17
ANALYTIC GEOMETRY of the EUCLIDEAN SPACE E 3
ANALYTIC GEOMETRY of the EUCLIDEAN SPACE E3
PLANE
Plane in the space is uniquely defined by
1. three different non-collinear points
2. two intersecting lines
3. two different parallel lines
4. line and point not on this line
5. point and direction perpendicular to the plane
Plane can be uniquely determined by an arbitrary point P and vector perpendicular to the plane.
Any non-zero vector , which is perpendicular to the plane, is called normal vector to the plane.
n
n
For an arbitrary point X of the plane holds
General equation of the plane
),,(
),,(
),,(
zyxX
zyxP
cba
PPP
n0.nn PXPX
0,0: 222 cbadczbyax
0,0]0,0,0[ 222 cbadO
Intercept form of the equation of plane given by three points X, Y, Z
0,,,1 rqpr
z
q
y
p
x
Special positions - plane perpendicular to the coordinate
plane (parallel to the coordinate axis not in this plane)
0,0,0
||,
cbdczby
xyzR
0,0,0
||,
cadczax
yxzR
0,0,0
||,
badbyax
zxyR
Special positions - plane parallel to the coordinate plane
0:
0,0:||
z
cdcz
xy
xy
R
R
0:
0,0:||
x
adax
yz
yz
R
R
0:
0,0:||
y
bdby
xz
xz
R
R
Plane can be uniquely defined by one arbitrary point P and two non-collinear direction vectors
Any point in the plane can be determined as particular
linear combination of the plane direction vectors
RststPX ,,.. vu
vu
,
Symbolic parametric equations of the plane
Parametric equations of the plane
RststPX
RststPXPX
,,..
,,..
vu
vu
RstzyxX
vsutzz
vsutyy
vsutxx
vvvuuuzyxP
P
P
P
PPP
,],,,[
..
..
..
),,(),,,(],,,[
33
22
11
321321 vu
Mutual position of 2 planes
• parallel
– perpendicular to the same direction of their normal vector , no common points
• Intersecting
– 1 common line, intersection – pierce line
),,(
0
0:
,0:
222
2
1
cba
cba
dczbyax
dczbyax
n
n
Line can be determined uniquely as intersection of two planes
21
2
2
2
2
2
222222
2
1
2
1
2
111111
0,0:
0,0:
p
cbadzcybxa
cbadzcybxa
Line can be uniquely determined by two different points, p = AB,
or one arbitrary point A and direction vector
Any non-zero vector parallel to the line is a line direction vector.
Any vector determined by 2 points on the line can be determined
also as scalar multiple of the line direction vector
u
RttAX ,. s
Symbolic parametric equations of the line
Parametric equations of the line
RttAX
RttAXAX
,.
,.
s
s
RtzyxX
stzz
styy
stxx
ssszyxA
A
A
A
AAA
],,,[
.
.
.
),,(],,,[
3
2
1
321s
Mutual position of 2 lines
• parallel
– collinear direction vectors
• intersecting
– one common point
• skew
– non-parallel, no common point
RvvBX
RuuAX
,.
,.
22
11
s
s
RvuvBuAM MMMM ,,.. 21 ss
RvvBX
RuuAX
,.
,.
22
11
s
s
Rkk ,. 12 ss
Rkk ,. 12 ss
Mutual position of line and plane
• parallel
– coplanar direction vectors
– perpendicular line direction vector s1
and plane normal vector n = s2 s3, s1.n = 0
• intersecting
– one common point
RvuvuBX
RttAX
,,..
,.
322
11
ss
s
Rvut
vuBtAM
MMM
MMM
,,
... 321 sss
Rlklk ,,.. 321 sss
Measuring distances • Distance of two points
– Distance of point A = [x0, y0, z0]
and plane ax + by + cz + d = 0
– Distance of point and line
– Distance of parallel lines
– Distance of parallel planes ax + by + cz + d1 = 0
ax + by + cz + d2 = 0
222
000),(cba
dczbyaxAd
),,(,),( 2121 cba
ddd n
n
Measuring angles
• Angle of two lines
– Angle of line
and plane ax + by + cz + d = 0
– Angle of two planes a1x + b1y + c1z + d1 = 0
a2x + b2y + c2z + d2 = 0
),,(,.
.)90cos( cban
nu
nu
vu
vuvu
.
.cos,,BA
u
A
),,(),,,(,.
.)cos( 22221111
21
21 cbacba nnnn
nn
