More on Planes and Systems

Lesson 2PM attached every plane in $\mathbb{R}^3$ to a single linear expression and identified its normal vector. That description is enough for asking whether a given point lies on the plane, but several natural questions involve the plane only incidentally. A computer rendering a scene has to decide, for every pair of objects, whether a flat surface between them blocks the line of sight; a parametric animation of a ball rolling along a tilted roof has to stay on the surface at every frame. Neither question is answered by finding a point of the plane. The first needs an algebraic test for which side of the plane a point lies on, and the second needs a description that produces points of the plane rather than one that checks them. Both live a short algebraic step from the material we already have, and this lesson walks the step.

Parametric Description of a Plane

The Parametric Equation of a Plane from Lesson 2PM wrote every point of the plane through three non-collinear points $A$, $B$, $C$ as a barycentric combination $\mathbf{x} = r\mathbf{a} + s\mathbf{b} + t\mathbf{c}$ with $r + s + t = 1$. Using the constraint to eliminate $r = 1 - s - t$ rearranges that combination into

with base point $\mathbf{p} = \mathbf{a}$ and direction vectors $\mathbf{e} = \mathbf{b} - \mathbf{a}$, $\mathbf{e}' = \mathbf{c} - \mathbf{a}$ lying inside the plane. The three-point form and this “base point plus two directions” form carry the same information, but the latter is the natural input whenever the plane comes to us through a point on it together with two independent directions it contains, rather than through three named vertices.

The pair $(s, t)$ is a coordinate system inside the plane, with axes pointing along $\mathbf{e}$ and $\mathbf{e}'$. These internal axes need not be perpendicular, and when the plane is viewed from a steep angle the induced grid can look distorted, in the same way that a tilted surface appears foreshortened from the side.

The figure above realises the example $x + 2y + 3z = 4$ from the next subsection: the base point is $P = (4, 0, 0)$, the direction arrows are $\mathbf{e} = (-2, 1, 0)$ and $\mathbf{e}' = (-3, 0, 1)$, and the sample point $Q = P + \mathbf{e} + \mathbf{e}' = (-1, 1, 1)$ is reached by the dashed parameter route $P \to P + \mathbf{e} \to P + \mathbf{e} + \mathbf{e}'$. In the internal $(s, t)$-coordinates of the plane, $Q$ sits at $(s, t) = (1, 1)$.

Translating Between Forms

The equational form $\mathbf{a} \cdot \mathbf{x} + c = 0$ and the parametric form $\mathbf{p} + s\mathbf{e} + t\mathbf{e}'$ answer different questions, and most practical work with planes involves passing from one to the other. The mechanical content of each conversion is short enough to absorb through a worked example.

The construction tacitly assumes that $A, B, C$ are non-collinear, and that hypothesis is easiest to check directly on the displacement vectors: three points $A, B, C$ lie on a common line precisely when one of $\overrightarrow{AB}, \overrightarrow{AC}$ is a scalar multiple of the other, with a positive multiple putting $B$ and $C$ on the same side of $A$ and a negative multiple putting them on opposite sides. Otherwise the two displacements span a genuine plane through $A$, and the procedure of the previous example goes through.

Starting from a Point and a Normal

A third natural input packages a plane as a point $P$ lying on it together with a vector $\mathbf{n}$ perpendicular to it. The equational form is immediate from the perpendicularity of $\mathbf{n}$ to every displacement inside the plane:

Passing to a parametric form requires two linearly independent directions orthogonal to $\mathbf{n}$. A reliable recipe, which sidesteps the cross product and so generalises verbatim to $\mathbb{R}^n$, is to read off two auxiliary points of the plane by fixing two of the three coordinates at convenient values (typically $0$ and $1$) and solving the equation for the third. Repeating with a different coordinate held constant gives a second auxiliary point, and a quick collinearity check on the two displacements from $P$ certifies that they are independent.

