Bisectors of Angles between Two Straight Lines
Jump to navigation
Jump to search
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:
- $h x^2 - \paren {a - b} x y - h y^2 = 0$