I am looking at the focus-focus definitions of the conics, i.e. defining them as the locus of points with the property that some function of the distance from the point to two foci is a constant.

For an ellipse the sum of distances is constant, and for a hyperbola the difference of distances is constant. The locus of points with the product of distances being constant is a Cassini oval, while the locus of points with the ratio of distances being constant is a circle. Is there a corresponding definition for a parabola? What other curves arise from simple functions of the distance to two points?

  • $\begingroup$ Since a parabola has only one focus, I think the short answer is no, there is no corresponding definition. $\endgroup$ – hardmath Dec 11 '13 at 14:47
  • $\begingroup$ See the circle's focus-focus definition. The foci in the new definition need not be the same as the focus in the focus-directrix definition. $\endgroup$ – Michael Biro Dec 11 '13 at 14:50
  • $\begingroup$ While the circle is a "degenerate" ellipse, the parabola is a boundary case between ellipse and hyperbola. Thus we are transitioning from a sum of distances to a difference of distances property, and using the same point "twice" as with the circle cannot help. $\endgroup$ – hardmath Dec 11 '13 at 15:00
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    $\begingroup$ Please take the time to actually read the question. The Apollonian definition for a circle does not use the same point "twice", it uses two different points - neither of which is the center of the circle. $\endgroup$ – Michael Biro Dec 11 '13 at 15:02
  • $\begingroup$ And neither of which is uniquely defined. I.e. although you can construct a circle in this way, the foci are no inherent property of the circle, are they? $\endgroup$ – MvG Dec 13 '13 at 16:25

Parabola is a special case of an ellipse or hyperbola, one focus is at infinity.

How do you define a "simple" function?

If the distances $d(P, F1)= u$ , $d(P, F2)= v$ , then apart from those you mentioned $( u+v, u-v, uv, \frac{u}{v} )$ there can be many functions that can be prescribed to remain constant to generate loci, it is up to your combinatorial imagination.

$u^2 + v^2$, $u^2 - v^2$, $\frac{u-v}{u+v}$, $u^2 + v^2 -u v$, $v^u$, $(u \log(v))$, $u e^{-v}$, $u^v + v^u$, $\frac{u+1}{v}$, $\frac{1}{u} +\frac{1}{v}$, $\frac{v-1}{u}$, $u \cos(v)$, $\frac{\cos(u)}{\cos(v)}$, $\frac{\sin(u)}{\sin(v)}$, .. &$c$., &$c$..

If $u$, $v$ are angles made with x-axis then trig relations may be also interesting I think.

EDIT 1: In the context you mentioned $ u + a\, v = b, (a,b) $ constant, are all Cartesian Ovals.


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