Equation of Horizontal Line

Theorem
Let $\mathcal L$ be a horizontal line embedded in the Cartesian plane $\mathcal C$.

Then the equation of $\mathcal L$ can be given by:
 * $y = b$

where $\tuple {0, b}$ is the point at which $\mathcal L$ intersects the $y$-axis.


 * Graph-of-horizontal-line.png

Proof
From the Normal Form of Equation of Straight Line in Plane, a general straight line can be expressed in the form:


 * $x \cos \alpha + y \sin \alpha = p$

where:
 * $p$ is the length of a perpendicular $\mathcal P$ from $\mathcal L$ to the origin
 * $\alpha$ is the angle made between $\mathcal P$ and the $x$-axis.

As $\mathcal L$ is horizontal, then by definition $\mathcal P$ is vertical.

By definition, the vertical line through the origin is the $y$-axis itself.

Thus:
 * $\alpha$ is a right angle, that is $\alpha = \dfrac \pi 2 = 90 \degrees$
 * $p = b$

Hence the equation of $\mathcal L$ becomes:

Hence the result.