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Aug 9th, 2012, 05:19 PM
#2
Re: draw this?
The lines are given by endpoints of the form (0, nw), ((m-n)w, 0) for constant m and w, and for n ranging from 1 to to m-1. In the drawn case, m=16, one more than the number of lines drawn, and I'd estimate w (the width) at around 20 pixels. The curve is a little harder to figure out. Suppose instead of discrete lines we had a continuum of lines, with endpoints (0, d), (D-d, 0) for some constant D, d ranging from 0 to D. The equation of a line of that form is...
y(d) = d - xd/(D-d)
Imagine focusing on lines slightly to the side of y(d), say y(d+e) for e small. Intersecting these two algebraically one finds the point of intersection is ((D-d)(D-(d+e))/D, d(d+e)/D). In the limit as e goes to 0 the intersection point is ((D-d)^2 / D, d^2 / D). One can show these x and y values are on the curve
y = (D - sqrt(Dx))^2 / D, x between 0 and D
The limiting curve, as you take arbitrarily many points, is then like this (unfortunately the x and y axes have different spacing there, so it's a little distorted).
The time you enjoy wasting is not wasted time.
Bertrand Russell
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