Plans

Figure 1.5: An example of plane in $\mathbb{R}^3$.

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

$$\mathbf{m}^{\top}\tilde{\mathbf{x}}=0$$ (1.48)

The scalar product between homogeneous coordinates always encodes hyperplanes.

Homogeneous coordinates are known up to a multiplicative factor, and therefore an optional constraint can be imposed: similar to lines, one can think that the first $n$ parameters of the homogeneous coordinate form a unit-length vector.

A generic plane, or hyperplane, is thus the set of points $\mathbf{x} \in \mathbb{R}^{n}$ that satisfy the condition

$$\mathbf{x} \cdot \mathbf{n} - \rho = 0$$ (1.49)

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

$$\frac{1}{\rho} \mathbf{p} \cdot \mathbf{n} = 1$$ (1.50)

and another expression of equation (1.49) that can be found in the literature is

$$(\mathbf{x} - \mathbf{x}_0) \cdot \mathbf{n} = 0$$ (1.51)

with $\mathbf{x}_0 \in \mathbb{R}^n$ a generic point of 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 and only $n$.

When introduced, the normalization constraint $\vert\hat{\mathbf{n}}\vert=1$ represents a particular case: under this condition, as in the case of lines, $\rho$ assumes the meaning of 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

$$d = \vert \mathbf{p} \cdot \hat{\mathbf{n}} - \rho \vert$$ (1.52)

otherwise, as in the case of lines, it is necessary to divide the distance by $\Vert\mathbf{n} \Vert$.

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

$$\left\{ \begin{array}{l} \mathbf{p} + t \mathbf{n} = \mathb... ...\\ \mathbf{x} \cdot \mathbf{n} = \rho \\ \end{array}\right.$$ (1.53)

or in

$$\mathbf{x} = \mathbf{p} - \frac{\mathbf{p} \cdot \mathbf{n} - \rho}{ \vert \mathbf{n} \vert^2 } \mathbf{n}$$ (1.54)

This formulation is also applicable to lines, as already seen.

As for the various methods for generation, in section 3.6.3 it will be shown how to obtain the least squares regression of a set of points to the plane equation.

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

$$x \sin \vartheta \cos \varphi + y \sin \vartheta \sin \varphi + z \cos \vartheta = \rho$$ (1.55)

the equation of the plane expressed in spherical polar coordinates (1.18).