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Conic sections

Conic sections · Diagrams · Complete reading

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Four plane cuts of a double cone

Four 3D schematics: a plane parallel to the base gives a circle; a tilted plane through one nappe gives an ellipse; a plane parallel to a generator gives a parabola; a vertical plane through both nappes gives two hyperbola branches. Blue marks each intersection and orange its cutting plane. The finite cones truncate open curves; circles appear foreshortened in the 3D projection.

Four 3D schematics: a plane parallel to the base gives a circle; a tilted plane through one nappe gives an ellipse; a plane parallel to a generator gives a parabola; a vertical plane through both nappes gives two hyperbola branches. Blue marks each intersection and orange its cutting plane. The finite cones truncate open curves; circles appear foreshortened in the 3D projection.

Vertical parabola sign cases

Vertical parabolas in both sign cases: positive a opens up, negative a opens down. Basic vertex (0,0), symmetry x=0.

Vertical parabolas in both sign cases: positive a opens up, negative a opens down. Basic vertex (0,0), symmetry x=0.

Vertical parabola translations

Vertical parabolas in both sign cases: positive a opens up, negative a opens down. Translated vertex (h,k), symmetry x=h. Illustration h=2,k=1.

Vertical parabolas in both sign cases: positive a opens up, negative a opens down. Translated vertex (h,k), symmetry x=h. Illustration h=2,k=1.

Horizontal parabola sign cases

Horizontal parabolas in both sign cases: positive a opens right, negative a opens left. Basic vertex (0,0), symmetry y=0.

Horizontal parabolas in both sign cases: positive a opens right, negative a opens left. Basic vertex (0,0), symmetry y=0.

Horizontal parabola translations

Horizontal parabolas in both sign cases: positive a opens right, negative a opens left. Translated vertex (h,k), symmetry y=k. Illustration h=2,k=1.

Horizontal parabolas in both sign cases: positive a opens right, negative a opens left. Translated vertex (h,k), symmetry y=k. Illustration h=2,k=1.

From a unit circle to an ellipse

Three stages: unit circle, horizontal stretch by positive a replacing x with x/a, then vertical stretch by positive b replacing y with y/b. Result x²/a²+y²/b²=1. Example factors a=2,b=3; equal axis scales.

Three stages: unit circle, horizontal stretch by positive a replacing x with x/a, then vertical stretch by positive b replacing y with y/b. Result x²/a²+y²/b²=1. Example factors a=2,b=3; equal axis scales.

Ellipse axes and translation

Two panels: origin-centred ellipse with A(a,0), A′(−a,0), B(0,b), B′(0,−b); lengths 2a,2b when a>b. Translated ellipse centred at (h,k), symmetry x=h,y=k, endpoints h±a,k±b and semi-axes a,b. Numerical examples explicitly labelled.

Two panels: origin-centred ellipse with A(a,0), A′(−a,0), B(0,b), B′(0,−b); lengths 2a,2b when a>b. Translated ellipse centred at (h,k), symmetry x=h,y=k, endpoints h±a,k±b and semi-axes a,b. Numerical examples explicitly labelled.

Horizontal hyperbola and translation

Two panels: x²/a²−y²/b²=1 has centre(0,0), vertices (±a,0), symmetry x=0,y=0 and asymptotes y=±(b/a)x. Translation by h,k gives centre(h,k), vertices (h±a,k), symmetry x=h,y=k and asymptotes y=k±(b/a)(x−h). Source illustration assumes h>0,k>0; figures explicitly use a=2,b=1,h=4,k=2.

Two panels: x²/a²−y²/b²=1 has centre(0,0), vertices (±a,0), symmetry x=0,y=0 and asymptotes y=±(b/a)x. Translation by h,k gives centre(h,k), vertices (h±a,k), symmetry x=h,y=k and asymptotes y=k±(b/a)(x−h). Source illustration assumes h>0,k>0; figures explicitly use a=2,b=1,h=4,k=2.

Vertical hyperbola and translation

Basic vertical hyperbola y²/b²−x²/a²=1 has centre(0,0), vertices (0,±b), symmetry x=0,y=0, asymptotes y=±(b/a)x. Translated form has centre(h,k), vertices (h,k±b), symmetry x=h,y=k and asymptotes y=k±(b/a)(x−h). Illustrations use a=3,b=1 and, in the translated case, positive h=3,k=2.

Basic vertical hyperbola y²/b²−x²/a²=1 has centre(0,0), vertices (0,±b), symmetry x=0,y=0, asymptotes y=±(b/a)x. Translated form has centre(h,k), vertices (h,k±b), symmetry x=h,y=k and asymptotes y=k±(b/a)(x−h). Illustrations use a=3,b=1 and, in the translated case, positive h=3,k=2.

Reciprocal model for the appendix

The rectangular hyperbola y=1/x, equivalently xy=1, has branches in quadrants I and III and asymptotes x=0,y=0. These asymptotes are approached in limits; they contain no curve points.

The rectangular hyperbola y=1/x, equivalently xy=1, has branches in quadrants I and III and asymptotes x=0,y=0. These asymptotes are approached in limits; they contain no curve points.

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