page 37

2.3.4 Planes, Lines, etc.

• The most fundamental mathematical geometry is a line. The basic relationships are given below,

y =	mx + b		defined with a slope and intercept
mperpendicular =		1	a slope perpendicular to a line
		---
		m
m =	y2 – y1		the slope using two points
	---------------
	x2 – x1

x	y	= 1	as defined by two intercepts
--	+ --
a	b

• If we assume a line is between two points in space, and that at one end we have a local reference frame, there are some basic relationships that can be derived.

page 38

y

θ β

d

(x2,y2 ,z2 )

(x1,y1 ,z1 )

d = (x	2	2	2	θ γ	θ α
	2 – x1 )	+ (y2 – y1 )	+ (z2 – z1 )

x

z (x0,y0 ,z0 )

The direction cosines of the angles are,

	x2 – x1			θ β		y2 – y1			z2 – z1
θ α = acos ---------------		d			=	acos ---------------	d		θ γ = acos --------------	d
(cos θ α	2	(cos θ β	2	(cos	2	= 1
	) +		) +		θ γ )
The equation of the line is,
x – x1	y – y1		z – z1						Explicit
x---------------2 – x1	= y---------------2 – y1		= z--------------2 – z1

(x,y ,z ) = (x1,y1 ,z1 )+ t((x2,y2 ,z2 )– (x	1,y1 ,z1 ))	Parametric t=[0,1]

• The relationships for a plane are,