ellipse

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Eccentricity of conic sectionsThe eccentricity of a conic section completely characterizes its shape. For example, all circles have zero eccentricity, and all parabolas have unit eccentricity; hence, all circles (and all parabolas) have the same shape, only varying in size. (Under appropriate magnification they are indistinguishable.) In contrast, ellipses and hyperbolas vary greatly in shape.

Projective conic sectionsThe conic sections (ellipse, parabola, and hyperbola) can be generated by projecting the circle formed by the intersection of a cone with a plane (the reality plane, or RP) perpendicular to the cone's central axis. The image of the circle is projected onto a plane (the projective plane, or PP) that is oriented at the same angle as the cutting plane (Ω) passing through the apex (“eye”) of the double cone. In this example, the orientation of Ω produces an ellipse in PP.

Figure 5: Ellipse and the major (or auxiliary) circle. The two foci of the ellipse are F and F′. A line that is drawn perpendicular to the x-axis at S cuts the ellipse and the circle at P and P′, respectively.

elliptic paraboloid

Figure 4: The standard form of the ellipse (see text).

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Shah Jahan. Taj Mahal. Mughal architecture. Emperor Shah Jahan fifth Mughal Emperor (reigned 1628-1658) India, Himachal Pradesh, Basohli or Jammu and Kashmir, Mankot, circa 1690 Drawings; Opaque watercolor, gold, and ink on paper (see notes)

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The Lost Colony of Roanoke, North Carolina, where 115 people mysteriously disappeared c. 1590. John White discovers the word Croatoan carved onto a tree upon his return to the deserted Roanoke Colony in 1590

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ellipse: Related Content

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Photos

Eccentricity of conic sectionsThe eccentricity of a conic section completely characterizes its shape. For example, all circles have zero eccentricity, and all parabolas have unit eccentricity; hence, all circles (and all parabolas) have the same shape, only varying in size. (Under appropriate magnification they are indistinguishable.) In contrast, ellipses and hyperbolas vary greatly in shape.

Projective conic sectionsThe conic sections (ellipse, parabola, and hyperbola) can be generated by projecting the circle formed by the intersection of a cone with a plane (the reality plane, or RP) perpendicular to the cone's central axis. The image of the circle is projected onto a plane (the projective plane, or PP) that is oriented at the same angle as the cutting plane (Ω) passing through the apex (“eye”) of the double cone. In this example, the orientation of Ω produces an ellipse in PP.

Figure 5: Ellipse and the major (or auxiliary) circle. The two foci of the ellipse are F and F′. A line that is drawn perpendicular to the x-axis at S cuts the ellipse and the circle at P and P′, respectively.

elliptic paraboloid

Figure 4: The standard form of the ellipse (see text).

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Topics

eccentric circle minor axis major axis semiminor axis semimajor axis apse conic section hyperbola parabola circle

Quiz

Equations written on blackboard

Numbers and Mathematics