Shortest Distance to Ellipse from Point

[androsnich]

Shortest Distance to Ellipse from Point,
Or: What is the Line Perpendicular to the Ellipse?


What is a good method to determine the shortest distance from a point to an Ellipse? This is similar to finding the line through a given point that is perpendicular to an ellipse.

To make it simple, I have the ellipse centered at the origin, (0,0), and the major axis along x. Also the point is always in quadrant I, and is always outside the ellipse.

Thanks for your help,
Harry

 

[TheTick]

One possible analytical approach:

Calculate formula for a circle that is tangent to the ellipse and centered on the point of interest. If you have some advanced CAD with sketch functions like SolidWorks available, it can do this for you.

Someday, someone may kill you with your own gun, but they should have to beat you to death with it because it is empty.

 

[androsnich]

I forgot to say, I will be solving this in Fortran (or some other language). I am looking for a closed form solution.


TheTick:

You give me an interesting idea - rolling a circle over the ellipse until it intersects.

I have tried calculating the formula for the normal line (to the ellipse) and solving for the normal line that intersects the point. But I got bogged down in the math.

Harry

 

[TheTick]

Mathematically, you are likely to get a pair of solutions when solving for intersection of line that is normal to ellipse. Still, it shouldn't be too hard to sort which one is closest.

Some offhand ideas about how to approach the problem:
The formula for a line tangent at any given point would be the derivative of the ellipse. The line perpendicular to the tangent has a slope =-1/(tangent slope).

I'll be back. Gotta tend to discontented baby.

 

[TheTick]

New approach:

reduce position on the ellipse to a single parameter function, t= 0 to 1. For each t, the distance from a point on the ellipse (x(t), y(t)) to the point of interest (X,Y) is sqrt((X-x(t))^2 + (Y-y(t))^2). The closest point is where the derivative of the distance function is zero.

Someday, someone may kill you with your own gun, but they should have to beat you to death with it because it is empty.