Conic section

A conic section, also called a conic or a quadratic curve, is a special kind of 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 and was sometimes thought of as a fourth type.

Long ago, around 200 BC, ancient Greek mathematicians began studying these shapes closely. A mathematician named Apollonius of Perga did important work to understand their properties.

In simple terms, a conic section can also be described by a special rule involving a point called a focus and a line called a directrix[/w/9]. The shape depends on how far the points of the curve are from this focus and this line, a ratio known as the eccentricity. In another way, these shapes can be described using equations with squared terms, called quadratic equations, which help us understand their properties and how they look.

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 through 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, creating two separate curves.

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 first appeared in the work of Menaechmus around 320 BC. He tried to solve a problem by using curves made from cones. These cones were created by spinning a triangle around one of its sides. By cutting these cones with a flat surface, different shapes called conic sections were formed. Depending on the angle of the cone, the shape could be an ellipse, a parabola, or a hyperbola.

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

Apollonius of Perga

The biggest advances in understanding conic sections were made by Apollonius of Perga around 190 BC. His eight books brought together all that was known and added new ideas. He showed that any cut through a double cone creates one of these special shapes, including circles. He also gave these shapes their names: ellipse, parabola, and hyperbola.

Islamic world

Apollonius's work was translated into Arabic, and many Islamic mathematicians used conics. Omar Khayyám, for example, used them to solve complex 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 further. René Descartes and Pierre Fermat applied algebra to these shapes, making the math easier. John Wallis defined conic sections using 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 two objects moving under the force of gravity are conic sections. If the 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 that appear when a plane cuts through a cone. They have interesting properties that become clearer when studied in a larger geometric system called the real projective plane. This system helps us understand why conic sections share similarities.

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

There are many ways to define and construct conic sections using geometry. One method uses two sets of lines and their intersections to create the curve. Another uses a grid of points to draw parts of the shape step by step. These constructions help us understand and work with conic sections in practical 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 is fully described by five points in a flat space, as long as no three points lie 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 known as 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 find where they meet or intersect. 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 equations of the two shapes in a certain way and solve for special values. This helps us find the lines that make up a simpler shape. By looking at where these lines meet the original shapes, we can discover all the meeting points.

Generalizations

Quadric surfaces generalize conic sections in three-dimensional space, including ellipsoids, paraboloids, and hyperboloids. Conics can also be defined in different types of geometries, but special attention is needed when working with certain number systems.

An oval is a special shape in a flat plane that shares important properties with conic sections, such as how lines intersect it. There are also special shapes called generalized conics that have more than two focus points. The meeting point of an elliptic cone and a sphere creates a spherical conic, which has many similar features to the conics found 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: