y = ax2 + bx + c
Click below to see one of the Associated curves.
|Definitions of the Associated curves||Evolute|
|Involute 1||Involute 2|
|Inverse curve wrt origin||Inverse wrt another circle|
|Pedal curve wrt origin||Pedal wrt another point|
|Negative pedal curve wrt origin||Negative pedal wrt another point|
|Caustic wrt horizontal rays||Caustic curve wrt another point|
In fact the geometrical methods of ruler and compass constructions cannot solve this (but Menaechmus did not know this). Menaechmus solved it by finding the intersection of the two parabolas x2 = y and y2 = 2x.
Euclid wrote about the parabola and it was given its present name by Apollonius. The focus and directrix of a parabola were considered by Pappus.
Pascal considered the parabola as a projection of a circle and Galileo showed that projectiles follow parabolic paths.
Gregory and Newton considered the properties of a parabola which bring parallel rays of light to a focus.
The pedal of the parabola with its vertex as pedal point is a cissoid. The pedal of the parabola with its focus as pedal point is a straight line. With the foot of the directrix as pedal point it is a right strophoid (an oblique strophoid for any other point of the directrix). The pedal curve when the pedal point is the image of the focus in the directrix is a Trisectrix of Maclaurin.
The evolute of the parabola is Neile's parabola. From a point above the evolute three normals can be drawn to the parabola, while only one normal can be drawn to the parabola from a point below the evolute.
If the focus of the parabola is taken as the centre of inversion, the parabola inverts to a cardioid. If the vertex of the parabola is taken as the centre of inversion, the parabola inverts to a Cissoid of Diocles.
The caustic of the parabola with the rays perpendicular to the axis of the parabola is Tschirnhaus's Cubic.
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