What is the equation of a straight line?

Question

I'm reading a book named "Geometry, A comprehensive course" by Dan Pedoe at the moment. But I don't understand when they talk about the equation of a straight line...

Here is what is written:

The lines of our plane are sets of points which satisfy equations of the form $$a_1X_1+a_2X_2+a_3=0$$

Could someone explain me what the variables of the equations stand for and what it actually is?

Thanks in advance

Answer 1

In a plane a line can be represented with various ways. Your question pertains to the fourth way below:

1. Slope & Intercept - Two constant coefficients $a$ and $b$ you used to define $y$ value for each $x$ $$y = a x + b$$
2. Linear Interpolation - Scalar value $t$ goes from $-\infty$ to $+\infty$ to define location between two constant points $[x_1,y_1]$ and $[x_2,y_2]$ $$[x,y] = (1-t) [x_1,y_1] + t [x_2,y_2]$$
3. Point & Direction - Scalar value $t$ goes from $-\infty$ to $+\infty$ to define location based on a constant point $[x_0,y_0]$ and a direction $[\cos \psi,\sin \psi]$ $$[x,y] = [x_0,y_0] + t [\cos\psi,\sin\psi]$$
4. Homogeneous Coordinates - Three constant coefficients $a$, $b$ and $c$ define a line with equation $$a x+ b y + c =0$$ Note that the coefficients $(a,b,c)$ are called the (homogeneous) coordinates of the line and that any scalar multiple, for example: $(2 a,2 b, 2 c)$, describes the same line. Also the vector $\hat{n} = \begin{pmatrix} \frac{a}{\sqrt{a^2+b^2}} & \frac{b}{\sqrt{a^2+b^2}} \end{pmatrix}$ is the unit vector normal (perpendicular) to the line, and the scalar $d=\frac{|c|}{\sqrt{a^2+b^2}}$ is the distance of the line to the origin.

In your case the three homogeneous coefficients are $(a_1,a_2,a_3)$ and your coordinate directions are $(X_1,X_2)$ instead of $(x,y)$.

Answer 2

Hint. I hope you do know the forms of writing the equation of a straight line. One of the forms is $ax+by=c.$ In your case, $X_1 = x, X_2 = y$ and $c=-a_3.$