Bisectors of Angles between Two Straight Lines/General Form
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Theorem
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:
- $\dfrac {l_1 x + m_1 y + n_1} {\sqrt { {l_1}^2 + {m_1}^2} } = \pm \dfrac {l_2 x + m_2 y + n_2} {\sqrt { {l_2}^2 + {m_2}^2} }$
Proof
First we convert $\LL_1$ and $\LL_2$ into normal form:
\(\ds \dfrac {l_1 x + m_1 y + n_1} {\sqrt { {l_1}^2 + {m_1}^2} }\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \dfrac {l_2 x + m_2 y + n_2} {\sqrt { {l_2}^2 + {m_2}^2} }\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x \cos \alpha + y \sin \alpha\) | \(=\) | \(\ds -\dfrac {n_1} {\sqrt { {l_1}^2 + {m_1}^2} }\) | where $\cos \alpha = \dfrac {l_1} {\sqrt { {l_1}^2 + {m_1}^2} }$ and $\sin \alpha = \dfrac {m_1} {\sqrt { {l_1}^2 + {m_1}^2} }$ | ||||||||||
\(\ds x \cos \beta + y \sin \beta\) | \(=\) | \(\ds -\dfrac {n_2} {\sqrt { {l_2}^2 + {m_2}^2} }\) | where $\cos \beta = \dfrac {l_2} {\sqrt { {l_2}^2 + {m_2}^2} }$ and $\sin \beta = \dfrac {m_2} {\sqrt { {l_2}^2 + {m_2}^2} }$ |
Then from Bisectors of Angles between Two Straight Lines: Normal Form, the angle bisectors of the angles formed at the point of intersection of $\LL_1$ and $\LL_2$ are given by:
\(\ds x \paren {\cos \alpha - \cos \beta} + y \paren {\sin \alpha - \sin \beta}\) | \(=\) | \(\ds -\dfrac {n_1} {\sqrt { {l_1}^2 + {m_1}^2} } + \dfrac {n_2} {\sqrt { {l_2}^2 + {m_2}^2} }\) | ||||||||||||
\(\ds x \paren {\cos \alpha + \cos \beta} + y \paren {\sin \alpha + \sin \beta}\) | \(=\) | \(\ds -\dfrac {n_1} {\sqrt { {l_1}^2 + {m_1}^2} } - \dfrac {n_2} {\sqrt { {l_2}^2 + {m_2}^2} }\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {x \cos \alpha + y \sin \alpha + \dfrac {n_1} {\sqrt { {l_1}^2 + {m_1}^2} } } - \paren {x \cos \beta + y \sin \beta + \dfrac {n_2} {\sqrt { {l_2}^2 + {m_2}^2} } }\) | \(=\) | \(\ds 0\) | |||||||||||
\(\ds \paren {x \cos \alpha + y \sin \alpha + \dfrac {n_1} {\sqrt { {l_1}^2 + {m_1}^2} } } + \paren {x \cos \beta + y \sin \beta + \dfrac {n_2} {\sqrt { {l_2}^2 + {m_2}^2} } }\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {l_1 x + m_1 y + n_1} {\sqrt { {l_1}^2 + {m_1}^2} }\) | \(=\) | \(\ds \dfrac {l_2 x + m_2 y + n_2} {\sqrt { {l_2}^2 + {m_2}^2} }\) | |||||||||||
\(\ds \dfrac {l_1 x + m_1 y + n_1} {\sqrt { {l_1}^2 + {m_1}^2} }\) | \(=\) | \(\ds -\dfrac {l_2 x + m_2 y + n_2} {\sqrt { {l_2}^2 + {m_2}^2} }\) | substituting back for $\cos \alpha$ and $\sin \alpha$ |
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $12$. Bisectors of the angles between two straight lines