3.8  Lines and planes

The line in vector form

A line \(L\) is fixed by a point \(P_0\) on it, with position vector \(\bm r_0\), and a direction vector \(\bm v\) along it, Figure 3.8.1. From \(P_0\) we reach any point \(P\) of the line by walking \(t\) steps of \(\bm v\), so the position vector of \(P\) is

\[ \boxed{\bm r = \bm r_0 + t\,\bm v} \qquad\Longleftrightarrow\qquad \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x_0 \\ y_0 \\ z_0 \end{bmatrix} + t \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix} \tag{3.8.1}\]

This is the line in vector form, also called parameter form. The line has no ends: letting the parameter \(t\) run through all real numbers gives all its points, negative \(t\) on the far side of \(P_0\), Figure 3.8.2. A step of \(t = 1\) is one length \(\lvert\bm v\rvert\), so \(t\) counts lengths only when \(\bm v\) is a unit vector.

In space a line has no single equation in \(x\), \(y\) and \(z\). It takes two, one for each of two planes that cross along it, while 3.8.1 does the job with one vector equation and also tells where on the line a point lies.

The line as linear interpolation

A line is most often fixed by two points \(P_1\) and \(P_2\). The vector between them lies along the line, so \(\bm r_0 = \bm r_1\) and \(\bm v = \bm r_2 - \bm r_1\) in 3.8.1, and collecting the terms of each point gives

\[ \boxed{\bm r = \bm r_1 + t\,(\bm r_2 - \bm r_1) = (1 - t)\,\bm r_1 + t\,\bm r_2} \tag{3.8.2}\]

The second form is a weighted average of the two points, with weights \(1 - t\) and \(t\) that add up to one, Figure 3.8.3. At \(t = 0\) all the weight sits on \(P_1\), at \(t = 1\) all of it on \(P_2\), and \(t = \tfrac12\) gives the midpoint. Values between 0 and 1 give the segment between the points. Outside that range one weight turns negative and the point is extrapolated along the rest of the line.

This is linear interpolation, and the parameter makes it a modelling tool. Nothing in 3.8.2 requires \(\bm r_1\) and \(\bm r_2\) to be positions. The same weights interpolate a temperature between two measurements, a value between two rows of a table, or a colour between two colours. When the parameter is a scaled time, \(t = \tau/T\) with \(\tau\) the time, the point travels from \(P_1\) to \(P_2\) in the time \(T\). A 3D printer or a CNC mill moves its tool this way, each straight move of its G-code interpolated between two end points, and the animations of this book move their points with the same formula. Curves traced in time return in Chapter 3.9.

Distance from a point to a line

A distance between two objects is always the shortest one, and the shortest way from a point to a line is the perpendicular. Let \(\bm a\) be the vector from \(P_0\) to the point \(A\) and \(\bm d\) the vector from the line to \(A\) that is perpendicular to the line, Figure 3.8.7. Walking \(t\) steps along the line and then along \(\bm d\) reaches \(A\), so

\[ \begin{aligned} \bm a &= t\,\bm v + \bm d \\ \bm d \perp \bm v \;\Rightarrow\; \bm d\cdot\bm v &= 0 \end{aligned} \]

Inserting \(\bm d = \bm a - t\bm v\) in the second row gives \((\bm a - t\bm v)\cdot\bm v = 0\), and

\[ t = \frac{\bm a\cdot\bm v}{\bm v\cdot\bm v}, \qquad \bm d = \bm a - t\,\bm v \tag{3.8.3}\]

The vector \(t\bm v\) is the projection of \(\bm a\) on \(\bm v\) (Chapter 3.6), its shadow on the line, Figure 3.8.6.

The same \(t\) follows from asking for the shortest \(\bm d\) directly. Its squared length

\[ \lvert\bm d\rvert^2 = (\bm a - t\bm v)\cdot(\bm a - t\bm v) = \bm a\cdot\bm a - 2t\,\bm a\cdot\bm v + t^2\,\bm v\cdot\bm v \]

is a parabola in \(t\), least where its derivative \(-2\,\bm a\cdot\bm v + 2t\,\bm v\cdot\bm v\) is zero, and that gives 3.8.3 again. Perpendicular and shortest are the same condition. The derivative route needs no picture, so it carries over to problems with many distances at once.

Distance between two lines

Two lines in space that neither meet nor are parallel are skew. Each has its own parameter, \(\bm r_1 = \bm r_{01} + s\,\bm v_1\) and \(\bm r_2 = \bm r_{02} + t\,\bm v_2\), and a pair \((s, t)\) picks one point on each line. The vector between them is

\[ \bm d = \bm r_2 - \bm r_1 \]

and it is shortest when it is perpendicular to both lines,

\[ \begin{aligned} \bm d\cdot\bm v_1 &= 0 \\ \bm d\cdot\bm v_2 &= 0 \end{aligned} \tag{3.8.4}\]

which are two linear equations for \(s\) and \(t\), Figure 3.8.9. Minimising \(\lvert\bm d\rvert^2\) over \(s\) and \(t\) gives the same pair, since \(\partial\lvert\bm d\rvert^2/\partial s = -2\,\bm d\cdot\bm v_1\) and \(\partial\lvert\bm d\rvert^2/\partial t = 2\,\bm d\cdot\bm v_2\). The two parameters must have different names: a shared \(t\) would tie the two points together and leave one unknown for two equations. Surveying and GPS positioning meet this problem, since two measured sight lines never meet exactly.

