General equation to the straight line

Question

Is the general equation to the straight line $Ax + By + C = 0$, or $Ax + By = C$. I have met both definitions in different books, and they obviously confer the same idea, but at most times it might be tricky dealing with both forms. For when equation 1 is algebraically rearanged we will find it to state that $Ax + By = -C$, which is different from our coveted result. This can serve to a disadvantage when substituting the constants for numbers. How am I to deal with these differences?

Answer 1

There are many forms describing a line. Some have more specific uses, and some are more general.

Between the two mentioned I think the first one is more common because it is used with homogeneous coordinates.

If the homogeneous coordinates (coefficients) of a line are $L = (A,B,C)$ and the coordinates of a point $P = (x,y,1)$ then the point is incident to the line when $L \cdot P = 0$ or expanded as

$$(A,B,C) \cdot (x,y,1) = 0 $$

$$ A x + B y + C = 0 $$

which is an equation of the lines since it is valid for all points $P=(x,y,1)$ that belong to the line.

Note that using the $(A,B,C)$ coordinates for lines, you can find the point where two lines intersect (in homogeneous coordinates) with

$$ P = L_1 \times L_2 $$

or the line that connects two points

$$ L = P_1 \times P_2 $$

where $\times$ is the regular 3-vector cross product.

Here are some other equations for a line

• $(-\sin \psi) x + (\cos \psi )y = d$ for the line with direction angle $\psi$ and perpendicular distance from the origin $d$.

• $\tfrac{x}{a} + \tfrac{y}{b} = 1$ Line with x-intercept $a$ and y-intercept $b$

• $y = m x + b$ Standard y-intercept $b$ and slope $m$

• $(x,y) = (1-t) (x_1,y_1) + t ( x_2,y_2) $ Parametric line between points $(x_1,y_1)$ and $(x_2,y_2)$ with parameter $t$

• $(x,y) = (x_0,y_0) + t (u,v)$ Ray starting at $(x_0,y_0)$ in the direction of $(u,v)$.

There are probably a lot more than the above. As I said, some are useful at specific cases and some are more general.