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Through the Looking Glass

Formula for Correction of Angles

I can look at far-away objects like the moon and the horizon through my crystal ball, but if I'm not looking directly though the center of the ball, then the positions that I see don't represent the true anglular distances between objects.  They are distorted a little bit.  Out of curiosity I decided to calculate what the difference is, so I've examined the case where one object is directly in the center of the view and another object is at an angle.  Here are my results.

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One-Point Folding

Graphing Parametric Surfaces in Octave

The graph is supposed to illustrate a surface that is formed by taking a piece of paper cut into a circle, which is then bent by putting a pencil tip at the center and pulling up on opposite points along a diameter.  The equation for the outer rim is calculated by first starting with spherical coordinates F(phi,theta) and a path of the form s(t)=(phi(t),theta(t)) where phi is longitude and theta is latitude.  The path is theta = t and phi(t) = -p cos(2t) + pi/4, with p to be determined later.  We compose F(s(t)) and compute arc length numerically for several values of p ranging from 0 to pi/4.  The length is determined to be approximately 2*pi when p = 0.5487.

Categories : mathematical

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