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