How to find an ellipse , given 2 passing points and the tangents at them?

Question

Please answer to a question , how to find an ellipse which passes the 2 given points and has the given tangents at them. And one related question is that the given condition can decide just one ellipse which satisfies it?

Thank you in advance.

Edit : May be , I can say the first answer (by Mr. André Nicolas) is for the general case. Is there no special case where only a finite set of ellipses satisfies the condition?

Edit : According to the answers and comments , I can compute an arbitrary chosen ellipse for my condition (using the method by Mr.Patrick Da Silva). But there are possibly many others which satisfy my conditions. Am I right?

Answer 1

Certainly such an ellipse is not uniquely determined. For example, there are lots of ellipses with tangent lines $y=1$ at $(0,1)$ and $y=-1$ at $(0,-1)$: any ellipse $a^2x^2+y^2=1$.

I have not checked that an ellipse with the desired properties always exists. But by a suitable projective transformation we can make the two points nearest neighbours on a pair of parallel lines. After the projection, there is a circle with the desired property, as well as infinitely many non-circular ellipses, and another infinity of hyperbolas. Transform back. It follows that there are infinitely many conics with the desired property.

Answer 2

EDIT: As Rahul's comment points out, I assumed implicitly that the ellipse was centered at the origin. This is important ; for everything to work out well here, you need the two points to be linearly independent vectors, the center must be assumed at the origin and obviously the tangent lines must not got through the origin. Since there is no ellipse over which two linearly independent points have parallel tangent lines, we must also assume that the tangent lines are intersecting somewhere else than at infinity.

Actually what you need is the normal vectors. An ellipse is completed characterized by $3$ coefficients, namely the $a$, $b$ and $c$ that define $$ ax^2 + bxy + cy^2 = 1. $$ The gradient of the function $f(x) = ax^2 + bxy + cy^2$ gives you the normal vectors to the ellipse by computing $$ \nabla f(x) = \begin{bmatrix} 2ax + by \\ bx + 2cy \end{bmatrix} = \begin{bmatrix} 2a & b \\ b & 2c \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}. $$ You know the normal vectors at two linearly independent points in the plane (this is very important, otherwise your condition can represent more than one ellipse). Therefore, you can compute them at $X_1 = (x_1, y_1)$ and at $X_2 = (x_2, y_2)$ to get $(\alpha_1, \beta_1)$ and $(\alpha_2, \beta_2)$ the normal vectors at $X_1$ and $X_2$. To make sure your normal vector points "outside of the ellipse", project it on its underlying point. In other words, if $(\alpha_1, \beta_1)$ is the normal vector at $(x_1, y_1)$, then compute $x_1 \alpha_1 + x_2 \beta_2$. This should be non-zero ; if it's positive, the sign is good for an ellipse. If it's negative, it's wrong ; change it. All this gives you the linear system $$ \begin{bmatrix} 2a & b \\ b & 2c \end{bmatrix} \begin{bmatrix} x_1 & x_2 \\ y_1 & y_2 \end{bmatrix} = \begin{bmatrix} \alpha_1 & \alpha_2 \\ \beta_1 & \beta_2 \end{bmatrix}. $$ If you invert the matrix with the $x$'s and $y$'s (which I assumed was possible), you get the $a$'s, $b$'s and $c$'s.

Hope that helps,

Answer 3

There are infinitely many ellipses, even assuming that the two tangent lines intersect.

This follows from the very nice property of an ellipse:

If line $t$ is a tangent to an ellipse at point $P$, and if the focii of the ellipse are $A$ and $B$, then the ray $\overrightarrow{BP}$ reflected on $t$ passes through $A$.

Basically, if you shoot a laser from $B$ to any point on the ellipse, and the ellipse is a perfect mirror, then the reflected ray will pass through $A$!

This can be proven using the physics principle: light takes the route which takes the shortest time. Using the fact that locus of an ellipse is $PA + PB = $ constant, if light from $B$ reflected on the ellipse has to take the shortest time, it can be shown that it has to pass through $A$.

