Hey, some thing I know about, unlike surfboards.

Math is all about definitions. So suppose you have a plane, regular x and y coordinates. Then you can define things like lines, eg a horizonal line that goes through the point x=0 y=2 is made up of all the points that have coordinates like (2,a) where “a” can be any number.

A hyperbola is defined like this:

Pick two points, say (-2,0) and (2,0) [but they could really be any two points] call these P and Q.

You then pick some number, say 5, call this K.

You can measure the distance from any point on the plane to any other, for let d(P,Q) be the distance from P to Q for example.

Then the hyperbola is defined as all of the points that satisfy this equation: d(T,Q)-d(T,P)=K

Or in english I “decide” that a point is on the parabola if the distance from that point to P minus the distance from that point to Q is equal to K.

This turns out to give two curves like the butterfly board you mentioned.

A parabola is a little different, and only defines one curve. It’s the (x,y) coordinates that satisfy (a*t*t, 2*a*t) where “a” is some number you chose to determine the shaper of the parabola and “t” is any positive number.

I don’t know if you actually wanted to know this, but here it is.

I used these pages to jog my memory, there’s some pictures is my explination didn’t make sense.

http://mathworld.wolfram.com/Hyperbola.html

http://mathworld.wolfram.com/Parabola.html