What is the geometric interpretation of a cartesian equation of a line?

Question

The geometric interpretation of the parametric equation of a line $\mathbf{w}+t\mathbf{u}$ is quite intuitive, you have a parameter that can expand, contract and change the direction of the line. How can I understand visually the cartesian equation of a line $ax + by = c$?

Answer 1

In Euclidean geometry (i.e with an inner product) take $(x_0,y_0)$ to be one point on the line, then $c=ax_0+by_0$ so the equation of the line rewrites as

$$a(x-x_0)+b(y-y_0)=0$$

And we can interpret that as

$$(a,b)\cdot \begin{pmatrix} x-x_0\\y-y_0 \end{pmatrix}=0$$

So the line passes through $(x_0,y_0)$ and has $(a,b)$ as normal vector

Answer 2

Try and rewrite the equation:

$ax + by = c$ as $y = (-\frac{a}{b})x + \frac{c}{b}$.

You can then simply see how $y$ changes as $x$ does (and see, for instance, that when $x = 0$, $y$ will take the value $\frac{c}{b}$, or that $y = 0$ when $x = \frac{c}{a}$. :)

Answer 3

Personally, I find it intuitive to visualize lines or planes by considering their gradients. One of the most useful aspects about the gradient of a function is that it tells us the direction of steepest ascent or increase of that given function.

To put this into perspective, let's consider your example. We can interpret $ax + by = c$ to be a particular instance of a general function $f(x, y) = ax + by$ where the $f$ evaluates to $c$. The gradient of $f$ can then be calculated as follows:

$$\nabla f = \left<\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right> = \left<a, b \right>$$

Because the gradient is a constant vector, we know that the function's direction of increase on a contour plot is going to be the same at any arbitrary point $(x, y)$ no matter where you evaluate the function. From this, we can intuit that the function is linear and perpendicular to the vector $\left<a, b\right>$.

Note that we can extend this idea to any number of higher dimensions. For example, we know that $f(x, y, z) = ax + by + cz$ is a plane that is orthogonal to the vector $\left<a, b, c \right>$. Such a vector is often referred to as the normal vector of the plane.

If anything is unclear, I'd be more than happy to elaborate.

Answer 4

It helps let you look at a circle as a line just substitute $a=x$, $b=y$ and you've got a circle around the origin with radius $\sqrt{c}$ . in general $a$ is the scaling of $x$, $b$ the scaling of $y$ . Scale each side by a factor, it's the same line. Negative $b$ to reflect across the $x$ axis, negative $a$ opposite slope through same point. negate both to negate the intercept , as will negating $c$ . make $a$ change magnitude, and it rotates around a point. etc.