This gives us a way of generating multiple ellipses.

Suppose the given points are $P$ and $Q$ and the tangents ($p$ and $q$ respectively) intersect at $O$, also assume that $\angle{POQ}$ is not acute.

The two tangents divide the plane into four regions. Consider the region $R$ which contains both $P$ and $Q$ on its border.

Now reflect $P$ on the line $q$ at point $Q$. This line intersects $R$ and we get a half-line $S_P$. Similarly reflect $Q$ on the line $p$ at $P$, and we get half-line $S_Q$.

Now given a point $B_i$ (candidate for a focus), we pick a corresponding point $A_i$ (candidate for the other focus) as follows (based on the above reflection property):

Shoot lasers from $B_i$ to $P$ and $Q$ (assuming lines $p$ and $q$ are mirrors) and if they intersect, we pick the point of intersection to be $A_i$.

If it turns out that $PB_i + PA_i = QB_i + QA_i$, then we have our ellipse.

Now given a point $B_1$ on $S_Q$ the corresponding $A_1$ we get is the point $Q$ itself.

We have that $B_1Q + A_1Q \lt B_1P + A_1P$

For a point $B_2$ on $S_P$, the corresponding $A_2$ we get is the point $P$ itself.

We have that $B_2Q + A_2Q \gt B_2P + A_2P$

Now if we pick $B_1$ and $B_2$ sufficiently far from $P$ and $Q$, as we move from $B_1$ to $B_2$, along the segment $B_1 B_2$, there will be a point $B_3$ for which $B_3Q + A_3Q = B_3P + A_3P$ (by continuity arguments) and giving us an ellipse we need.

Now we can pick multiple lines parallel to $B_1B_2$ and use that to get multiple ellipses (at most two parallel lines can give the same ellipse).

If $\angle{POQ}$ is acute, we can pick $B_1$ and $B_2$ on $p$ and $q$ itself, with $P$ lying between $O$ and $B_1$ and $Q$ lying between $O$ and $B_2$ and $B_1$, $B_2$ being sufficiently far from $O$.

Answer 4

Here is a complete characterization of all possible ellipses that satisfy the given criteria.

Case 1: The two tangents through the specified points $P$ and $Q$ intersect at a point $O$ distinct from $P$ and $Q$. Then there is an affine transformation mapping $O$, $P$, and $Q$ to $(0,0)$, $(1,0)$, and $(0,1)$ respectively. Under this transformation, any ellipse satisfying the original criteria is mapped to an ellipse passing through $(1,0)$ and $(0,1)$ and tangent to the coordinate axes at those points. All such ellipses are of the form $(x-1)^2+(y-1)^2+2axy = 1$ with $-1 < a < 1$.

Here are some figures from my answer to a duplicate question that I had not realized was a duplicate. The ellipses are shown for $a=\frac12,0,-\frac12$.

Case 2: The two tangents are parallel and neither passes through both points. Then there is an affine transformation mapping $P$ and $Q$ to $(-1,0)$ and $(1,0)$, and the tangents to lines parallel to the $x$-axis. Now there is an infinite family of ellipses given by André Nicolas which satisfy these conditions, namely $ax^2 + y^2 = 1$ with $a > 0$.

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Update: See these two previous posts for a way to choose the best ellipse from this one-dimensional family of ellipses, where by "best" we mean the one with the least eccentricity:

• "...how to find the ellipse with minimal eccentricity"
• Roundest ellipse with specified tangents

Answer 5

A conic curve (as an ellipse) is in general given by 5 points. If 2 of these points get very close together, we can see that this condition in the limit is equivalent to give a point and the tangent at that point. So there are in general infinite solutions (2 points and 2 tangents are "equivalent" to 4 points and then we have another point left)

David Vergés Barcelona, Catalonia

Added: Projective transformations are powerful tools to treat this type of problems.