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I've been trying to figure out how to create an ellipse mathematically given the following with no luck.

The ellipse should be tangent to a given circle with known center(h,k) and radius(r). The ellipse is also tangent to a vertical line at point Q. The center of the ellipse's center will have the same y coordinate as point Q. I am given that a/b=e so I do not know exactly what a or b are of the ellipse. I also know the ellipse will be inside of the given circle and the tangent point will happen in the 1st quadrant of the ellipse somewhere.

The problem I keep running into is the fact I don't know what a or b are, just the ratio.

Some point P is where the ellipse and circle intersect.

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  • $\begingroup$ Hint: if $Q=(x_0,y_0)$, then the center of the ellipse would be $(h,y_0)$. $\endgroup$ – Isko10986 Nov 15 '16 at 21:54
  • $\begingroup$ Edit: Misread your response. The ellipse and the circle are not tangent at the peak of the circle $\endgroup$ – United13 Nov 15 '16 at 21:58
  • $\begingroup$ From there, just get the distance of $(h,y_0)$ from Q and from the nearest point on the circle. Those will be your semi-major/minor axes, hence your a/b. $\endgroup$ – Isko10986 Nov 15 '16 at 22:01
  • $\begingroup$ and btw, if $Q$ is inside[outside] the circle, your ellipse is inside[outside] the circle; if $Q$ is on the circle, then your ellipse is the circle. $\endgroup$ – Isko10986 Nov 15 '16 at 22:04
  • $\begingroup$ Q is not on the circle but it also is inside the circle. The point of intersection between the ellipse and the circle is at some other point P. $\endgroup$ – United13 Nov 15 '16 at 22:10
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Let's see what we have.

  • The circle is $(X-h)^2+(Y-k)^2=r^2$.
  • The ellipse is $((X-c)/b)^2+((Y-y_Q)/a)^2=1$, i.e. axis-aligned with $b$ as the horizontal semiaxis and $a$ as the vertical semi-axis, and with $y_Q$ as the $y$-coordinate of the center. $c$ is the unknown $x$ coordinate of the center.
  • As $Q$ lies on the ellipse, $((x_Q-c)/b)^2=1$ so $(x_Q-c)^2=b^2$ and $c=x_Q\pm b$.
  • The equation of the ellipse can also be written as a polynomial $$a^2X^2-2a^2cX+a^2c^2+b^2Y^2-2b^2y_QY+b^2y_Q^2-a^2b^2=0$$ or using a matrix and homogeneous coordinates as $$(X,Y,1)\cdot\begin{pmatrix} a^2 & 0 & -a^2c \\ 0 & b^2 & -b^2y_Q \\ -a^2c & -b^2y_Q & a^2c^2+b^2y_Q^2 - a^2b^2 \end{pmatrix}\cdot\begin{pmatrix}X\\Y\\1\end{pmatrix}=0\;.$$ If $P=(x_P,y_P)$ is a point on that conic, then another point $(X,Y)$ lies on the tangent in $P$ if $$(X,Y,1)\cdot\begin{pmatrix} a^2 & 0 & -a^2c \\ 0 & b^2 & -b^2y_Q \\ -a^2c & -b^2y_Q & a^2c^2+b^2y_Q^2 - a^2b^2 \end{pmatrix}\cdot\begin{pmatrix}x_P\\y_P\\1\end{pmatrix}=0\;$$ so that tangent can be written as the equation $$a^2(x_P-c)X + b^2(y_P-y_Q)Y = a^2c(x_P-c)+b^2y_Q(y_P-y_Q)+a^2b^2\;.$$
  • The same can be done for the circle, to obtain the tangent to the circle in point $P$. I'll leave this as an exercise.
  • Here we have two equations of a line, which I'll abbreviate to $\lambda_1X+\mu_1Y=\tau_1$ and $\lambda_2X+\mu_2Y=\tau_2$. If these are to describe the same line, then they have to be multiples of one another. Which means $\lambda_1\tau_2=\lambda_2\tau_1$ and $\mu_1\tau_2=\mu_2\tau_1$. On the other hand, as both of these tangents pass through the same point $P$, it is enough to check that their directions agree, which can be accomplished by $\lambda_1\mu_2=\lambda_2\mu_1$ ignoring the right hand sides.
  • You also have equations saying that $P$ is a point on the ellipse resp. circle. Simply plug $P$ in the corresponding equation. (Without this, the line I called the tangent would only be the polar).

