Hello there, I've just started diving into the mysterious world of 3D math, and I'm getting stuck at a certain point. I was wondering if some of you who are obviously much more knowledged than I am on these matters could explain something to me. In most texts, it is stated that "the normal [of a plane] points away from the origin, so the distance is negative. If the normal pointed toward the origin, the distance would be a positive value. Obviously if the plane goes through the origin then the distance would be equal to zero." The last bit, I understand. But why is the distance negative when the normal points away from the origin? And why does the cross product sometimes point one way, and sometimes point the other way? Isn't wether it points towards or away from the origin all dependent on the angle between the two vectors you perform the cross product on? (It seems to be when playing with this applet: http://physics.syr.edu/courses/java-suite/crosspro.html) So how can a plane face one way or the other then? If you take two vectors in the plane, it may face one way, but if you take two other vectors in the same plane, it may suddenly point the other way. Do I need to understand this? Is this just one of the things that happens, or am I completely thinking the wrong way? This matter is also confusing me on the matter of determining wether a point is in front of or behind a plane. I don't see -why- V (dot) N + D turns negative or positive... Anyone who can make me see the light, so to speak? Thanks in advance! [edited by - Bas Paap on July 11, 2002 7:18:28 PM]

Hold your right hand out in front of your body, thumb up, fingers outstretched. Now imaging three vectors... one coming up out of your thumb, one coming out of your fingers and one coming out of your palm. Let them all be orthogonal to each other (essentially meaning at right angles). You have now defined a right hand vector basis. Now imagine one of those vectors has its direction reversed... this is a now a left hand basis, which you can form with your left hand!

The reason why the cross product operation gives different signs sometimes is because of the order in which you consider the vectors in the operation. AxB =-(BxA ). If you reverse the order of the vectors, this is equivalent to computing the cross product (on the original order of the vectors) in a left handed basis. The cross product is though, a right handed operation, meaning that given your two vectors, align the first with your thumb and the second with your fingers... the result will point out of your palm.

As to your plane questions...

Define two vectors that lie in the plane (or define your plane by these two vectors). Using a right hand basis, determine the direction of the normal vector to the plane by taking the cross product of the two vectors. The direction will depend on the order in which you took the vectors and will therefore define the facing of the plane.

If you are computing a distance in the direction of the normal to the origin, from the plane, then that distance must be negative if the normal is pointing away from the origin and must be positive if the normal is pointing toward the origin.

I hope this helps.

Cheers,

Timkin

[edited by - Timkin on July 11, 2002 8:24:40 PM]

the sign of the distance also depends on the form of the plane equation you're using, i.e.

Ax+By+Cz+D=0 OR
Ax+By+Cz=D

where (A,B,C) is the normal and D is the distance from origin
and to state the obvious, both forms are perfectly valid with the only difference being the sign of D

[edited by - joanusdmentia on July 11, 2002 8:31:27 PM]

Excellent. I had gotten ahead a bit on this matter since posting this (one is on one''s own when the forums are down...), but your replies have confirmed my findings and cleared up what was still a mystery to me. Thank you both very much!

Expect me to be back, though... This 3D math basics chapter is big, and I''m not even halfway through yet... :-)