Algebra :: nklein software

Finding the Perfect Hyperbola February 15th, 2010
Patrick Stein

For an application that I’m working on, I needed a way to scale quantities that range (theoretically) over the real numbers (though practically are probably between plus and minus three) into positive numbers. I wanted the function to be everywhere increasing, I wanted $f(0) = 1$ , and I wanted control of the derivative at $x = 0$ .

The easy choice is: $f(x) = e^{\alpha x}$ . This is monotonically increasing. $f(0) = 1$ and $f^\prime(0) = \alpha$ .

I needed to scale three such quantities and mush them together. I thought it’d be spiffy then to have three different functions that satisfy my criteria. The next logical choice was $f(x) = 1 + \mathrm{tanh} (\alpha x)$ . It is everywhere positive and increasing. And, it has $f^\prime(0) = \alpha$ .

Now, I needed third function that was always positive, always increasing, had $f(0) = 1$ and $f^\prime(0) = \alpha$ . One choice was: $f(x) = e^{e^{\alpha x} - 1}$ . But, that seemed like overkill. It also meant that I really had to keep my $\alpha$ tiny if I didn’t want to scale things into the stratosphere.

Playing with hyperbolas

So, I thought… why don’t I make a hyperbola, rotate it, and shift it so that the apex of one side of the hyperbola is at $(0,1)$ . And, I can adjust the parameters of the hyperbola so that $f'(0) = \alpha$ . After a variety of false starts where I tried to keep the hyperbola general until the very end ( $\frac{x^2}{a^2} - \frac{y^2}{b^2} = r^2$ , rotated by $\theta$ degrees, and shifted by $\beta$ ), I quickly got bogged down in six or seven incredibly ugly equations in eight or nine variables.

So, it was time to start trying to make it easy from the beginning. I noticed that if was going to rotate it by an angle $\theta$ in the clockwise direction, then I needed $\phi = \frac{\pi}{2} - \theta$ to be such that $\tan \phi = \alpha$ if my slope was going to work out right in the end. So, I’m looking at the basic triangle on the right then to determine all of my sines and cosines and such.

Based on that triangle, it was also obvious that the asymptote for my starting hyperbola had to have $\frac{b}{a} = \frac{1}{\alpha}$ . I played around a bit then with making $a = \alpha$ and $b = 1$ . In the end, I found things simplified sooner if I started with $a = 1$ and $b = \frac{1}{\alpha}$ .

I also needed $r$ to be such that the point $(-r,0)$ rotate up so that its $y$ -coordinate was $1$ . This meant that $r \sin \theta = 1$ or $r = \frac{1}{\sin \theta} = \sqrt{1 + \alpha^2}$ .

So, my starting hyperbola then was: $x^2 - \alpha^2 y^2 = 1 + \alpha^2$ .

From there, I had to rotate the $x$ and $y$ by $\theta$ in the clockwise direction. This gave me:

$(x\cos\theta - y\sin\theta)^2 - \alpha^2 (x\sin\theta + y\cos\theta)^2 = 1 + \alpha^2$

A little multiplying out leads to:

$(x^2\cos^2\theta - 2xy\sin\theta\cos\theta + y^2\sin^2\theta) \\ - \alpha^2(x^2\sin^2\theta + 2xy\sin\theta\cos\theta + y^2\cos^2\theta) = 1 + \alpha^2$

From there, using $cos^2\theta = \frac{\alpha^2}{1 + \alpha^2}$ , $sin^2\theta = \frac{1}{1 + \alpha^2}$ , and $\sin\theta\cos\theta = \frac{\alpha}{1 + \alpha^2}$ , we come to:

$\frac{-2\alpha(1 + \alpha^2)xy + ( 1 - \alpha^4 )y^2}{1 + \alpha^2} = -2\alpha{}xy + (1 - \alpha^2)y^2 = 1 + \alpha^2$

The only step remaining was to shift it all over so that when $y = 1$ , we end up with $x = 0$ . Plugging in $y = 1$ , we see that $-2x\alpha + 1 - \alpha^2 = 1 + \alpha$ . That equation balances when $x = -\alpha$ . We want it to balance when $x = 0$ , so we’re going to substitute $(x - \alpha)$ in place of the $x$ in the above equation to get:

$-2\alpha(x - \alpha)y + (1 - \alpha^2)y^2 = 1 + \alpha^2$

We can easily verify that the point $(0,1)$ is on the curve. And, we can implicitly differentiate the above to get:

$-2\alpha(x - \alpha)\frac{dy}{dx} - 2\alpha{}y + 2(1 - \alpha^2)y\frac{dy}{dx} = 0$
Plopping $x = 0$ and $y = 1$ in there, we find that $\frac{dy}{dx} = \alpha$ .

