6/19/2015©Zachary Wartell 1 3D Coordinate Systems and Transformations Revision 1.1 Copyright Zachary Wartell, University of North Carolina at Charlotte,

04/18/23©Zachary Wartell 1

3D Coordinate Systemsand Transformations

Revision 1.1Copyright Zachary Wartell, University of North Carolina at Charlotte, 2006

All Rights Reserved

Textbook:Chapter 5


3D Coordinate Systems

Z

X

YY

X

Z

Right-handed coordinate

system

Left-handedcoordinate

system

, Larry F. Hodges


Points & Vectors

• 3D affine space: points (locations), vectors (alignment with signed magnitude)

• operations:

– Points: pi - pj = v, pi + v = pj (p + 0 = p)

– Vectors: vi ±vj = vk , vj = a ∙ vk (a R)

length(v), |v| =

dot product: vivj =

2 2 2x y z

cos

j

i i i j i j i j i j

j

i j

x

x y z y x x y y z z

z

v v


• operations:– vectors:

• cross product: vi×vj = vk

.

Also

sin

i i i j j j i j j i j i i j i j j i

i j i j

x y z x y z y z y z x z x z x y x y

v v v v

Points & Vectors (cont.)


Parametric Definition of a Line

Given two points p1 = (x1, y1, z1), p2 = ( x2, y2, z2)

x = x1 + t (x2 - x1)

y = y1 + t (y2 - y1)

z = z1 + t (z2 - z1)

Given a point p1 and a vector v = (xv, yv, zv)

x = x1 + t xv, y = y1 + t yv , z = z1 + t zv

COMPACT FORM: p = p1 + t [p2 - p1] or p = p1 + vt

p1

p2

t = 1

t = 0

t > 1

t < 0

, Larry F. Hodges


Equation of a plane

Ax + By + Cz + D = 0

Alternate Form: A'x + B'y + C'z +D' = 0

where A' = A/d, B' = B/d, C' = C/d, D' = D/d

d =

Distance between a point and the plane is given by

A'x + B'y + C'z +D' (sign indicates which side)

Given Ax + By + Cz + D = 0 Then (A, B, C) is a normal vector

Pf: Given two points p1 and p2 in the plane, the vector (p2 - p1) is in the plane and (A,B,C) • (p2 - p1) = (Ax2 + By2 +Cz2) - (Ax1 + B y1 + Cz1) = ( - D ) - ( - D ) = 0

2 2 2A B C

, Larry F. Hodges


Derivation of Plane Equation

P1

P2

P3

• To derive equation of the plane given three points: p1, p2 , p3

• (p3 - p1) x (p2 - p1) = n, orthogonal vector

• Given a general point p = (x,y,z)

• n • [p - p1] = 0 if p is in the plane.

Or given a point (x,y,z) in the plane and normal vector n then n • (x,y,z) = -D

p

, Larry F. Hodges


Basic Transformations

•Translation

•Scale

•Rotation

•Shear


Translation

1 0 0

0 1 0{1,0}

0 0 1

0 0 0 1

x x

y y

z z

x wt t x

y wt t yw

z wt t z

w w

Point or vector


Scale (about origin)

0 0 0

0 0 0{1,0}

0 0 0

0 0 0 1

x x

y y

z z

x s s x

y s s yw

z s s z

w w


Rotation (about coordinate axis)

• Positive Rotations are defined as follows

Axis of rotation is Direction of positive rotation is

x y to z

y z to x

z x to y

, Larry F. Hodges

Z

Y

X


Rotations

cos sin 0 0

sin cos 0 0( ) :

0 0 1 0

0 0 0 1

1 0 0 0

0 cos sin 0( ) :

0 sin cos 0

0 0 0 1

cos 0 sin 0

0 1 0 0( ) :

sin 0 cos 0

0 0 0 1

z

x

y

R

R

R


Shears

1 0 0

0 1 0{1,0}

0 0 1 0

0 0 0 1

xz xz

yz yz

x SH z SH x

y SH z SH yw

z z

w w

XY sheared on Z

Of course other combinations occur too, e.g. ZX sheared on Y, X sheared on Y,etc.

x

y

z


Rotation About An Arbitrary Axis

Z

p1

p2

Y

X

u

θ



Z

p1

p2

Y

X

Translate p1 to origin

u'

T(-p1)

u



Z

Y

X

b

a

c

ux

uy

uz

ß

u'

Rotate u' to u'' in YZ plane (rotation about Y axis, Ry(-β))

2 2

2 2

2 2

cos /

sin /

x z

x y

y z

z

x

a u u

b u u

c u u

u a

u a

Ry(-β)) ∙ T(-p1)

, Larry F. Hodges



Z

Y

X

b

a

c

ux

uy

uz

ß

u'

2 2

2 2

2 2

cos /

sin /

x z

x y

y z

z

x

a u u

b u u

c u u

u a

u a

u''

Rotate u' to u'' in YZ plane (rotation about Y axis, Ry(-β))

Ry(-β)) ∙ T(-p1)

, Larry F. Hodges



Z

Y

X

a

c

ux

uy

uz

α

Next, Rotate u'' to Z axis (u''') (rotation about X axis, Rx(α))

cos /

sin /y

a

u

u

u

u''

After Ry(-ß), angle α and u'' lies in the y-z plane.

