Carrotsticks
Retired
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- Jun 29, 2009
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So I know most people here have drawn this thing
at SOME point in their lives out of pure boredom.
The other day I was just thinking about it and I thought 'Suppose I were to make the interval K approach 0, and thus have an infinitely large number of lines and intervals, what smooth continuous curve does it form?'
My instincts point towards an astroid (http://en.wikipedia.org/wiki/Astroid) but I can't think of a way to prove this.
Also, suppose that the square in the diagram had side length a. Assuming it is an astroid, if I wanted to find the ratio of Red:White, I'm guessing I would integrate the function parametrically, and put it over a^2?
Any ideas on how to find the equation of a curve. defined by the point of intersections of infinitely large number of lines?
The only idea I have so far is to utilise the Method of Exhaustion (http://en.wikipedia.org/wiki/Method_of_exhaustion) and to take a limiting case as k -> 0, but I have no idea how to set up the initial equation.
![Shape.jpg](/data/attachments/17/17861-5562c1da03b420f5ff1815dbb10890e9.jpg)
The other day I was just thinking about it and I thought 'Suppose I were to make the interval K approach 0, and thus have an infinitely large number of lines and intervals, what smooth continuous curve does it form?'
My instincts point towards an astroid (http://en.wikipedia.org/wiki/Astroid) but I can't think of a way to prove this.
Also, suppose that the square in the diagram had side length a. Assuming it is an astroid, if I wanted to find the ratio of Red:White, I'm guessing I would integrate the function parametrically, and put it over a^2?
Any ideas on how to find the equation of a curve. defined by the point of intersections of infinitely large number of lines?
The only idea I have so far is to utilise the Method of Exhaustion (http://en.wikipedia.org/wiki/Method_of_exhaustion) and to take a limiting case as k -> 0, but I have no idea how to set up the initial equation.