Planes

Figure 1.5: Example of a plane in $\mathbb{R}^3$.

The discussion of lines can be generalized to planes and hyperplanes in space $\mathbb{R}^{n}$. As with lines, an implicit homogeneous form of the equation of a plane exists, understood as the locus of points expressed by the homogeneous coordinate $\tilde{\mathbf{x}} \in \mathbb{R}^{n+1}$ with respect to $\mathbf{x}\in\mathbb{R}^{n}$:

$$\mathbf{m}^{\top}\tilde{\mathbf{x}}=0$$ (1.84)

The dot product between homogeneous coordinates always encodes hyperplanes.

Homogeneous coordinates are defined only up to a multiplicative factor, and it is therefore possible to impose an optional constraint: as for lines, the first $n$ parameters of the homogeneous coordinate may be regarded as forming a unit-length vector.

A generic plane, or hyperplane, is therefore the locus of points $\mathbf{x}\in\mathbb{R}^{n}$ satisfying the condition

$$\mathbf{x} \cdot \mathbf{n} - \rho = 0$$ (1.85)

where $\mathbf{n} \in \mathbb{R}^{n}$ is the normal to the plane and $\rho=0$ if and only if the plane passes through the origin. In the case of $\rho \neq 0$, an alternative representation of the plane is

$$\frac{1}{\rho} \mathbf{p} \cdot \mathbf{n} = 1$$ (1.86)

and another form of equation (1.85) found in the literature is

$$(\mathbf{x} - \mathbf{x}_0) \cdot \mathbf{n} = 0$$ (1.87)

with $\mathbf{x}_0 \in \mathbb{R}^n$ a generic point on the plane, from which the correspondence $\rho = \mathbf{x}_0 \cdot \mathbf{n}$ can be derived.

It should be remembered that the degrees of freedom are always exactly $n$.

When introduced, the normalization constraint $\vert\hat{\mathbf{n}}\vert=1$ represents a special case: under this condition, as in the case of lines, $\rho$ represents the minimum Euclidean distance between the plane and the origin.

If the plane (or hyperplane) is normalized, the distance between a generic point $\mathbf {p}$ and the plane is measured as

$$d = \vert \mathbf{p} \cdot \hat{\mathbf{n}} - \rho \vert$$ (1.88)

otherwise, as in the case of lines, the distance must be divided by $\Vert\mathbf{n} \Vert$.

The point $\mathbf {x}$ closest to a generic point $\mathbf {p}$ belonging to the hyperplane lies at the intersection of the direction line $\mathbf{n}$ passing through $\mathbf {p}$ and the plane itself:

$$\left\{ \begin{array}{l} \mathbf{p} + t \mathbf{n} = \mathb... ...\\ \mathbf{x} \cdot \mathbf{n} = \rho \\ \end{array}\right.$$ (1.89)

that is,

$$\mathbf{x} = \mathbf{p} - \frac{\mathbf{p} \cdot \mathbf{n} - \rho}{ \vert \mathbf{n} \vert^2 } \mathbf{n}$$ (1.90)

This formulation also applies to lines, as already seen.

As for the various generation methods, Section 4.6.3 will show how to obtain the least-squares regression of a set of points onto the equation of a plane.

As in the case of a line, the parameters of a plane in $\mathbb{R}^3$ can also be expressed using 3 polar coordinates (azimuth, zenith, and $\rho$):

$$x \sin \vartheta \cos \varphi + y \sin \vartheta \sin \varphi + z \cos \vartheta = \rho$$ (1.91)

the equation of the plane expressed in spherical polar coordinates (1.54).