Relativistic effects & Non-collinear magnetism (WIEN2k / WIENncm)
ANALYTIC GEOMETRY of the EUCLIDEAN SPACE Evelichova/MII_ENGLISH/Lectures/P2_Linear figu… ·...
Transcript of ANALYTIC GEOMETRY of the EUCLIDEAN SPACE Evelichova/MII_ENGLISH/Lectures/P2_Linear figu… ·...
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ANALYTIC GEOMETRY of the EUCLIDEAN SPACE E3
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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
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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
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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
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Intercept form of the equation of plane given by three points X, Y, Z
0,,,1 rqpr
z
q
y
p
x
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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
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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
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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
,
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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
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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
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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
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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
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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
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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
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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
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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
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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