Equation of Straight Line in Plane/Normal Form

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Theorem

Let $\LL$ be a straight line such that:

the perpendicular distance from $\LL$ to the origin is $p$
the angle made between that perpendicular and the $x$-axis is $\alpha$.


Then $\LL$ can be defined by the equation:

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


Polar Form

Let $\LL$ be defined in normal form:

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

Then $\LL$ can be presented in polar coordinates as:

$r \map \cos {\theta - \alpha} = p$


Proof 1

Straight-line-normal-form.png


Let $A$ be the $x$-intercept of $\LL$.

Let $B$ be the $y$-intercept of $\LL$.


Let $A = \tuple {a, 0}$ and $B = \tuple {0, b}$.

From the Equation of Straight Line in Plane: Two-Intercept Form, $\LL$ can be expressed in the form:

$(1): \quad \dfrac x a + \dfrac y a = 1$


Then:

\(\ds p\) \(=\) \(\ds a \cos \alpha\) Definition of Cosine of Angle
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds \dfrac p {\cos \alpha}\)
\(\ds p\) \(=\) \(\ds b \sin \alpha\) Definition of Sine of Angle
\(\ds \leadsto \ \ \) \(\ds b\) \(=\) \(\ds \dfrac p {\sin \alpha}\)


Substituting for $a$ and $b$ in $(1)$:

\(\ds \dfrac x a + \dfrac y a\) \(=\) \(\ds 1\)
\(\ds \leadsto \ \ \) \(\ds \dfrac {x \cos \alpha} p + \dfrac {y \sin \alpha} p\) \(=\) \(\ds 1\)
\(\ds \leadsto \ \ \) \(\ds x \cos \alpha + y \sin \alpha\) \(=\) \(\ds p\)

$\blacksquare$


Proof 2

Straight-line-normal-form-Proof-2.png

Let $P = \tuple {x, y}$ be an arbitrary point on $\LL$.

Let $O$ be the origin of the Cartesian plane in which $\LL$ is embedded.

Let $PQ$ be the perpendicular dropped from $P$ to the $x$-axis.

Let $QS$ be the perpendicular dropped from $Q$ to the line $ON$.

Let $PR$ be the perpendicular dropped from $P$ to the line $QS$.

By definition of cosine:

$OS = OQ \cos \alpha$

By definition of sine:

$PR = PQ \sin \alpha$

Then:

\(\ds p\) \(=\) \(\ds ON\)
\(\ds \) \(=\) \(\ds OS + SN\)
\(\ds \) \(=\) \(\ds OS + PR\)
\(\ds \) \(=\) \(\ds OQ \cos \alpha + PQ \sin \alpha\)
\(\ds \) \(=\) \(\ds x \cos \alpha + y \sin \alpha\)

$\blacksquare$


Also known as

The normal form of a straight line in the plane is also known as the canonical form.

Some sources refer to it as the $\tuple {p, \alpha}$ form, after the symbols conventionally used to define it.


Sources


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