h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

it´s incredible what can be found at the guide.
i´m serious..i love your article,well, at least i can use it

Glad you find it useful.

Perhaps you could add a section on how to derive formulas like this using integral calculus, if you get time.

By the way, it looks like π doesn't work any more, they seem to have got rid of the ability to do HTML entries so you have to use GuideML. There's probably a GuideML equivalent - or you could use an image.

I don't have any control over this entry any more... we'll just have to wait until the editor notices...

I know your formula is correct, but I always have a problem imagining the toroid. It's obviously not as far around the inside as it is around the outside. You'd think, if you cut it and straightened it, you'd have angled ends. But then, if you cut it into infinite slices, I suppose they are each perfect prisms?

Kind of related, I could never understand (although I accepted it) that an oblique section through a cone is a circle. I would have expected an egg shape.

Maybe it's just me.

Any way - gret little article. It will help someone with their homework I'm sure.
Awu.

No. If you cut the toroid and straightened it, you'd have a cylinder, because in straightening it the outside shortens by the same amount as the inside lengthens by.

Neh?

An oblique section through a cone is an ellipse isn't it?
I agree though, I always thought it would be an egg-shape (one big end, one little end, like a cam).

What is an 'egg-shape' anyway? Is there a formula for it?

A related question...
What shaped cam produces a sine wave linear motion?

Just wonderin'
Dan

Yes - I meant ellipse. Glad I'm not the only one who imagined it to be "egg shaped".

Are you thinking of a cam that raises and lowers, say, a stylus vertically, with the paper moving horizontally? I often walk down the corridor swinging my arms, wondering what the locus of my hand would be!

Maybe we could reconstruct it, graphically, by starting with a sine wave and projecting it on to the cam shape. I'll play with that.

An ellipse is a deformed circle.

You can draw a graph of a circle using two parametric equations. By modifying the parametric equations, you can deform the circle into an ellipse. You can then integrate to find its area.

I don't know anything beyond that.

Another way of representing an ellipse is with the equation x^2/a^2 + y^2/b^2 = 1 (where ^ represents an exponent... I don't feel like looking up the GuideML to do it.)

a and b are the maximum width and height of the ellipse, if they're the same then it's a circle (and you can easily multiply through by a^2 to get the more familiar equation for a circle.) There's a more general formula for an ellipse on rotated and/or moved axes, and a still more general formula which holds for any conic section, but I don't trust my memory on those.

One way to draw or think about an ellipse is to have two points (nails or pins perhaps) which are the foci, and have a string with each end attached to a focus. Then if you move a pencil around so that it's always holding the string taut, you'll draw an ellipse - the length of the string, and hence the sum of the distances between the pencil and the two foci are always the same.