This has been on my favorites lists for a long time. I've forgotten why.

Famous Curves Index

http://www-history.mcs.st-and.ac.uk/~history/Curves/Curves.html

This includes, for Halloween, the Witch of Agnesi curve

Y(X^2 + a^2) = a^3

-- SteveH - 31 Oct 2005

KtmGuest^?, thanks for the explanation!

-- CarolynJohnston - 30 Oct 2005

I am

-- CatherineJohnson - 30 Oct 2005

The logistic map is usually used to model ecological , economic, and other growth processes where there's some kind of limiting factor, e.g. rabbit populations which are limited by the amount of food available. x_n-1 is the initial population and has a value between 0 and 1, x_n is the value in the following time period and is also between 0 and 1, and r is the growth factor and can be any positive number.

What the map shows is that for any starting value of x:

• for 0 <= r < 1, the population will go to zero over time,
• for 1 < r <~ 3, the population will go to a single stable value,
• for 3 <~ r <~ 3.45 the population will oscillate between two stable values,
• for 3.45 <~ r <~ 3.57 the population will oscillate between 4, then 8, then 16, then 32, then ... stable values,
• for r ~ 3.57 the population becomes chaotic, i.e. there are no stable values for population, except for small pockets of stability.

-- KtmGuest - 30 Oct 2005

I LOVE THESE!!!!!!!!!!

-- CatherineJohnson - 30 Oct 2005

It's usually done with a loop of string that forms a triangle with one of the end points on the minor axis and the foci pins.

-- KDeRosa - 30 Oct 2005

The elipse trick is used by carpenters to make accurate elipses for archways and the like.

-- KDeRosa - 30 Oct 2005

some cool math animations

Check out this page of cool math animations by Lingfa Yang.

Most of the animations are of constructions of some fairly obscure algebraic curves, but they are cool anyway.

This one, the "Pursuit Curve", I've never heard of, but from the animation you can see pretty clearly what a Pursuit curve is about. The red dot is 'pursuing' the black dot that is moving on the y-axis (the red dot will never actually touch the y-axis though).

Does anyone know the significance of this one, the "Logistic Map'?

And check out this ellipse one. This illustrates one of the characteristics of an ellipse -- that if you tie a string to the two foci, and trace a pencil around the inside of the string (keeping it taut), then the shape you'll draw will be an ellipse.

(hat tip, Bernie)

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• Posted by: CarolynJohnston
• LogDate: Oct 29, 2005 @ 23:24

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