Straight Lines & Pair of Lines

Linear Geometry Foundation

A linear equation in two variables $ax+by+c=0$ ($a^2+b^2\ne 0$) geometrically represents a straight line in the Cartesian plane ${\mathbb{R}}^2$.

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1. Distance Metrics & Angle Between Lines

📊Visual Mathematical Intuition

Straight Lines, Acute/Obtuse Angle Bisectors & Pair Homogenization

Angle Bisectors:$\frac{a_1 x + b_1 y + c_1}{\sqrt{a_1^2 + b_1^2}} = \pm \frac{a_2 x + b_2 y + c_2}{\sqrt{a_2^2 + b_2^2}}$

Orthogonal Bisectors:$B_1 \perp B_2 \iff m_1 m_2 = -1$

Pair of Lines Angle:$\tan\theta = \frac{2\sqrt{h^2 - ab}}{|a + b|} \implies a + b = 0 \iff \theta = 90^\circ$

Intersecting straight lines $L_1, L_2$, mutually perpendicular acute and obtuse angle bisectors $B_1 \perp B_2$, and pair-of-lines angle formula $\tan\theta = \frac{2\sqrt{h^2-ab}}{|a+b|}$.

Perpendicular Distance from $(x_1,y_1)$ to $ax+by+c=0$

$$d=\frac{|a{x}_1+b{y}_1+c|}{\sqrt{a^2+b^2}}$$

Distance Between Parallel Lines $ax+by+c_1=0$ and $ax+by+c_2=0$

$$d=\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}$$

Angle $\theta$ Between Lines with Slopes $m_1,m_2$

$$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$$

• Lines are parallel $⟺m_1=m_2$.
• Lines are perpendicular $⟺m_1{m}_2=-1$.

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2. Equations of Angle Bisectors

For lines $L_1:a_{1}x+b_{1}y+c_1=0$ and $L_2:a_{2}x+b_{2}y+c_2=0$ (make $c_1,c_2>0$):

$$\frac{a_{1}x+b_{1}y+c_1}{\sqrt{a_1^2+b_1^2}}=\pm \frac{a_{2}x+b_{2}y+c_2}{\sqrt{a_2^2+b_2^2}}$$

• Bisector Containing Origin: Choose $+$ sign.
• Acute vs Obtuse Angle Bisector:
  • If $a_1{a}_2+b_1{b}_2>0⟹+$ sign is Obtuse Bisector, $-$ sign is Acute Bisector.
  • If $a_1{a}_2+b_1{b}_2<0⟹+$ sign is Acute Bisector, $-$ sign is Obtuse Bisector.

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3. Homogenization of Second-Degree Curves

Homogenization Method

The joint equation of lines joining the origin to the points of intersection of a second-degree curve:

$$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$$

and a line $lx+my+n=0$ (written as $\frac{lx+my}{-n}=1$) is obtained by homogenizing to degree 2:

$$a{x}^2+2hxy+b{y}^2+2(gx+fy)\left(\frac{lx+my}{-n}\right)+c{\left(\frac{lx+my}{-n}\right)}^2=0$$

These lines are mutually perpendicular if and only if:

$$\text{Coefficient of }{x}^2+\text{Coefficient of }{y}^2=0$$