This is pretty good as it goes. The only step is to complete the square to find a nicer expression for $y$ in terms of $x$ . We start by adding $\left[ \frac{\alpha}{\sqrt{1 - \alpha^2}} (x - \alpha) \right]^2$ to both sides to get:

$\left[ \sqrt(1 - \alpha^2)y - \frac{\alpha}{\sqrt{1 - \alpha^2}(x - \alpha)} \right]^2 = 1 + \alpha^2 + \frac{\alpha}{1 - \alpha^2} (x-\alpha)^2$

This is easy enough to solve for $y$ by taking the square root of both sides and shuffling some things about:

$y = \frac{\alpha}{1 - \alpha^2}(x - \alpha) + \frac{1}{\sqrt{1 - \alpha^2}} \sqrt{1 + \alpha^2 + \frac{\alpha^2}{1 - \alpha^2}(x - \alpha)^2}$

Here are all three curves with $\alpha = \frac{1}{4}$ . The exponential is in black, the hyperbolic tangent is in red, and the hyperbola is in blue:

The first image on the page here was made with Zach’s Vecto library with some post-processing in the GIMP. (Here is the source file: hyperbola.lisp.) The second image was made entirely within the GIMP. And, the last image was made using Foo Plot and the GIMP.

Clifford Algebras for the Non-Mathie May 17th, 2009
Patrick Stein

I was at a party last night. I mentioned to someone that I was a math geek.

She asked, What kind of math are you into now?

I said, I really want to learn about Clifford Algebras.

She replied, What are they like?

Me, I did the deer in headlights thing. I had no idea of her math level. I had my doubts that she’d ever done Calculus. I would guess that the quadratic equation was the defining aspect of what she thought of as Algebra. I didn’t know where to start.

In thinking back now, I could have at least said something constructive.

What I Could Have Said

So, the algebra you learned about in high school was just the tip of a huge body of mathematics. If you take away the idea that you have to plug a number in for x and just look at the what you can do with the x’s still in there, there is a whole structure going on. At more advanced levels, mathematicians work with those structures and other structures like them.

Do you know what a vector is? One of the easiest ways to think about it is this. Suppose you’ve got a number on the number line. You can kinda think of that number as a one-dimensional vector. It tells you which direction to go (positive or negative) from zero and how far to go. Now, if you take something with more dimensions than a line, like a a two-dimensional surface or three-dimensional space, you can still have the idea of what direction to go from zero and how far to go. You just have to broaden your idea of which direction to go.

So, the algebra that you did through high school is all centered on having x represent a number (a one-dimensional vector). But, things get a lot hairier if you let x represent a three-dimensional or an eight-dimensional vector. People generally agree about how to add vectors. Now, you’ve got to pick some way to multiply two vectors together.

Clifford Algebras are one system for multiplying vectors together.

If No One is Hyperventilating, Continue…

You can multiply a number by a vector to get another vector. You can also multiply a vector by a vector to get a bivector. You can multiply a vector by a bivector to get a trivector. Etc. Actually, the etc. is misleading there. It doesn’t go on forever. With Clifford Algebras, you have to pick how many dimensions your vector has. You can’t multiply a three-dimensional vector by a four-dimensional vector.

Plus, if you started with a two-dimensional vector, then bivectors are as big as you get. If you multiply a two-dimensional bivector by a two-dimensional vector, you get a two-dimensional vector. If you multiply a two-dimensional bivector by another two-dimensional bivector, you just get a number.

But, all of this is probably more than you wanted to know…. at a party….

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