# What exactly is a plane?

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Hi, I understand a plane is a flat surface that is orientated in 3d space. But does a plane have position, eg can two planes can be parallel? Or is a plane have just an orientation, and goes on to infinity? thanks

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Well, you can have a finite or infinite plane. You need at least 3 points to identify a plane in 3D space. Two planes can be parallel, if you think about it.

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AFAIK planes are always considered to be infinite. A plane has the property that it divides 3d space into two half-spaces, to the front and back of the plane respectively. A hyperplane is a generalization of a plane to higher dimensions; it has the same property in that a hyperplane of dimension N divides RN into two half-spaces.

A line in 2d has the same property; I'm not sure though whether the term hyperplane can be used for a 2d line (math experts?).

There are a couple of different ways to represent a (hyper)plane. One is as the equation:

ax+by+cz+d = 0

All points [x, y, z] that satisfy the equation are on the plane.

You might recognize the 'ax+by+cz' part as the dot product. The vector [a, b, c] is the normal to the plane. So another way to represent the plane is:

Dot(p, n) = d

Where p is a point, n is the normal, and d is the negative of the 'd' value in the other equation. All points p which satisfy this equation are on the plane. This leads to a useful function:

f(p) = Dot(p, n) - d

Which returns the signed distance from p to the plane. Note that we're assuming that n is unit-length. If it is not, the plane is still valid, but the distance is scaled.

A plane equation of either type can be easily derived from either a point and a normal, or three points that form two linearly independent vectors.

Anyway, as with any topic in math there's more that could be said, but maybe that will help clear things up a little.

[Edit: Looking at your post again, I didn't actually answer all your questions. A plane can have any orientation and be positioned anywhere in space. If you think of the plane in normal-distance form, the normal can point in any direction, and the plane can be 'pushed' any distance along that normal. As silver suggested, two planes with colinear normals are parallel and perhaps coincident. If they are not parallel, they intersect in a line.]

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A plane is defined with a normal, and the shortest distance from the plane to the origin. So you can have two parallel infinite planes but with a different distance to the origin.
The normal of the plane basically defines two perpendicular vectors which are in the plane. By taking a linear combination of these two vectors, you can specify each point in the plane.

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Two planes can be parallel, because you can have two planes at the same orientation but passing through different points. Think about being in an infinite room. The floor and ceiling are both then planes which are parallel.

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As for finite planes, these are usually refered to as "polygons", apart from the ones with curvy bits, which are refered to as "shapes" [wink]

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Quote:
 Original post by johnnyBravoHi, I understand a plane is a flat surface that is orientated in 3d space.But does a plane have position, eg can two planes can be parallel?Or is a plane have just an orientation, and goes on to infinity?

It is a good question, and both answers happen to be true. They just define different entities that are both commonly named "planes".

In a vector space, all you have to go around are vectors. No points. When you think about it, a vector isn't so much defined by its beginning and end points rather than the difference between the two. You can translate a vector as much as you want, it won't affect the vector's coordinates. So, in a vector space, you aren't concerned with endpoint issues, only with the vector's inherent coordinates. Or, if you wish, one of the endpoints is always fixed in (0,0,0...).

In a vector space, a "plane" is defined as the set of vectors that can be produced out of a linear combination of two (different) vectors:

P = { w = λ*u + μ*v | u ∈ Rn, u ∈ Rn, u != v, λ ∈ R, &mu ∈ R }

Since (0,0,...) is always in the plane (λ = μ = 0), all the plane has to go for itself is its orientation (or, dually, its base vectors). Note that if you had more than two vectors forming the set's base, then you would have an "hyperplane" ( a plane is merely an hyperplane of dimension 2 ). An hyperplane is a lower-dimensional "slice" of your whole (n-dimensional) space, that goes on to infinity.

Now, in an affine space, there are such a things as points (in fact, an affine space can be defined as a vector space plus an origin point). Vectors are still defined as in a vector space (obviously), but a point plus a vector produces a point (which a vector plus a vector is still a vector). Since you do have an origin point as a given, you can make more points.

In an affine space, planes are more what you would expect of them. They'll be defined by a point and a pair of base vectors (or a normal vector). NOW, you can have two parallel, non-identical planes, having the same normal vector, but not sharing any points.

Hope that wasn't too confusing. Algebra is fun. [smile]

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