https://www.youtube.com/watch?v=cfrQSzDfFoI
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The Cartesian equation of a hyperboloid is:
(x^2 + y^2)/a^2 - z^2/b^2 = 1
With the origin set at the center of the figure, define the x direction to be left/right and the z direction up/down. The y direction is then perpendicular to the board. To obtain the equation of the cuts in the board, take y to be zero, i.e., the equation in the plane of the board.
What results is the Cartesian equation of an hyperbola.
x^2/a^2 - z^2/b^2 = 1
When z = 0 , corresponding to a horizontal line through the center of the board, we have x = a, which determines the separation of the two curved lines forming the hyperpola.
For z nonzero, we have:
x^2 = a^2 * (1 + z^2/b^2)
so, as z increases the separation between the +x value and the -x value becomes greater and greater thus creating the characteristic flare of the hyperbola.
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When I next hear someone speaking in Hyperboles this will be the vision in my mind.Quote:
Originally Posted by Jon
Watching a college majorette doing her routine on stage in a cloud of vapor under ultraviolet light while she spins up holding her baton at a certain angle makes a lasting impression of a 3 dimensional hyperbola being created in the vapor