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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.
transformation of a circular region into an approximately rectangular region
Figure 11: Beltrami's circle model.
Concentric circles and infinity
Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated.
A diagram of a double cone intersected by two green planes labeled PP (picture plane) and RP (reality plane), illustrating projected conic sections in projective geometry, with the apex and a point Omega marked.
Figure 5: Intersecting circles. If a circle with centre A has a point X inside, as well as a point Y outside, a second circle with centre B, and if X and Y lie on AB, then the circles have at least one common point on each side of AB.
Chord in a circle
Pascal's projective theoremThe 17th-century French mathematician Blaise Pascal proved that the three points (x, y, z) formed by intersecting the six lines that connect any six distinct points (A, B, C, D, E, F) on a circle are collinear.
Figure 10: Poincaré's circle model.
Understand how to calculate a circle's area with the help of chain and ruler
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John Machin
English mathematician

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