4
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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
  • 3
    $\begingroup$ 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
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.