From a typical point inside an ellipse, how many points on the ellipse have normal to the ellipse? Someone asked me at school many years ago but I don't think I worked it out.
For every ellipse , there is a curve called ellipse evolute associated with it. The ellipse evolute is the locus of centers of curvature for . It is also the envelope of the normal lines of .
For a point inside , the number of points on where normal to can be either or . It depends on whether is lying outside, on or inside the ellipse evolute.
The graph below illustrates what happens when lies on the ellipse evolute (the black star shaped curve) and the three such that normal to .
Following is a brief analysis of the problem. For an alternate and more complete treatment, take a look at the paper Apollonius' ellipse and evolute revisited by Frederick Hartmann and Robert Jantzen.
Let use choose a coordinate system such that is given by the equation
Let be a point inside and be a point on . It is known that the normal of at is along the direction . The condition for normal to is given by
This is the equation for a hyperbola ( the orange hyperbola in above graph ) with lying on it. Since is inside and the two arms of the branch of holding extends to infinity. Each of the arm will intersect at least once. This means and intersected at least twice. Since five points determine a conic, the number of intersections between and is at most .
To determine the actual number of intersections, let us introduce a new coordinate system
We have 3 possible cases:
- When is near , . The quartic equation has either or real roots. Since we know intersect at least twice, there are four on such that is normal to .
- When is far away from , . The quartic equation has only real roots and hence there are only two that make normal to .
- On the special case , the quartic equation has 3 distinct real roots. One of them is a double root which corresponds to is tangent to at some points. There are three that make normal to .
As mentioned before, the picture above illustrates the case. The black star shaped curve is the ellipse evolute where . When is lying on it, one branch of the hyperbola will be touching the ellipse . If you move inside the ellipse evolute, the bottom branch of will move inwards too and will start to intersect at four places.
To obtain a simpler expression for the ellipse evolute, let , we have
Notes
ellipse evolute is a special case of a kind of curve called astroid.
The wiki page of evolute has the definition of center of curvature. It also has a nice animation showing the ellipse evolute as an envelop of the normals.
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Thankyou for the explanation. I remember concave diamonds, but I don't think I got this far before. – Empy2 Dec 19 '13 at 14:57