Bisectors of Angles between Two Straight Lines

Theorem

Normal Form

Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, expressed in normal form as:

\(\ds \LL_1: \ \ \)

\(\ds x \cos \alpha + y \sin \alpha\)

\(=\)

\(\ds p\)

\(\ds \LL_2: \ \ \)

\(\ds x \cos \beta + y \sin \beta\)

\(=\)

\(\ds q\)

The angle bisectors of the angles formed at the point of intersection of $\LL_1$ and $\LL_2$ are given by:

\(\ds x \paren {\cos \alpha - \cos \beta} + y \paren {\sin \alpha - \sin \beta}\)

\(=\)

\(\ds p - q\)

\(\ds x \paren {\cos \alpha + \cos \beta} + y \paren {\sin \alpha + \sin \beta}\)

\(=\)

\(\ds p + q\)

General Form

Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, expressed in general form as:

\(\ds \LL_1: \ \ \)

\(\ds l_1 x + m_1 y + n_1\)

\(=\)

\(\ds 0\)

\(\ds \LL_2: \ \ \)

\(\ds l_2 x + m_2 y + n_2\)

\(=\)

\(\ds 0\)

The angle bisectors of the angles formed at the point of intersection of $\LL_1$ and $\LL_2$ are given by:

$\dfrac {l_1 x + m_1 y + n_1} {\sqrt { {l_1}^2 + {m_1}^2} } = \pm \dfrac {l_2 x + m_2 y + n_2} {\sqrt { {l_2}^2 + {m_2}^2} }$

Homogeneous Quadratic Equation Form

Consider the homogeneous quadratic equation:

$(1): \quad a x^2 + 2 h x y + b y^2 = 0$

representing two straight lines through the origin.

Then the homogeneous quadratic equation which represents the angle bisectors of the angles formed at their point of intersection is given by:

$h x^2 - \paren {a - b} x y - h y^2 = 0$