Rx(α) ∙ Ry(-β) ∙ T(-p1)

, Larry F. Hodges



Z

Y

X

u''

After Rx(α), u'' lies in the Z axis as u'''.

Next, rotate about Z by θ (Rz(θ))

u'''

θ

Rz(θ) ∙ Rx(α) ∙ Ry(-β) ∙ T(-p1)


R((p1,p2),θ) = T(p1) ∙ Ry(β) ∙ Rx(-α) ∙ Rz(θ) ∙ Rx(α) ∙ Ry(-β) ∙ T(-p1)


Y

X

Now we’ve rotated about U.

Next, apply the inverse transformations to place u''' back on u

p1

p2

uZ

θ



Y

X

p1

p2

UZθ

Y

X

p1

p2

UZθ

2

2

2

More compactly:

(( , ), ) = 1

1 1 1

, 1 1 1

1 1 1

where cos , sin

x x y z x z y

y x z y y z x

z x y z y x z

u c c u u c u s u u c u s

u u c u s u c c u u c u s

u u c u s u u c u s u c c

c s

R 1 1 R

1 2

R

M p p MM p p

0

M u


Rotation Representations So Far….

Euler angles-Rx Ry Rz - note order is arbitrary, 6 possible-3 real numbers-interpolation problems-gimbal lock-compose rotation matrix: 27 *’s and 18 +’s

Matrix-9 real numbers-no good way to interpolate rotations-compose rotation matrix: 27 *’s and 18 +’s

Axis-Angle-4 real numbers-again unclear how to interpolate rotations-no clear way to compose rotations (must convert to matrix)


Euler Angles: Gimbal Lock

Occurs when two rotation axes are aligned by an intermediate rotation

Example: Assume Euler order Rz ∙ Ry ∙ Rx

If Ry = 90 degrees, then x-axis aligns with y-axis and

Rz = -Rx

X'

Z

Y

X


Rotation Matrix: Interpolation

a b c a b c

d e f d e f

g h i g h i

?


Axis-Angle: Composition

R(?,?) = R(uk,θk) ∙ R(uk-1,θk-1) ….. R(u1,θ1)


Quaternions

Alternative rotation representation:

4 real numbersinterpolation between quaternions well defined

(“slerp”)no gimbal lockcomposition of quaternions: 16 *’s, 9+’s

(matrix 27 *’s and 18 +’s)


Quaternions

• Rotation representation:

( , ) ( , ( , , ))

cos , sin2 2

x y zs s v v v

s

q v

v u

Relation to axis-angle representation (u,θ) is:

x

y

Z

u

θ


Quaternion Rotation: Basic Idea

• To rotate point p:– represent p as quaternion:

– compute rotated quaternion point representation

– extract standard coordinate representation p' from qp'

(0, )pq p

1 p pq qq q

(0, ) pq p

x

y

Z

u

pp'


Details of Rotation

• How do you multiply quaternions:

• How do you find inverse of quaternion:

• For rotations we assume unit quaternions, hence:

1 2 1 2 1 2 1 2 2 1 1 2( , )s s s s q q v v v v v v

2

12

magnitude:

1inverse: ,

s

s

q v v

q vq

1 , when 1s q v q


Quaternion Rotation: Complete Equation

• To rotate point p:1 1

2

, where ( , )

.... after some algebra .....

( ) 2 ( ) ( )

Note, 26 *'s, 14 +'s versus 9 *'s, 6 +'s for 3x3 matrix-vector oper.

s

s s

p pq qq q q v

p p v p v v p v v p

x

y

Z

u

pp'


1 2 1 2 1 2 1 2 2 1 1 2( , )s s s s q q v v v v v v

Quaternion Rotation Examples:

0

1

2

1

( ) :

(cos( / 2),sin( / 2)(1,0,0)) (0, (1,0,0))

( ) :

(cos( / 2),sin( / 2)(0,1,0)) 0, (0,1,0))

( ) :

(cos( / 2),sin( / 2)(0,0,1)) 0, (0,0,1))

( ) ( ) :

(0, (0,1,0) (0,0,1)) (0, (1,0,0)) ( )

y

x

y

z

z

2 x

R

q

R

q = (

R

q = (

R R

q q R

Recall:

We have examples:


How do quaternions help?

• Quaternion composition:

requires 16 *’s, 12+’s while 3x3 matrix composition needs 27*’s and 18+’s

• No gimbal lock• interpolation between two orientations using “spherical

linear interpolation” (slerp) of their quaternions:

-constant angular velocity-unique interpolation-interpolation that is same regardless of coordinatesystem used for computation

1 2 1 2 1 2 1 2 2 1 1 2( , )s s s s q q v v v v v v

6/19/2015©Zachary Wartell 1 3D Coordinate Systems and Transformations Revision 1.1 Copyright Zachary Wartell, University of North Carolina at Charlotte,

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