Dear Jayaram,

please tell something more about the initial data in the problem.

Best regards,Yuri

The problem is depth profiling of pot holes. Only chord lenghs are available.

You may need to determine the focus points on the ellipse, which are drawn from the upper vertex of the minor axis and stand on the major axis. The length of the major axis is the sum of the distances from the focus points to any point on the ellipse, and also to the vertices of the chord.

Dear Jayaram,

Firstly you may find the coordinates (of intersection points) of the chord and the ellipse, and use two point form for find for length of the chord. 

Best regards,

Wiwat Wanicharpichat

Dear Wiwat Wanicharpichat,

The chord length is known, therefore, the coordinates of the points that intersect the ellipse is also known. Now  what needed to be known is the length of the major axis parallel to the chord of the known length.

Dear  Jayaram,

There are a lot of solutions (not uniqe). We may find the mid-point of the given chord and find the line with perpendicular to the chord. The center of this ellipse must appears in this line.

By definition of ellipse: Ellipse is a plane curve such that the sums of the distances of each point in its periphery from two fixed points, the foci, are equal.

If we given a center (point) on the perpendicular line (above) we can find two foci on the line which containing the center and parallel to the given chord, then we have a major axis and also have the length of the major axis of an ellipse.

Regards,

Wiwat Wanicharpichat

Dear Jayaram,

if you know lengths of at least 3 chords parallel to major axis and the distances between these chords you can solve the problem. Using the standard equation x^2/a^2+y^2/b^2=1 (a stands for the major axis) you can write down three conditions.

Say, you have lengths 2x_1>2x_2>2x_3 of chords (on the one side from major axis) and the distances d between first and second chord, D between first and third chord. Assuming that y_1 is the distance (unknown) between center and the first chord you can write down:

y_1^2=b^2-b^2/a^2.x_1^2;

(y_1+d)^2=b^2-b^2/a^2.x_2^2;

(y_1+D)^2=b^2-b^2/a^2.x_3^2.

Then you exclude y_1 (subtracting works good) and get 2 equations on 2 parameters a,b. Theoretically it will help, I hope.

Regards, Yuri

Thank you. The problem is, i know only chord length. Now an ellipse should be cpnstructed.