=r^{2}$.

In the simplest case of a circle whose center is at the origin, the equation is simply a restatement of the Pythagorean Theorem:

$x^{2}+y^{2}=r^{2}$

General form

The general form of a circle equation is

$x^{2}+y^{2}+2gx+2fy+c=0$, where

<-g,-f> is the center of the circle.

Polar Coordinates

In the case of a circle centered at the origin, the polar equation of a circle is very simple because polar coordinates are essentially based on circles. For a circle with radius $a$,

$r=a$.

In the more complicated case of a circle with an arbitrary location, the equation is

$r^{2}-2rr_{0}\cos(\theta -\varphi )+r_{0}^{2}=a^{2}$,
where $r_{0}$ is the distance from the circle's center to the origin and $\varphi$ is the angle pointing to the circle.

There are many cases that allow the equation to be simplified. If a point on the circle is touching the origin, its polar equation may consist of a single trig function.

Parametric Equations

When the circle's equation is parametrized with respect to $t$, the equation becomes

$x=h+r\cos t$,
$y=k+r\sin t$.

Resources

• Introduction to Conic Sections, Lumen Learning
• Conic section, Wikipedia
• Circle, Wikibooks: Conic Sections
• Ellipse, Wikibooks: Conic Sections
• Circle, Wikibooks: Conic Sections
• Circle, Wikibooks: Conic Sections