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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Mona Lisa, oil on wood panel by Leonardo da Vinci, c. 1503-06; in the Louvre, Paris, France. 77 x 53 cm.

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Artist interpretation of a Space meteoroid impact. Meteor impact. Asteroid, End of the world, danger, destruction, dinosaur extinct, Judgement Day, Planet Earth, Doomsday Predictions, comet

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Chinese pictograph, calligraphy tablet of Huang Tingjian, a famous calligrapher in Ancient Song Dynasty. The background of Chinese cultural elements.

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Marriage. A couple getting married during a church wedding ceremony. (religion, ceremony, bride, groom)

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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

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

Quiz

Equations written on blackboard

Numbers and Mathematics