The Second Octant

Date: 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/