Conic section

A conic section, also called a conic or a quadratic curve, is a special shape made when a plane cuts through a cone's surface. There are three main types of conic sections: the hyperbola, the parabola, and the ellipse. A circle is a special kind of ellipse.

Long ago, around 200 BC, ancient Greek mathematicians studied these shapes. A mathematician named Apollonius of Perga did important work on them.

In simple terms, a conic section can be described by a point called a focus and a line called a directrix[/w/9]_. The shape depends on how far points of the curve are from this focus and this line. These shapes can also be described using special math rules called quadratic equations.

Euclidean geometry

Conic sections have been studied for thousands of years and are an important part of Euclidean geometry.

A conic is a curve formed when a plane cuts through a cone. There are three main types of conics: the ellipse, the parabola, and the hyperbola. A circle is a special type of ellipse. Ellipses happen when the plane cuts the cone in a closed loop. A parabola forms when the plane is parallel to one of the cone’s sides. A hyperbola occurs when the plane cuts through both halves of the cone.

These shapes have many interesting properties and are used in many areas of science and engineering.

conic section	equation	eccentricity (e)	linear eccentricity (c)	semi-latus rectum (ℓ)	focal parameter (p)
circle	x 2 + y 2 = a 2 {\displaystyle x^{2}+y^{2}=a^{2}\,}	0 {\displaystyle 0\,}	0 {\displaystyle 0\,}	a {\displaystyle a\,}	∞ {\displaystyle \infty }
ellipse	x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}	1 − b 2 a 2 {\displaystyle {\sqrt {1-{\frac {b^{2}}{a^{2}}}}}}	a 2 − b 2 {\displaystyle {\sqrt {a^{2}-b^{2}}}}	b 2 a {\displaystyle {\frac {b^{2}}{a}}}	b 2 a 2 − b 2 {\displaystyle {\frac {b^{2}}{\sqrt {a^{2}-b^{2}}}}}
parabola	y 2 = 4 a x {\displaystyle y^{2}=4ax\,}	1 {\displaystyle 1\,}	N/A	2 a {\displaystyle 2a\,}	2 a {\displaystyle 2a\,}
hyperbola	x 2 a 2 − y 2 b 2 = 1 {\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}	1 + b 2 a 2 {\displaystyle {\sqrt {1+{\frac {b^{2}}{a^{2}}}}}}	a 2 + b 2 {\displaystyle {\sqrt {a^{2}+b^{2}}}}	b 2 a {\displaystyle {\frac {b^{2}}{a}}}	b 2 a 2 + b 2 {\displaystyle {\frac {b^{2}}{\sqrt {a^{2}+b^{2}}}}}

History

Menaechmus and early works

The idea of a conic section started with Menaechmus around 320 BC. He used curves from cones to solve problems. These cones were made by spinning a triangle around one of its sides. Cutting the cones with a flat surface made different shapes, called conic sections. The shape could be an ellipse, a parabola, or a hyperbola, depending on the angle of the cone.

Later, Euclid and Archimedes also studied these shapes, but much of their work was lost. Archimedes used conics to find areas and volumes of some shapes.

Apollonius of Perga

The biggest progress in understanding conic sections was made by Apollonius of Perga around 190 BC. His eight books collected all known ideas and added new ones. He showed that any cut through a double cone makes one of these special shapes, including circles. He also named these shapes: ellipse, parabola, and hyperbola.

Islamic world

Apollonius's work was translated into Arabic, and many Islamic mathematicians used conics. Omar Khayyám used them to solve hard math problems. An instrument for drawing these shapes was described as early as 1000 AD.

Europe

In Europe, Johannes Kepler expanded the theory of conics. Later, mathematicians like Girard Desargues and Blaise Pascal used new geometry ideas to study them. René Descartes and Pierre Fermat used algebra to make the math easier. John Wallis defined conic sections with equations, and Jan de Witt wrote the first textbook on the subject.

Applications

For specific applications of each type of conic section, see Circle, Ellipse, Parabola, and Hyperbola.

Conic sections are important in astronomy. The paths of objects moving under gravity are conic sections. If objects are bound together, they move in ellipses. If they are moving apart, they follow parabolas or hyperbolas. See the two-body problem.

The special reflective properties of conic sections are used in tools like searchlights, radio-telescopes, and some optical telescopes. A searchlight uses a parabolic mirror with a light bulb at its focus. A similar design is used for a parabolic microphone. The 4.2 meter Herschel optical telescope uses a parabolic mirror to reflect light to a hyperbolic mirror, which then focuses the light.

In the real projective plane

Conic sections are curves you get when a plane cuts through a cone. They have special properties that are easier to see in a larger geometric system called the real projective plane. This system helps us see why conic sections often look similar.

In this bigger system, conic sections can be described with special equations that use three variables. The type of conic—ellipse, parabola, or hyperbola—depends on how the plane cuts the cone, especially concerning a special line called the "line at infinity."

There are many ways to define and draw conic sections using geometry. One way uses two sets of lines and where they cross to make the curve. Another way uses a grid of points to draw parts of the shape step by step. These methods help us understand and use conic sections in useful ways.

In the complex geometry

In the complex coordinate plane C², shapes like ellipses and hyperbolas can look very similar when you use imaginary numbers. For example, an ellipse can change into a hyperbola by using a special kind of rotation.

When we look at these shapes in a bigger space called the complex projective plane CP², they all start to look the same because you can change one into another using special math rules.

Two of these shapes can share up to four points, and depending on how these points overlap, the shapes might just touch or even look almost the same. Every straight line will cross one of these shapes twice, and there are special cases where the line just touches the shape at one point.

Degenerate cases

Further information: Degenerate conic

Sometimes, special cases of conic sections are called "degenerate." These happen when the plane cuts through the tip, or apex, of the cone.

In simple terms, the degenerate cases can be:

A single point, when the plane just touches the tip of the cone.
A straight line, when the plane touches the cone along one of its sides.
Two intersecting lines, when the plane cuts through the cone in a special way.

These are like the edge cases of the usual shapes (ellipse, parabola, hyperbola).

Pencil of conics

Main article: Pencil (mathematics) § Pencil of conics

A special type of curve called a conic can be fully described by five points in a flat space, as long as no three points are in a straight line. When we look at all the conic curves that pass through four fixed points (also in a flat space, with no three in a straight line), we call this group a "pencil of conics." These four points are called the base points of the pencil. For any other point that is not one of these base points, there is exactly one conic curve from the pencil that passes through it. This idea is a broader version of a group of circles called a pencil of circles.

Intersecting two conics

When we look at two shapes called conic sections, we can see where they touch or meet. These special shapes can meet at no points, two points, or four points.

One way to find where they meet uses special math tools. We mix the rules of the two shapes in a certain way and solve for special numbers. This helps us find lines that make up a simpler shape. By looking at where these lines touch the original shapes, we can discover all the meeting points.

Generalizations

Quadric surfaces are shapes in 3D space that are related to conic sections. They include things like ellipsoids, paraboloids, and hyperboloids. Conics can also be studied in different types of math, but some care is needed with certain number systems.

An oval is a special flat shape that behaves in ways similar to conic sections. There are also shapes called generalized conics that have more than two focus points. When an elliptic cone meets a sphere, it forms a spherical conic, which shares many features with conics on flat surfaces.

In other areas of mathematics

The way we group shapes like ellipses, parabolas, and hyperbolas is used in many parts of mathematics. It helps divide big topics into smaller, easier ones to study.

This grouping comes from something called a "quadratic form," which is a special way of writing equations. In two variables, these forms tell us about the shape: