Equation of one branch of a hyperbola in general position

Question

Given a generic expression of a conic:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F=0$$

is there a way to write an expression for one of the branches as a function of the coefficients? I tried using the quadratic formula to get an expression for $y$: $$y=\frac{-(Bx+E)\pm \sqrt{(Bx+E)^2 - 4(C)(Ax^2 + Dx + F)}}{2C}$$

but this doesn't always work. Consider:

$$xy=1$$

Here, $A=0, B=1, C=0, D=0, E=0, F=0$, so $y=\frac{\cdot}{0}$, which isn't particularly helpful. In other cases it is not as bad, but still not what I'm looking for. E.g.

$$x^2 - y^2 - 1=0$$

Using the formula above, we get $$y=\pm \sqrt{x^2-1}$$ which seems nice, but $y=\sqrt{x^2-1}$ it is actually one half of each branch rather than one entire branch, as can be seen here.

http://www.wolframalpha.com/input/?i=plot%28x^2+-+y^2+-+1%3D0%29

http://www.wolframalpha.com/input/?i=plot%28y%3Dsqrt%28x^2-1%29%29

I am trying to draw one of these branches, so I need an ordered set of points along a predefined "grid" of either of the variables. Is it possible to do this?

Answer 1

First of all, if you want to draw a portion of the hyperbola, you typically just need parametric equations of the form $x = x(t)$, $y = y(t)$; you don't necessarily need to get $y$ as a function of $x$.

But, ignoring that quibble, I'll try to answer the question you asked.

The key is to first eliminate the $xy$ term in the implicit equation. In effect, you do this by rotating the coordinate system. This is really the same idea as Will Jagy used in his answer, but it might be easier to understand if I don't mention eigenvectors.

Suppose we introduce a $uv$ cooordinate system that is rotated by an angle $\theta$ (counterclockwise) from the $xy$ one. Then we have

$$x = u\cos\theta - v\sin\theta \quad ; \quad y = u\sin\theta + v\cos\theta$$

You can plug these $x$ and $y$ expressions into your original equation, and you'll get

$$\bar A u^2 + \bar B uv + \bar C v^2 + \bar Du + \bar Ev + \bar F = 0 $$

where

$$\bar A = A \cos^2\theta + 2B \sin \theta\cos\theta + C \sin^2\theta $$

$$\bar B = 2 B \cos 2\theta - (A - C) \sin 2\theta $$

$$\bar C = A \sin^2\theta - 2B \sin\theta\cos\theta + C \cos^2\theta $$

and so on. Now we just have to cleverly choose $\theta$ so that the $uv$ term disappears. Clearly, we'll get what we want if we choose $\theta$ so that $\bar B = 0$, and this means

$$ \tan2\theta = \frac{2B}{A - C}$$

After using this technique, we can now assume that the equation has the form

$$a u^2 + c v^2 - 2adu - 2cev + f = 0 $$

Note that I'm using $-2ad$ in place of $\bar D$ and $-2ce$ in place of $\bar E$, just to make the next step more convenient. And the next step is just some "complete the square" tricks. The equation can be written:

$$a (u^2 - 2du) + c (v^2 - 2ev) + f = 0 $$

In other words:

$$a (u^2 - 2du + d^2) + c (v^2 - 2ev + e^2) + f - ad^2 - ce^2 = 0 $$

which is

$$a (u - d)^2 + c (v - e)^2 = ad^2 + ce^2 - f $$

Now it should be clear that we have either an ellipse (if the signs of $a$ and $c$ are the same) or a hyperbola (if the signs are different). I'm ignoring the parabola case and several degenerate cases. The ellipse/hyperbola has its center at the point $(d,e)$ in the $uv$ coordinate system, and its axes of symmetry are parallel to the $u$ and $v$ axes.

At this point, if our curve turns out to be a hyperbola, you can use standard parametric equations (like the ones Wil Jagy gave) to trace out one branch or the other.

Answer 2

I think you are going to have a more satisfactory experience doing this: find the center $\vec{P} = (x_0, y_0)$ of your hyperbola. Find the eigenvectors of the matrix $$ \left( \begin{array}{cc} A & B/2 \\ B/2 & C \end{array} \right) $$ and normalize and choose order and $\pm$ so that with basis $\vec{u},\vec{v}$ you can then write your branch as $$ g(t) = \vec{P} + \vec{u} \cosh t + \vec{v} \sinh t. $$ Note that while $\vec{u},\vec{v}$ are perpendicular to each other they are not particularly of any length, in the way I write it above. If you prefer an orthonormal basis you then just put one constant scalar multiplication in front of the $\cosh$ term and one in front of the $\sinh$ term. It will all work out if you actually do have a hyperbola, which happens when $B^2 > 4 A C.$

I did teach this about 20 years ago, a whole section on translations and rotations of conic sections for a engineering calculus course. The memories are dim, but it does appear that it is best to find an orthonormal basis for the above matrix first, then find the center expressed in that basis, and end up with constant scalar coefficients of the hyperbolic trig terms.

Answer 3

Following Will Jagy's suggestion, here are some examples:

Example #1

Consider

$$x^2 - y^2 -1 = 0$$

($A=1$, $B=0$, $C=-1$, $D=0$, $E=0$, $F=-1$). From

$$p_c = \begin{pmatrix}x_c\\y_c\end{pmatrix} = \begin{pmatrix}\frac{BE-2CD}{4AC-B^2}\\ \frac{DB-2AE}{4AC-B^2}\end{pmatrix}$$

we have

$$p_c = \begin{pmatrix}0\\0\end{pmatrix}$$

Now, the eigenvalues/vectors of the matrix $$\begin{pmatrix}A & \frac{B}{2}\\\frac{B}{2} & C\end{pmatrix} = \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}$$ are $\lambda_1 = -1$, $\lambda_2 = 1$, $v_1 = \begin{pmatrix}0\\1\end{pmatrix}$, $v_2 = \begin{pmatrix}1\\0\end{pmatrix}$

From the parametric form

$$g(t)=p_c + v_1 cosh(t) + v_2 sinh(t)$$

we have

$$g(t) = \begin{pmatrix}0\\1\end{pmatrix} cosh(t) + \begin{pmatrix}1\\0\end{pmatrix} sinh(t)$$

which looks like

Now how would we draw the other branch? That is, how do you determine the values of the parameter that constitute each branch?

Example #2

Consider

$$7x^2 - 3y^2 - 25 = 0$$

($A=7$, $B=0$, $C=-3$, $D=0$, $E=0$, $F=-25$). From

$$p_c = \begin{pmatrix}x_c\\y_c\end{pmatrix} = \begin{pmatrix}\frac{BE-2CD}{4AC-B^2}\\ \frac{DB-2AE}{4AC-B^2}\end{pmatrix}$$

we have

$$p_c = \begin{pmatrix}0\\\frac{-25}{6}\end{pmatrix}$$

Now, the eigenvalues/vectors of the matrix $$\begin{pmatrix}A & \frac{B}{2}\\\frac{B}{2} & C\end{pmatrix} = \begin{pmatrix}7 & 0 \\ 0 & -3 \end{pmatrix}$$ are $\lambda_1 = -3$, $\lambda_2 = 7$, $v_1 = \begin{pmatrix}0\\1\end{pmatrix}$, $v_2 = \begin{pmatrix}1\\0\end{pmatrix}$

From the parametric form

$$g(t)=p_c + v_1 cosh(t) + v_2 sinh(t)$$

we have

$$g(t) = \begin{pmatrix}0\\ \frac{-25}{6}\end{pmatrix} + \begin{pmatrix}0\\1\end{pmatrix} cosh(t) + \begin{pmatrix}1\\0\end{pmatrix} sinh(t)$$