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Planes


A plane is the mathematical concept of an infinite, flat 2-dimensional 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, two-dimensional field within a three-dimensional space. An example of this would be a single triangle component of a 3-D 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:

\(Ax+By+Cz+D=0\)


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):

\(lx+my+nz=p\)

The equivalent vector form is \(r\cdot n=p\), where \(r\) is the position of a point on the plane, and \(n\) is a unit vector normal to the plane.


Surface Normal
Surface Normal

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:

\(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\)

Where \(a\), \(b\), and \(c\) are the intercepts on the \(x\), \(y\), and \(z\) axes, respectively.


Three-point 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):

\( \begin{vmatrix} x &y &z &1\\ x_1 &y_1 &z_1 &1\\ x_2 &y_2 &z_2 &1\\ x_3 &y_3 &z_3 &1\\ \end{vmatrix} =0\)


Parametric Form of Three-Point Equation

A plane that passes through the three points \(a,b,\) and \(c\) has the parametric equation:

\(r=a+λ(b-a)+μ(c-a)\)


Notes

(1)

A point \(x,y,z\) is in a plane containing point \((x_1,y_1,z_1)\) and perpendicular to a line with direction numbers {\(A,B,C\)} if and only if it satisfies the standard form equation:

\(A(x-x_1)+\)\(B(y-y_1)+\)\(C(z-z_1)=0\)

(2)

Where the angle between two planes is the angle between normals to those planes, the planes are parallel if and only if the dot product of the normals is \(\pm 1\). We can also state this in terms of the cosine of the the relative angle:

\(cos\,θ=\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\)



Analytic Geometry: Vectors