Now you have 5 polynomial equations:

  1. $a=eb$ (known ratio $e$)
  2. $(x_Q-c)^2=b^2$ ($Q$ lies on ellipse)
  3. $(x_P-h)^2+(y_P-k)^2=r^2$ ($P$ lies on circle)
  4. $a^2x_P^2-2a^2cx_P\cdots-a^2b^2=0$ ($P$ lies on ellipse)
  5. $\lambda_1\mu_2=\lambda_2\mu_1$ (same tangent directions)

As I understand your question, these involve 6 known quantities ($h,k,r,x_Q,y_Q,e$) and 5 unknowns ($a,b,c,x_P,y_P$). So I'd say that there is a good chance that the above equations will lead to a few distinct solutions. As many of them look non-linear, I'd not expect a single unique solution, though. One can now use a computer algebra system to enumerate solutions satisfying all of these polynomial equations.

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    $\begingroup$ Good and thorough work ! A little remark: the OP had no idea of the meaning of $a$, $b$ and $e$, which is a problem, because he has given a formula $a/b=e$; but if $a$, $b$ are interpreted as the half lengths of the axes, $a/b$ cannot be interpreted as the eccentricity (the formula linking $a$ and $b$ and $e$ is $e=c/a=\sqrt{a^2-b^2}/a$). But it is not harmful if we consider that the ratio $a/b$, called $e$, is a constant, without calling it "eccentricity". $\endgroup$ – Jean Marie Nov 20 '16 at 22:00
  • $\begingroup$ @JeanMarie: You are right, thanks for pointing this out. I've modified my answer to avoid calling this eccentricity. In case of actual eccentricity, the polynomial equation in question should read $a^2e^2=a^2-b^2$ as your $c$ is again different from my $c$. $\endgroup$ – MvG Nov 20 '16 at 23:02
  • $\begingroup$ I have written a somewhat simplified solution. I would appreciate to have your checking/opinion on it. $\endgroup$ – Jean Marie Nov 21 '16 at 13:52
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The solution of @MvG is general. I propose here a simplified version in order to improve our understanding and thus the degrees of freedom we have.

Let us refer to the figure below. I have considered WLOG that the circle is the unit circle: center in $(0,0)$, radius 1.

I have taken the following conventions, slightly different from the OP and MvG. I call $(x_C,y_0)$ the coordinates of the center of the ellipse; I attribute the names $a$ and $b$ respectively, to the horizontal and vertical semi-major axis lengths, with $\rho$ defined by $$\tag{0}\rho:=\dfrac{a}{b}.$$ ($\rho$ has been taken in replacement of $e$ in order to avoid ambiguities with the classical use of $e$ for eccentricity).

Take note of the relationship:

$$\tag{1}x_c+a=x_0.$$

Let $P$ the point of contact of the circle and the ellipse. If $\theta$ is the polar angle of $P$, its coordinates are $(\cos\theta,\sin\theta)$.

Let us reformulate the problem as follows:

Show that, being fixed $x_0,y_0$ and $x_c$, for each value of $\theta$, there exist an ellipse centered in $C(x_c=x_0-a, y_0)$ with horizontal semi-major axis $CQ$, that is internally tangent to the circle in $P$.

Proof: The equation of the ellipse is (by taking $(0)$ into account): $$\tag{2}\dfrac{(x-x_c)^2}{a^2}+\dfrac{(y-y_0)^2}{b^2}=1 \ \ \ \ \iff \ \ \ \ (x-x_c)^2+\rho^2(y-y_0)^2=\rho^2 b^2.$$

The fact that, at point $P(x,y)=(\cos\theta,\sin\theta)$, the gradient $V=\pmatrix{(x-x_c)\\ \rho^2(y-y_0)}$ of the ellipse is proportional to the gradient of the circle is expressed in the following way:

$$\begin{vmatrix}\cos\theta-x_c&\cos\theta\\ \rho^2(\sin\theta-y_0)&\sin\theta\end{vmatrix}=0 \ \ \iff $$

$$\tag{3}(\cos\theta-x_c)\sin\theta=\rho^2 \cos\theta(\sin\theta-y_0)$$

Besides, we have to express that point $P$ belongs to the ellipse. This gives (using $(2)$) the constraint

$$(\cos\theta-x_c)^2+\rho^2(\sin\theta-y_0)^2=\rho^2 b^2 \ \ \ \ \iff$$

$$\tag{4}(\cos\theta-x_c)^2=\rho^2 (b^2-(\sin\theta-y_0)^2).$$ Eliminating $e^2$ by taking the quotient of (4) by (3), we obtain a relationship where every parameter has a value, giving the following formula for $b$:

$$\tag{5}b^2=(\sin\theta-y_0)^2\left(1+\dfrac{\cos\theta-x_c}{\tan \theta (\sin\theta-y_0)}\right).$$

Thus, knowing $b$, we have the lacking information for building our ellipse.

Remark: One can wonder if the RHS of $(5)$ is always positive. A little analysis shows that this positivity constraint is equivalent to $x_0 \cos \theta+ y_0 \sin \theta \leq 1$ which is true. enter image description here

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