Math 2.0

Geometry Definition

Parabola

Builds on Euclidean planeEuclidean distanceLineDistance from a point to a line

Definition

Let FF be a point of the Euclidean plane ℝ2\mathbb{R}^2, and let ℓ\ell be a line that does not contain FF. Let d(P,F)d(P, F) be the Euclidean distance, and let d(P,ℓ)d(P, \ell) be the distance from PP to ℓ\ell. The parabola with the focus FF and the directrix ℓ\ell is the set { P∈ℝ2  :  d(P,F)=d(P,ℓ) }.\{\, P \in \mathbb{R}^2 \;:\; d(P, F) = d(P, \ell) \,\}.

Remarks

For F=(a,0)F = (a, 0) with a>0a > 0 and ℓ\ell the line x=−ax = -a, the parabola is the set of the points with y2=4axy^2 = 4 a x. For real numbers p≠0p \neq 0, qq and rr, the graph of y=px2+qx+ry = p x^2 + q x + r is a parabola. The eccentricity of each parabola is 11. Any two parabolas are similar. Menaechmus studied conic sections in the 4th century BC. Archimedes found the area of a parabolic segment, and the name is due to Apollonius.

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To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Euclidean distance, concept, no evidence levelEuclidean distanceconceptEuclidean plane, object, no evidence levelEuclidean planeobjectLine, concept, no evidence levelLineconceptNormal vector, concept, no evidence levelNormal vectorconceptDistance from a point to a line, statement, no evidence levelDistance from a point…statementParabola, definition, no evidence levelParaboladefinition
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  1. Euclidean distance concept
  2. Euclidean plane object
  3. Line concept
  4. Normal vector concept
  5. Distance from a point to a line statement
  6. Parabola definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Parabola, Definition as a locus of points
  2. Web https://en.wikipedia.org/wiki/Parabola, History
  3. Web https://en.wikipedia.org/wiki/Conic_section, Standard forms in Cartesian coordinates
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WritingW1 stated
MSC51N20
DefinesParabola
UsesEuclidean plane, Euclidean distance, Line, Distance from a point to a line

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