PlanesA plane is the mathematical concept of an infinite, flat 2dimensional field. For example, the computer screen is aligned on a plane. What we want to talk about here though, are planes in space; that is, an arbitrarily oriented, twodimensional field within a threedimensional space. An example of this would be a single triangle component of a 3D graphic model, which lies along a plane. This is especially important for gaming and graphics systems such as collision detection and lighting. We need to be able to determine the exact orientation of the plane in space in order to add realistic effects. General Form of a Plane
The general form of a plane in Cartesian coordinates is given by the equation:
Normal Form and Vector Form of a Plane
Generally, when we refer to a normal, we refer to a unit vector that is perpendicular to the specified plane. These are oftern called
surface normals
or simply normals.The normal form equation of a plane specifies the plane in terms of the direction cosines \({l, n, m}\)
of the normal and the length \(p\) of the normal (assumed to be 1):
Any point \(q\) is on the plane if and only if the dot product of a unit vector normal to the plane and the normalized vector \(\overrightarrow{rq}\) is zero. Often, some acceptable limit of closeness to zero (ε) is used. Intercept Form of a Plane
The intercept form of a plane is:
Threepoint Form of a Plane
The equation of a plane that passes through the three points \(x_1, y_1, z_1\), \(x_2, y_2, z_2\), and \(x_3, y_3, z_3\) is easily
expressed in matrix form (refer to the linear algebra section on determinants):
Parametric Form of ThreePoint Equation
A plane that passes through the three points \(a,b,\) and \(c\) has the parametric equation:
Notes
(1)


Analytic Geometry: Vectors 