The lattice in the figure is the image of the integer $(s, t)$-grid under the parametrisation $\mathbf{x}(s, t) = \mathbf{p} + s\mathbf{e} + t\mathbf{e}'$ of the plane $x + 2y + 3z = 4$ from Example 41. The two families of gridlines are the fibres $\{s = \text{const}\}$ and $\{t = \text{const}\}$, each itself a line inside the plane, and they meet at the lattice points.

Traces Inside a Plane

One payoff of the parametric form is that it constrains motion to the plane for free. For any continuous curve $\tau \mapsto (s(\tau), t(\tau))$ in the $(s, t)$-parameter space, the composition

is automatically a continuous curve in $\mathbb{R}^3$ lying entirely on the plane. The equational form $\mathbf{a} \cdot \mathbf{x} + c = 0$ offers no such guarantee, so whenever the task is to generate points on a plane, for example when rendering a path along a surface, the parametric form is the right tool, whereas whenever the task is to test points already in hand, the equational form is.

Sides of a Plane

A plane partitions $\mathbb{R}^3$ into three disjoint pieces: the plane itself and the two open regions on either side. The linear expression whose zero set defines the plane already contains the information needed to tell these regions apart: the only thing that changes between them is the sign of that expression.

Every point of $\mathbb{R}^3$ belongs to exactly one of $E$, $H^+$, $H^-$, since $f$ takes a unique real value at each point and the three cases $f > 0$, $f = 0$, $f < 0$ exhaust the possibilities. The geometric question of which side of $E$ a point lies on therefore reduces to a single sign comparison, and the algebraic shortcut extends to the related question of when two points are separated by the plane.

The test turns the geometric question of whether a plane separates two points into one arithmetic evaluation per point. That is the mechanism behind one of the basic primitives in three-dimensional rendering, and it also sits at the algebraic heart of a much larger family of questions across dimensions.

Lines in Three-Dimensional Space

A line in $\mathbb{R}^3$ is not the zero set of a single linear expression, since that zero set cuts out a plane. The parametric form, which carried two direction vectors alongside a base point when describing a plane, collapses to the natural description of a line once a single direction is kept.

When the line is presented through two distinct points $P, Q$, the displacement $\mathbf{d} = \mathbf{q} - \mathbf{p}$ supplies the direction, and the parametric form becomes

which recovers $P$ at $t = 0$, $Q$ at $t = 1$, and the midpoint of $\overline{PQ}$ at $t = 1/2$. For $t \neq 1$, the Directed Ratio from Lesson 2PM is

so the parameter $t$ determines the directed ratio, but is not itself the directed ratio.

The single-parameter description of a line and the two-parameter description of a plane are algebraic shadows of their geometric dimensions, and the number of free parameters needed to sweep out a figure will be our first handle on the notion of dimension in later lessons, where it extends far beyond the $\mathbb{R}^3$ of daily experience.

Assembling a Plane from Parallel Slices

Throughout this lesson and the last, every equation $ax + by + cz = d$ with $(a, b, c) \neq \mathbf{0}$ has been treated as the zero set of a plane, without a derivation of that fact from anything more primitive. One way to see why the claim is true is to assemble the surface from its slices by vertical planes, building on the familiar fact that a linear equation in two variables cuts out a line.

Relabelling axes if necessary, arrange that $c \neq 0$ and divide through to put the equation in the form

Fixing $x = h$ collapses the equation to $z = \beta y + (\alpha h + \gamma)$, a line of slope $\beta$ in the $(y, z)$-plane sitting at $\{x = h\}$, with vertical intercept $f(h) = \alpha h + \gamma$. As $h$ varies, these parallel-sloped lines stack to form the full surface, and the intercept $f(h)$ is linear in $h$, so its graph is a straight ruling and the stacked slices fit together flat.

