hyperbola
Sign in to savethumb|A hyperbola is an open curve with two branches, the intersection of a plane (geometry)|plane with both halves of a double cone. The plane does not have to be parallel to the axis of the cone; the hyperbola will be symmetrical in any case.|alt=The image shows a double cone in which a geometrical plane has sliced off parts of the top and bottom half; the boundary curve of the slice on the cone is the hyperbola. A double cone consists of two cones stacked point-to-point and sharing the same axis of rotation; it may be generated by rotating a line about an axis that passes through a point of
AI overview
A hyperbola is an open curve with two separate branches that forms when a plane cuts through both halves of a double cone at any angle. This shape is important in mathematics and science because it appears naturally in various physical phenomena and serves as a fundamental curve in geometry alongside circles, ellipses, and parabolas.
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Article
62 sectionsContents
- Etymology and history
- Definitions
- As locus of points
- Hyperbola with equation {{math|1=''y'' = ''A''/''x''}}
- By the directrix property
- Proof
- Construction of a directrix
- As plane section of a cone
- Pin and string construction
- Steiner generation of a hyperbola
- Inscribed angles for hyperbolas {{math|1=''y'' = ''a''/(''x'' − ''b'') + ''c''}} and the 3-point-form
- As an affine image of the unit hyperbola {{math|1=''x''<sup>2</sup> − ''y''<sup>2</sup> = 1}}
- Parametric representation
- Vertices
- Implicit representation
- Hyperbola in space
- As an affine image of the hyperbola {{math|1=''y'' = 1/''x''}}
- Tangent construction
- Point construction
- Tangent–asymptotes triangle
- Reciprocation of a circle
- Quadratic equation
- In Cartesian coordinates
- Equation
- Eccentricity
- Asymptotes
- Semi-latus rectum
- Tangent
- Rectangular hyperbola
- Parametric representation with hyperbolic sine/cosine
- Conjugate hyperbola
- In polar coordinates
- Origin at the focus
- Origin at the center
- Eccentricity
- Parametric equations
- Hyperbolic functions
- Properties
- Reflection property
- Midpoints of parallel chords
- Orthogonal tangents – orthoptic
- Pole-polar relation for a hyperbola
- Other properties
- Arc length
- Derived curves
- Elliptic coordinates
- Conic section analysis of the hyperbolic appearance of circles
- Applications
- Sundials
- Multilateration
- Path followed by a particle
- Korteweg–de Vries equation
- Angle trisection
- Efficient portfolio frontier
- Biochemistry
- Hyperbolas as plane sections of quadrics
- See also
- Other conic sections
- Other related topics
- Notes
- References
- External links
thumb|A hyperbola is an open curve with two branches, the intersection of a plane (geometry)|plane with both halves of a double cone. The plane does not have to be parallel to the axis of the cone; the hyperbola will be symmetrical in any case.|alt=The image shows a double cone in which a geometrical plane has sliced off parts of the top and bottom half; the boundary curve of the slice on the cone is the hyperbola. A double cone consists of two cones stacked point-to-point and sharing the same axis of rotation; it may be generated by rotating a line about an axis that passes through a point of the line. thumb|Hyperbola (red): features thumb|A lampshade with a circular rim casts a hyperbola-shaped shadow on a vertical wall. In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.
Besides being a conic section, a hyperbola can arise as the locus of points whose difference of distances to two fixed foci is constant, as a curve for each point of which the rays to two fixed foci are reflections across the tangent line at that point, or as the solution of certain bivariate quadratic equations such as the reciprocal relationship xy = 1. In practical applications, a hyperbola can arise as the path followed by the shadow of the tip of a sundial's gnomon, the shape of an open orbit such as that of a celestial object exceeding the escape velocity of the nearest gravitational body, or the scattering trajectory of a subatomic particle, among others.