# Are the first 3 parameters that describe a 3D plane actually a 3d vector?

A 3d plane is typically defined as `a,b,c,d`. Are `a,b,c` actually the `x,y,z` coordinates of a 3d vector, with `d` defining the rotation of the plane, something like axis-angle rotation data?

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The four-variable representation of a plane is the coefficients in the equality

ax + by + cz = d

This can be seen as N = (a, b, c) being a normal vector and d being a distance from the coordinate origin (in units of N), and we can also write this equation as N·P = d, where P = (x, y, z).

This representation does not allow defining a specific “origin of the plane” — mathematical planes don't have origins.

If you change the = to < or >, you describe a "half-space", which can be used for things such as an infinite floor in a physics engine; the opposite half-space is obtained by negating both N and d.

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Good answer. Just one correction: d is more like a squared distance (unless (a,b,c) has length 1, which is often the case but not guaranteed). –  Sam Hocevar Sep 21 '12 at 14:02
@SamHocevar Squared? This is all linear. I think you're thinking of what I phrased as that d is in units of N. –  Kevin Reid Sep 21 '12 at 14:07
oh, it's all right then, sorry! –  Sam Hocevar Sep 21 '12 at 14:09

"Typically" is a quite subjective word, in my experience there are different way to describe a plane in a 3D space that are more common because of the properties that such constructions show.

About your question, there is away to use 4 real values to determine a plane in a 3D space. As you pointed out, a,b,c may be the components of a vector that is perpendicular to the desired plane. If N =(a,b,c) is our perpendicular vector, you may find a point in your plane that is P = d N for some d real and positive. Here you say that d is the distance from the origin in term of N; if N is a unit vector, then d is the distance between the origin and your plane in the way that the term "distance" is commonly meant.

Surprisingly you can define any possible oriented plane bacause you can use a negative values of d; doing so you loose the direct meaning of d as distance until you put it in an absolute value (|d|).

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"Typically" is a subjective word. "Tipically" isn't a word of any kind. (Sorry for nitpicking, but I couldn't resist since you even went and emphasized it.) –  Ilmari Karonen Sep 21 '12 at 18:52