The Second OctantDate: 04/03/2002 at 09:37:32 From: Kjetil Subject: Geometry Hi, Where is the second octant? We are always talking about the first octant, where x, y, and z are positive. But no one seems to know how to count the next octants. Is it when x > 0, y < 0 and z > 0 or x < 0, y > 0 and z > 0 ? Kjetil Date: 04/05/2002 at 14:10:49 From: Doctor Douglas Subject: Re: Geometry Hi, Kjetil, thanks for submitting your question to the Math Forum. By analogy with the two-dimensional case (x,y), we have the first coordinate being inverted first: (x,y) = (+,+) first quadrant (-,+) second quadrant (-,-) third quadrant [not (+,-) as in binary counting, (+,-) fourth quadrant because we want a continuous path] For the three-dimensional case: (x,y,z) = (+,+,+) first octant (-,+,+) second octant (-,-,+) third octant [convention is same as in 2D case] (+,-,+) fourth octant (+,+,-) fifth octant [here's a natural choice - for (-,+,-) sixth octant fifth through eighth just repeat (-,-,-) seventh octant first through fourth for negative (+,-,-) eighth octant z values] I think that this is the most common convention, and thus the second octant is identified with {x < 0 and y,z > 0}. However, there are other conventions that could be adopted, particularly for the fifth through the eight octants. For example, if it is important to preserve the "continous path" character among all eight octants (e.g., the fourth octant touches the third and the fifth octant), then the sequence might go like this: +++ : -++ : --+ : +-+ : +-- : --- : -+- : ++- In this last sequence, we see that we flip exactly one sign in going from one octant to the next. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/ |
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