| Anna |
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. |
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| David
Loeffler |
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 |
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| Matthew
Smith |
And if you're familiar with spherical polar co-ordinates, these three pieces of information can just be r, q and f, 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 q and f 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
. q, the polar angle, is the angle ZOP, where Z is on the z-axis, so that q = cos-1(z/r). f, 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 f = tan-1(y/x). Matthew. |