parabola
Sign in to savethumb|right|upright=1.36|Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward. thumb|The parabola is a member of the family of conic sections.
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A parabola is a curved shape that you get when you slice through a cone at a certain angle, and it's one of the fundamental curves studied in mathematics because it appears throughout nature and engineering. The curve extends infinitely in certain directions and has special mathematical properties that make it useful for applications like satellite dishes and projectile paths.
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Article
63 sectionsContents
- History
- Definition as a locus of points
- In a Cartesian coordinate system
- Axis of symmetry parallel to the ''y'' axis
- General position
- As a graph of a function
- Similarity to the unit parabola
- As a special conic section
- In polar coordinates
- Conic section and quadratic form
- Diagram, description, and definitions
- Derivation of quadratic equation
- Focal length
- Position of the focus
- Alternative proof with Dandelin spheres
- Proof of the reflective property
- Construction and definitions
- Deductions
- Other consequences
- Tangent bisection property
- Intersection of a tangent and perpendicular from focus
- Reflection of light striking the convex side
- Alternative proofs
- Pin and string construction
- Properties related to Pascal's theorem
- 4-points property
- 3-points–1-tangent property
- 2-points–2-tangents property
- Axis direction
- Steiner generation
- Parabola
- Dual parabola
- Inscribed angles and the 3-point form
- Pole–polar relation
- Tangent properties
- Two tangent properties related to the latus rectum
- Orthoptic property
- Lambert's theorem
- Facts related to chords and arcs {{anchor|Chords|Arcs}}
- Focal length calculated from parameters of a chord
- Area enclosed between a parabola and a chord
- Corollary concerning midpoints and endpoints of chords
- Arc length
- A geometrical construction to find a sector area
- Focal length and radius of curvature at the vertex
- As the affine image of the unit parabola
- Parametric representation
- Vertex
- Focal length and focus
- Implicit representation
- Parabola in space
- As quadratic Bézier curve
- Numerical integration
- As plane section of quadric
- As trisectrix
- Generalizations
- In the physical world
- Gallery
- See also
- Footnotes
- References
- Further reading
- External links
thumb|right|upright=1.36|Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward. thumb|The parabola is a member of the family of conic sections.
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.