Prove a property of conics connecting ( variables on arc) focal radius $r$, pedal length /normal $d$ dropped onto tangent from focus and (constants) semi latus-rectum $p$ and eccentricity $e$ :

$$ \frac{2p}{ r}-\frac{p^2}{d^2} = 1- e^2 \tag1$$

which is a new $ (r-\,d) $ relation.

For a parabola special case $$ p= \frac{2 d^2}{r} $$

Are these properties hitherto known ?

The above (1) is intrinsic, based earlier curvature properties I had derived. We can relate to its particular canonical Newton polar form $ (r\,- \theta) , $ when major axis not along $x-$ axis $ \alpha \ne 0$ .

$$ \dfrac{p}r = 1-e \cos(\theta - \alpha) \,\tag2 $$

We obtain a $ (d-\theta) $ relation:

$$ \dfrac{p^2}{d^2} = e^2 -2e \cos (\theta-\alpha) +1\,\tag3 $$

The three equations take two variables at a time from $ (r,\theta, d\,)$ and link them with two constants $ (p,e) $.

John Bentin points out that Equn(1) the Dark Kepler Problem/Petr Blashke Ex 4 conics equation in pedal coordinates from a most recent publication AIP Wiki reference ( June 2017) !

$$ \dfrac{L^2}{d^2} - \dfrac{2 M}{r} = c \tag4 $$

which can be geometrically interpreted to (1) by adjusting physical (gravitational) terms.

Equation (1) is general for a conic.

In Pedal Equn Wiki assigned values $ (-1,0,1)$ give particular cases (ellipse,hyperbola and parabola) respectively to the following expression.

$$ (\frac{b^2}{d^2}-\frac{2a}{r}) \tag5 $$



This is a very nice result. It is un-obvious enough to have been overlooked hitherto—but I am no authority on this! I have sketched my proof below, for the case of an ellipse (the case of a hyperbola may be treated similarly), in confirmation of your claim.

Choosing cartesian axes as the axes of the ellipse, we may write a general point $T$ on an ellipse as $(a\cos\phi,b\sin\phi)$, where $a$ and $b$ are constants with $a\geqslant b$. (The parameter $\phi$ is chosen to avoid confusion with a polar angle $\theta$ or $\theta-\alpha$ in the question.) We have the usual relationships $b^2=a^2(1-e^2)$ and $p=b^2/a$. Using the standard formula for the distance $d$ of a point, here the focus $(ae,0)$, from a line, here the tangent at $T$, namely $bx\cos\phi+ay\sin\phi-ab=0$, we get, after some simplification, $$d^2=\frac{p^2}{1-e^2}\left(\frac2{1+e\cos\phi}-1\right).$$Also, a calculation of $r^2$ using Pythagoras, and more simplification, yields $$\frac pr=\frac{1-e^2}{1-e\cos\phi}.$$ Combining these formulae with further (algebraic) simplification finally gives the result $$\frac{2p}r-\frac{p^2}{d^2}=1-e^2.$$

Added later Some searching has dug up this page for the pedal equation of a conic. So it has to be said that your result isn't new.

  • $\begingroup$ Thanks. Blaschke's paper is published just a month and a half before so may be relatively new? ! :) $\endgroup$ – Narasimham Jul 13 '17 at 8:28
  • $\begingroup$ @Narasimham: It's not clear from the Wikipedia article whether the pedal equation for the ellipse was taken from Edwards (1892), Yates (1947), or (originally) Blaschke (2017); and the latter paper doesn't make this clear, either. If the equation isn't in Yates' book (which I could probably get hold of), then it points to Blaschke as the originator. In fact, Blaschke may well have written the Wikipedia article. Why not try emailing him? $\endgroup$ – John Bentin Jul 13 '17 at 13:13

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.