How to determine general form of line equation in 3D from 2 points without using vectors, matrices, etc

Question

For a 2D line equation in General Form ($ax + by + c = 0$) it is possible to calculate all coefficients from two given points as follows:

$a = y_1-y_2$

$b = x_2-x_1$

$c = (x_1-x_2) y_1 + (y_2-y_1) x_1$

For a 3D line equation in General Form ($ax + by + cz + d = 0$) is there an equivalent method of calculating the coefficients?

P.S. I took the equations above from here.

Answer 1

As comments already told you, $ax+by+cz+d=0$ is a plane. There are various ways to describe lines in 3d. The one which might be closest to what you do is using Plücker coordinates.

Let's look at the planar case for comparison. Start by appending a 1 (homogenization constant) to both coordinate vectors. Then compute all $2\times2$ determinants of that.

$$ \begin{pmatrix}x_1&x_2\\y_1&y_2\\1&1\end{pmatrix} \\ \begin{vmatrix}y_1&y_2\\1&1\end{vmatrix} = y_1-y_2 = a \quad \begin{vmatrix}x_1&x_2\\1&1\end{vmatrix} = x_1-x_2 = -b \quad \begin{vmatrix}x_1&x_2\\y_1&y_2\end{vmatrix} = x_1y_2-x_2y_1 = c $$

My $c$ looks different from yours because you can cancel one $x_1y_1$ after expanding things, so there is no real difference. The key messages here are

1. The $2\times2$ determinants contain all the information you need to describe the line.
2. Scaling all three numbers together by any non-zero number still represents the same line (homogeneous).

Now do the same for 3d. Start with

\begin{pmatrix}x_1&x_2\\y_1&y_2\\z_1&z_2\\1&1\end{pmatrix}

and compute all $2\times2$ determinants of that. There are 6 of these, and the convention I'm used to is using two indices for them, like $a_{12},a_{13},a_{14},a_{23},a_{24},a_{34}$. Or $a_{xy},a_{xz},a_{xw},a_{yz},a_{yw},a_{zw}$ if you prefer. Doesn't really matter. There also exist various conventions on how to assign signs to these. I know a formalism which would leave all six signs just as they come out of the determinant (and therefore would lead to an opposite sign in the $b$ of the planar case), putting the signs into the operations instead.

The above six numbers are not independent. They will always satisfy the equation $a_{12}a_{34}-a_{13}a_{24}+a_{14}a_{23}=0$. So if you know 5 of the numbers, you can derive the sixth unless its partner in this equation happens to be zero. And the coordinates are again homogeneous: scaling all numbers by the same factor (e.g. by $-1$ when swapping all signs) still represents the same line.

There are things you can compute using the Plücker coordinates of a line: compute the plane spanned by that and a third point, or intersect it with a plane to get a point, or check whether two lines in space intersect. But providing all the details here is too much for a single answer. Unfortunately the book describing this the way I learned it is in German.