Geometry for Games

Lines in 2D

General equation

The general equation of a line has the form

ax + by + c = 0

where a and b cannot be simultaneously 0.

 

Parametric Equation

y = y₀t + y_f
x = x₀t + x_f

Slope equation

For any non-vertical line y = ax+b, where a is the slope of the line. or, if you know the slope φ and b, then y = tg(φ)x + b

 

Vector equation of a line

Let r_p be the radus-vector of a point p laying on the line. Let also d be the directional vector of the line : a radius vector with the same slope as the line and (usually) with a norm (length) of 1. 

Then the equation of the line in vector form can be expressed as:

r(t) = r_p + d.t, where t varies continuously between -∞ and +∞. This equation is valid in any number of dimensions.

Expressing the vector equation of a line in cartesian coordinates, we arrive at exactly the parametric equation:

 

Line through two points

If x1 ≠x2 a = tg(φ) = (y2-y1)/(x2-x1) b = y2- ax2 = y2- x2(y2-y1)/(x2-x1) To retrieve the slope, φ  = Math.atan2(y2-y1,x2-x1) If x1= x2 the equation is x = x1 (or 1.x + 0.y - x1 = 0). The slope is π/2

 

Line crossing the axis at (x₀,0)  and (0,y₀)

This is a particular case of the previous situation a = tg(φ) = -y0/x0 b = y0 To retrieve the slope, φ  = Math.atan2(-y0,x0)   [in Java]

Line characteristics

Form	Slope (φ)	intersects X at	Intersects Y at
y = ax + b	arctan (a)	x  = -b/a  (never, if a = 0)	y = b
y = y0t + yf x = x0t + xf	If If x0 ≠ 0, arctan( y0/x0) If x0= 0, π/2	x =xf - x0 yf/y0 (never, if y0 = 0)	y =yf - y0 xf/x0 (never, if x0 = 0)
ax+by+c = 0	if b ≠ 0 arctan (-a/b), if b = 0, π/2	x = -c/a  (never, if a = 0)	y = -c/b (never if b = 0)
x1,y1,x2,y2 (two points)	arctan((y2-y1)/(x2-x1)) If x2 = x1, π/2	(never, if y1=y2)	(never, if x1=x2)

 

Conversions between different forms

From	To	Transformation
ax + by + c = 0	y = fx + g	If b ≠0,  f = -a/b,  g = -c/b If b = 0,  not possible
y = fx+g	ax+by+c = 0	a = -f, b = 1, c = -g
y = y0t + yf x = x0t + xf	y = fx+g	If x0 ≠ 0 ⇒f = y0/x0, g =yf - xf/x0 If x0 = 0 ⇒ not possible
y = fx+g	y = y0t + yf x = x0t + xf	xf = 0 yf = g
y = y0t + yf x = x0t + xf	ax+by+c  = 0	If x0 ≠ 0 ⇒ c= (xfy0 - x0yf), a = x0, b = -y0 If x0 = 0 ⇒c = -xf, a = -1, b = 0
x1,y1,x2,y2 (two points)	y = ax + b	If x1 ≠x2, a = (y2-y1)/(x2-x1),  b = y2- x2(y2-y1)/(x2-x1) If x1= x2 , not possible
x1,y1,x2,y2 (two points)	y = y0t + yf x = x0t + xf	Let . Then yf = y1 xf = x1
x1,y1,x2,y2 (two points)	ax+by+c = 0	a =(y2--y1) b =(x1-x2) c = (x2y1--y2x1)

 

 

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