Least squares

Two measured sight lines toward the same target miss each other, and three lines have no point in common at all. When the measurements leave no exact answer, we choose the unknowns that make the sum of the squared distances least. This is the method of least squares, and the derivative route of the last two sections carries it out: write each squared distance in the unknowns, add them up, and set every partial derivative to zero. The squares keep the derivatives linear in the unknowns, so the conditions form a linear system.

For a point \(\bm p\) and several lines, each line \(i\) contributes \(\lvert\bm d_i\rvert^2\), with \(\bm d_i\) the distance vector from the line to \(\bm p\) of 3.8.3. Only the part of a move of \(\bm p\) across the line changes that distance, and the gradient of \(\lvert\bm d_i\rvert^2\) with respect to \(\bm p\) works out to \(2\bm d_i\). The best point therefore satisfies

\[ \sum_i \bm d_i = \bm 0 \tag{3.8.5}\]

so the distance vectors from the lines to the best point balance, Figure 3.8.11.

The plane in vector form

A plane is fixed by a point \(P_0\) in it, with position vector \(\bm r_0\), and two directions \(\bm a\) and \(\bm b\) along it that are not parallel. From \(P_0\) we reach any point of the plane by \(s\) steps of \(\bm a\) and \(t\) steps of \(\bm b\),

\[ \boxed{\bm r = \bm r_0 + s\,\bm a + t\,\bm b} \tag{3.8.6}\]

which is the plane in parameter form, Figure 3.8.13. The plane is two-dimensional, so it takes two parameters where the line took one, and the pair \((s, t)\) works as a coordinate system drawn on the plane.

The normal \(\bm n = \bm a\times\bm b\) is perpendicular to both directions (Chapter 3.7) and therefore to every vector \(\bm r - \bm r_0\) from \(P_0\) to a point \(P\) in the plane, Figure 3.8.14. That gives the plane in vector form

\[ \boxed{(\bm r - \bm r_0)\cdot\bm n = 0} \tag{3.8.7}\]

Expanding the dot product gives the normal form, also called the affine form,

\[ \bigl((x, y, z) - (x_0, y_0, z_0)\bigr)\cdot(n_x, n_y, n_z) = 0 \quad\Longleftrightarrow\quad n_x x + n_y y + n_z z \underbrace{- (x_0 n_x + y_0 n_y + z_0 n_z)}_{\text{constant}} = 0 \]

The coefficients of \(x\), \(y\) and \(z\) in the normal form are the components of a normal vector. The planes we know first are the coordinate planes: the \(xy\), \(xz\) and \(yz\) planes all pass through the origin and have \(\bm k\), \(\bm j\) and \(\bm i\) as normals, so the \(xy\) plane is \(z = 0\).

Every pair \((s, t)\) in the parameter form gives a point, which is what we need to draw, mesh or interpolate a plane. The vector form answers the opposite question: \((\bm r - \bm r_0)\cdot\bm n\) is zero for a point in the plane, positive on the side \(\bm n\) points to and negative on the other. Examples 9 and 10 use that test.

With three points \(P_1\), \(P_2\) and \(P_3\) of the plane, \(\bm a = \bm r_2 - \bm r_1\) and \(\bm b = \bm r_3 - \bm r_1\) turn the parameter form into linear interpolation between three points,

\[ \bm r = (1 - s - t)\,\bm r_1 + s\,\bm r_2 + t\,\bm r_3 \tag{3.8.8}\]

with three weights that add up to one, like the two of 3.8.2. All three weights are positive inside the triangle \(P_1P_2P_3\), and one of them turns negative outside it, so \(s \ge 0\), \(t \ge 0\), \(s + t \le 1\) is the triangle itself, Figure 3.8.15. Every surface in a CAD viewer, an STL file for 3D printing or a computer game is built from such triangles, and 3.8.8 interpolates positions, colours or temperatures over each of them.

Distance from a point to a plane

The construction is the one for a point and a line, with the roles turned. Let \(\bm a\) be the vector from \(P_0\) to the point \(A\). The distance vector \(\bm d\) is perpendicular to the plane, so it lies along \(\bm n\) and is the projection of \(\bm a\) on \(\bm n\) (Chapter 3.6), Figure 3.8.18,

\[ \bm d = \frac{\bm a\cdot\bm n}{\bm n\cdot\bm n}\,\bm n, \qquad d = \lvert\bm d\rvert = \frac{\lvert\bm a\cdot\bm n\rvert}{\lvert\bm n\rvert} \tag{3.8.9}\]

The sign of \(\bm a\cdot\bm n\) tells the side: positive when \(A\) lies on the side \(\bm n\) points to.

Fitting a plane to measured points

A probe that measures the height of a machined surface at a few points returns points that lie near a plane without lying in it, and the least-squares plane is the reference the deviations are judged against. With the plane written as \(z = c_0 + c_1 x + c_2 y\), the measured height \(z_i\) above the point \((x_i, y_i)\) misses the plane by the residual

\[ e_i = z_i - (c_0 + c_1 x_i + c_2 y_i) \]

and minimising \(\sum e_i^2\) gives three linear equations for \(c_0\), \(c_1\) and \(c_2\). Moving \(c_1 x + c_2 y\) to the left side gives the normal form \(-c_1 x - c_2 y + z - c_0 = 0\), so the normal \(\bm n = (-c_1, -c_2, 1)\) is read off as in Example 7. The derivative with respect to \(c_0\) is \(-2\sum e_i\), so the residuals of the fit add up to zero.

The residual \(e_i\) is measured vertically. The distance from the point to the plane is \(e_i\,n_z/\lvert\bm n\rvert\) by 3.8.9, and for a nearly horizontal plane the factor is close to one and the two agree. Fitting the perpendicular distances themselves leads to an eigenvalue problem, which returns with the matrices.