The contrast with the right-hand surface is instructive. Both pictures show a surface whose every slice $\{x = h\}$ is a line of slope $-1$ in the $(y, z)$-plane. On the left the intercept $f(h) = -2h + 3$ depends linearly on $h$ and the stacked slices produce the plane $z = -2x - y + 3$; on the right $f(h) = 2 - \tfrac{1}{2}h^3 + \tfrac{1}{6}h^2 + \tfrac{7}{6}h$ is a cubic, and the same parallel-slope lines assemble instead into a wavy ruled surface. Linearity of $f$ in $h$ is the structural ingredient that turns a family of parallel slices into a flat plane, and its absence is what allows an assembly of parallel-slope lines to buckle.

Toward Systems of Linear Equations

Every equation $a_1 x_1 + a_2 x_2 + a_3 x_3 = b$ encountered in this lesson is a single linear equation in three unknowns, cutting out a plane in $\mathbb{R}^3$. Problem 29 already nudged past that one-equation setting: finding where a line meets a plane is the problem of jointly satisfying the line’s three parametric equations and the plane’s single equation, four scalar constraints tying together four unknowns $(x, y, z, t)$. The intersection of two non-parallel planes is the joint solution set of their two defining equations in three unknowns, and the intersection of three generic planes is the joint solution set of three. Each configuration is a small instance of one general object: a system of linear equations.

The general system of $m$ linear equations in $n$ unknowns has the form

with coefficients $a_{ij}$ and right-hand sides $b_i$. Equivalently, the $i$th equation may be written in the compact summation form

When every $b_i = 0$ the system is called homogeneous, and otherwise inhomogeneous. Both the $a_{ij}$ and the $b_i$ are drawn from a chosen field, and we write $\mathbb{F}$ for such a generic field: a set carrying addition, subtraction, multiplication, and division subject to the nine field axioms (A1)-(A9) laid out for the real numbers in the MCEA lesson “Predicates, Numbers, and Sets”. Those nine axioms, and only those, are what every manipulation in the chapters ahead is ultimately allowed to invoke, so any statement proved from them applies uniformly to every choice of $\mathbb{F}$; the familiar ones are $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$, and unless stated otherwise $\mathbb{F}$ will mean one of these three with the default reading $\mathbb{F} = \mathbb{R}$.

Systems of this form have been studied since the beginnings of algebra in its modern sense. Newton’s Arithmetica Universalis, compiled from his Cambridge lectures of the 1670s and published in 1707, already set out systematic elimination for small linear systems; and the matrix notation that packages the coefficients $a_{ij}$ into a rectangular array was introduced by Arthur Cayley in his 1858 Memoir on the Theory of Matrices, building on the determinantal calculus developed a few years earlier by his Cambridge colleague James Joseph Sylvester. The linear-algebraic investigation of such systems has since grown into the hub of the whole subject, dragging along with it the theories of linear spaces, linear transformations, and matrices.

Three basic questions organise everything that follows.

1. Existence. Does the system have any solution at all?
2. Uniqueness. When a solution exists, is it the only one?
3. Structure. When there are infinitely many solutions, what geometric shape does the solution set have, and how is it parametrised?

For the tiny cases visited in this lesson and the last, geometry in $\mathbb{R}^2$ and $\mathbb{R}^3$ already answers all three at a glance: two non-parallel lines in the plane $\mathbb{R}^2$ meet in a single point; two non-parallel planes in $\mathbb{R}^3$ meet in a line, a one-parameter family of solutions; three generic planes in $\mathbb{R}^3$ meet in a point; and two parallel but distinct planes meet in the empty set. In higher dimensions, and with more than three equations, that intuition runs out, and we will need a systematic algebraic procedure together with the language of matrices and ranks to reduce existence, uniqueness, and structure to bookkeeping. That procedure, and the vocabulary that goes with it, is the subject of the next few lessons, where we work out row reduction for arbitrary $m \times n$ systems and see how it settles all three questions uniformly.