This is a kind of simple question, but it gives me hard time:
An ellipse is given in coordinate system. It passes points $(a, 0)$, $(0, b)$, $(-a, 0)$, $(0, -b)$, where $a$ and $b$ are positive constants. A point $P$ on this ellipse is at the distance $l$ from $(a, 0)$, measured along ellipse, clockwise.
(i) What are coordinates of the point P?
(ii) What is the equation of the normal to the ellipse passing though point P?
$l$ must obviously be between $0$ and the circumference of the ellipse.
This problem arose in the context of certain data visualization problem (visualization of tree-like structures), and an explicit formulas involving only math functions available in general purpose programming language would be of great help. However, it looks to me it is not possible.
The original problem is about getting elliptical version of the diagram below:
To find the point that's a distance $l$ along the ellipse, you'll have to use numerical methods. The coordinates of the point can't be obtained by simple formulas.
Since this is just for graphics, I suppose the point doesn't need to be very accurate. So, you could just approximate the ellipse by a polyline, and calculate arc lengths on the polyline.
At the point $(a\cos\theta, b\sin\theta)$ on the ellipse, the normal vector is in the direction $(b\cos\theta, a\sin\theta)$.