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Topic: Curves Paradox? (Read 2294 times) |
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towr
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Re: Curves Paradox?
« Reply #25 on: Aug 4th, 2006, 1:34pm » |
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Something that occured to me. If you have 9 points that are on the intersection of three parallel lines with three other parallel lines, can be described in 8 numbers.
You only need the direction of both sets of lines (2 numbers), and the distances of the line to a the origin (2 x 3 numbers).
So that's 8 degrees of freedom, one too few for uniquely defining a cubic curve.
Similarly any set of 9 points which takes less than 9 degrees of freedom would, I propose, not uniquely define a cubic curve.
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| « Last Edit: Aug 4th, 2006, 1:37pm by towr » |
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Deedlit
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Re: Curves Paradox?
« Reply #26 on: Aug 4th, 2006, 3:51pm » |
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Keep in mind that whether a set of points defines a unique cubic curve is a property of that specific set of points, whereas degrees of freedom depend on which particular subspace we are saying that set of points belongs. For example, any set of nine points can be said to belong to a subspace in which 5 of the points are always present, and the other 4 can be arbitrary. This has 8 degrees of freedom.
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Barukh
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Re: Curves Paradox?
« Reply #27 on: Sep 22nd, 2006, 11:27pm » |
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I forgot to mention that this problem is called Cramer's Paradox.
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| « Last Edit: Sep 22nd, 2006, 11:28pm by Barukh » |
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