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I recently encountered this problem wich I couldn't find the solution of, so I thought that maybe someone could help me. The problem is:

"Given an ellipse, $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ and the point A, $$A(x_a,y_a)$$ find the closest point (of the ellipse) to A"

The center of the ellipse is the origin, and also the ellipse it's not rotated, so this simplifies the question. To solve it, I started by going on Geogebra and represent the situation, and I came up with this:

https://www.geogebra.org/classic/dfgwpuqm

I added the point B, that lies on the perimeter of the ellipse, and I also added the tangent passing through that point. I then drew the perpendicular line to the tangent passing through A, and I found the point C, the intersection of the tangent and the perpendicular. If I'm not mistaken, the point C touches the ellipse when C and B are in the same spot, or in other words, when the perpendicular passes through B. After that I could not go any further, and I'll really appreciate if someone could help me solving this problem.

I'm just a guy that likes to solve problem like this one, and maybe it's easier than I thought. Anyway, thanks everybody that will try to help me.

P.S.: it's not requested by the problem, but is there a curve that describes the path of the point C? Again, thank you very much.

federico
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2 Answers2

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The problem is not analytically solvable, therefore I suggest numerical approach. The parametric equation of the ellipse is: \begin{align} \mathbf{x}=(a\cos t, b\sin t)~~~t\in [0,2\pi) \end{align} The tangential vector equals to \begin{align} \mathbf{t}=(-a\sin t, b\cos t) \end{align} the vector $\mathbf{o}$ connecting the point on the ellipse and the point $A$ yields \begin{align} \mathbf{o}=(a\cos t-x_a, b\sin t-y_a) \end{align} The proper value of parameter $t$, which defines the desired closest point, is obtainable from orthogonality condition: \begin{align} \mathbf{o}^{\text{T}}\mathbf{t}=(b^2-a^2)\sin t\cos t+ax_a\sin t - b y_a\cos t=0 \end{align} which has to be solved numerically for $t$, using Newton's method for example. For a good initial value an approximative formulae can be used (https://www.researchgate.net/publication/318875654_Point-to-ellipse_and_point-to-ellipsoid_distance_equation_analysis).

Vítězslav Štembera
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Denote $T(x_t,y_t)$ the nearest point to the given point $A,$ and lying on the ellipse.

To find $T$ by construction in GeoGebra, each of the following helps. I used 1. in the enclosed construction.

  1. The line $AT$ is normal to the ellipse at $T,$ perpendicular to the tangent at $T.$

  2. For an ellipse, normal line is the angle bissector of $\angle FTF',$ where $F,F'$ are foci of the ellipse.

enter image description here

user376343
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