Anna

Posted on Saturday, 01 May, 2004 - 01:43 pm:

I'm interested to know what the mimimum number of pieces of imformation needed to discribe a plane in 3D space is. I was told 4, using polar
co-ordinates. Could someone explain to me what these are, or tell me if it possible to describe a plane in 4 numbers without them.

David Loeffler

Posted on Saturday, 01 May, 2004 - 02:04 pm:

For all planes not passing through the origin you can get away with 3. A plane not containing the origin is uniquely specified by knowing the location of the point on it nearest the origin.

David

Matthew Smith

Posted on Saturday, 01 May, 2004 - 04:06 pm:

And if you're familiar with spherical polar co-ordinates, these three pieces of information can just be

r

θ

and

ϕ

, the co-ordinates of this point nearest to the origin. In fact, this system takes care of planes passing through the origin as well, as

r = 0

but

θ

and

ϕ

could define the direction of the normal to the plane.

Spherical polar co-ordinates are an extension of polar co-ordinates to three dimensions. Let $P$ , the point of interest, be at the point specified by Cartesian co-ordinates $(x, y, z)$ . Let $O$ be the origin, $(0, 0, 0)$ . Then $r$ , the radial distance, is the distance $OP$ , so that $r = \sqrt{(x^{2} + y^{2} + z^{2})}$ . $θ$ , the polar angle, is the angle $ZOP$ , where $Z$ is on the $z$ -axis, so that $θ = \cos^{- 1} (z / r)$ . $ϕ$ , the azimuthal angle, is the angle $XOQ$ , where $Q$ is the projection of $P$ on to the $x$ - $y$ plane (and $X$ on the $x$ -axis), so that $ϕ = \tan^{- 1} (y / x)$ .

Matthew.