**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Focus (geometry)

Summary

In geometry, focuses or foci (ˈfəʊkaɪ; : focus) are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an n-ellipse.
Conic section#Eccentricity, focus and directrix Ellipse#FocusParabola#Position of the focusHyperbola#Directrix and focus and Confocal conic sections
An ellipse can be defined as the locus of points for which the sum of the distances to two given foci is constant.
A circle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be more simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as the circle of Apollonius, in terms of two different foci, as the locus of points having a fixed ratio of distances to the two foci.
A parabola is a limiting case of an ellipse in which one of the foci is a point at infinity.
A hyperbola can be defined as the locus of points for which the absolute value of the difference between the distances to two given foci is constant.
It is also possible to describe all conic sections in terms of a single focus and a single directrix, which is a given line not containing the focus. A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a fixed positive constant, called the eccentricity e. If 0 < e < 1 the conic is an ellipse, if e = 1 the conic is a parabola, and if e > 1 the conic is a hyperbola. If the distance to the focus is fixed and the directrix is a line at infinity, so the eccentricity is zero, then the conic is a circle.
It is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (48)

Related courses (7)

Related lectures (78)

Focus (geometry)

In geometry, focuses or foci (ˈfəʊkaɪ; : focus) are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an n-ellipse.

Conic section

A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting 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 it was sometimes called as 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.

Two-body problem

In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each object arises from the other one, and all other objects are ignored. The most prominent case of the classical two-body problem is the gravitational case (see also Kepler problem), arising in astronomy for predicting the orbits (or escapes from orbit) of objects such as satellites, planets, and stars.

MATH-126: Geometry for architects II

Ce cours traite des 3 sujets suivants : la perspective, la géométrie descriptive, et une initiation à la géométrie projective.

ENG-639: Dynamic programming and optimal control

This course provides an introduction to stochastic optimal control and dynamic programming (DP), with a variety of engineering
applications. The course focuses on the DP principle of optimality, and i

MATH-189: Mathematics

Ce cours a pour but de donner les fondements de mathématiques nécessaires à l'architecte contemporain évoluant dans une école polytechnique.

General Solution of Central Force MotionPHYS-101(k): General physics : mechanics

Explores the general solution of motion in a central force field, conservation of energy, Kepler's problem, and trajectory.

Dandelin Theorem: Ellipse ConstructionMATH-126: Geometry for architects II

Explores ellipse construction using the Dandelin theorem and properties of conics in space.

Geometry: Passage to SpaceMATH-126: Geometry for architects II

Explores the transition from 2D to 3D geometry, covering conical sections, spatial curves, and equivalent shapes.