
Re: Hyperbolas in nature
Posted:
May 13, 1996 9:11 PM


In article <Pine.3.89.9605121846.A63120100000@oregano.Princeton.EDU>, conway@math.Princeton.EDU (John Conway) writes: > > > On 10 May 1996, C. Kenneth Fan wrote: > > > I think I've got another one: > > > > Hold a large hoola hoop around yourself, look straight forward, and close one of > > your eyes. (Don't look down directly at the hoop, just view it vertically > > askance.) Then, chances are you'll have a hyperbola across your field of > > vision...If you've got just incredibly overexagerated posture, you might > > catch a parabola, or even an ellipse, but for me, I'd end up cheating by > > tilting my head down or I'd break my back to see the ellipse this way. > > > > Ken > > > This one supposes that one's visual field has been identified with > a plane, which admittedly is the usual assumption, but usually it's > really being made only of a small portion of the visual field, and here > it's a large one, which would more naturally be identified with a > hemisphere (or, at least, SOME portion of a sphere). > > John Conwaqy >
I was thinking about this, and thought it would be reasonably close so long as one didn't demand that one see a large section of a hyperbola.
But, it also made me wonder about drawing circles on a canvas in perspective. Let's suppose that the observer's vantage is such that the circle appears as an ellipse on the canvas. What is the simplest way of constructing that ellipse?
I thought of one way, which isn't very elegant, and is specifically for the case where the line of sight is perpendicular to the canvas through the center of the circle. It involves constructing a geometric mean.
Another way, is to use a pen light and shine it on the canvas.
The one book I saw on perspective by artists said to, "eyeball it."
I think it's neat how if one draws just the outline of a tilted circle on a canvas, the observer cannot tell whether the circle is tilted up or down. However, if you just add the center of the circle, it becomes possible to tell because the center of the circle is not located at the center of the ellipse.
While on this subject, I might as well add that the book had a "Renaissance" solution to the problem of drawing a brick wall in perspective so that the bricks all had the same dimensions. I think it's beautiful and goes like this: Mark off equal intervals on one vertical edge of the brick wall and connect these to a vanishing point. Now draw a diagonal on the wall and see where it intersects the radial lines just drawn.
In fact, the whole subject of perspective in art would probably make a good topic to study in high school...it could culminate in proving that conic sections appear as conic sections no matter from where observed.
Ken

