Composite 3D Transformations

Example of Composite 3D Transformations

Try to transform the line segments P1P2 and P1P3 from their start position in (a) to their ending position in (b).

The first solution is to compose the primitive transformations T, Rx, Ry, and Rz. This approach is easier to illustrate and does offer help on building an understanding. The 2nd, more abstract approach is to use the properties of special orthogonal matrices.

y

xz

y

xz

P1 P2

P3

P1

P2

P3

(a) (b)

Composition of 3D Transformations

Breaking a difficult problem into simpler sub-problems:1.Translate P1 to the origin.2. Rotate about the y axis such that P1P2 lies in the (y, z) plane.3. Rotate about the x axis such that P1P2 lies on the z axis.4. Rotate about the z axis such that P1P3 lies in the (y, z) plane.

y

xz

y

xz

y

xz

P1 P2

P3

P1

P2

P3

y

xz

P1 P2

P3

y

xz

P1P2

P3P3

P2 P1

1

2

34

Composition of 3D Transformations

1.

2.

1000100010001

),,(

1

1

1

111

zyx

zyxT

10000sin0cos00100cos0sin

))90((

yR

T

T

T

zzyyxxPzyxTP

zzyyxxPzyxTP

PzyxTP

]1[),,(

]1[),,(

]1000[),,(

1313133111'3

1212122111'2

1111'1

y

xz

P1 P2

P3

D1

Composition of 3D Transformations

3

4.

10000cossin00sincos00001

)(

xRy

xz

P1P2 D2

y

xz

P1

P2

D3

P3

)(zR

TRzyxTRRR yxz ),,()90()()( 111

Finally, we have the composite matrix:

Vector Rotation

x

y

x

y

Rotate the vector

01

u

sincos

01

cossinsincos

u

The unit vector along the x axis is [1, 0]T. After rotating about the origin by , the resulting vector is

x

Vector Rotation (cont.)

y

Rotate the vector

10

cossin

10

cossinsincos

v

x

y

v

The above results states that if we try to rotate a vector, originally pointing the direction of the x (or y) axis, toward a new direction, u (or v), the rotation matrix, R, could be simply written as [u | v] without the need of any explicit knowledge of , the actual rotation angle.

Similarly, the unit vector along the y axis is [0, 1]T. After rotating about the origin by , the resulting vector is

Vector Rotation (cont.)

The reversed operation of the above rotation is to rotate a vector that is not originally pointing the x (or y) direction into the direction of the positive x or y axis. The rotation matrix in this case is R(- ), expressed by R-1( )

where T denotes the transpose.

)(cossinsincos

)cos()sin()sin()cos(

)(1

TT

T

Rvu

R

x x

yy

Rotate the vector u u

Example

what is the rotation matrix if one wants the vector T in the left figure to be rotated to the direction of u.

T

(2, 3)

u

TT

uu

133

132

32

32|| 22

132

133

133

132

| vuR

If, on the other hand, one wants the vector u to be rotated to the direction of the positive x axis, the rotation matrix should be

132

133

133

132

T

T

vu

R

Rotation Matrices

Rotation matrix is orthonormal:• Each row is a unit vector

• Each row is perpendicular to the other, i.e. their dot product is zero.

• Each vector will be rotated by R() to lie on the positive x and y axes, respectively. The two column vectors are those into which vectors along the positive x and y axes are rotated.

• For orthonormal matrices, we have

1sincos1)sin(cos

cossinsincos

22

22

R

0)sin(cossincos

)()(1 TRR

Cross Product

• The cross product or vector product of two vectors, v1 and v2, is another vector:

• The cross product of two vectors is orthogonal to both• Right-hand rule dictates direction of cross product.

1221

1221

1221

21 )(y x y xz x z x

z y z yvv

v1

v2v1 v2

u2

Extension to 3D Cases

The above examples can be extended to 3D cases….

In 2D, we need to know u, which will be

rotated to the direction of the positive x axis.uv

x

y

z u1

v=u1u2

In 3D, however, we need to know more than

one vector. See in the left figure, for example,

two vectors, u1 and u2 are given. If after

rotation, u1 is aligned to the positive z axis, this

will only give us the third column in the rotation

matrix. What about the other two columns?

3D Rotation

In many cases in 3D, only one vector will be aligned to one of the coordinate axes, and the others are often not explicitly given. Let’s see the example:

y

xz

y

xz

P1 P2

P3

P1

P2

P3

Note, in this example, vector P1P2 will be

rotated to the positive z direction. Hence the

fist column vector in the rotation matrix is the

normalised P1P2. But what about the other

two columns? After all, P1P3 is not perpendi-

cular to P1P2. Well, we can find it by taking

the cross product of P1P2 and P1P3. Since

P1P2 P1P3 is perpendicular to both P1P2

and P1P3, it will be aligned into the direction

of the positive x axis.

And the third direction is decide by the cross

product of the other two directions, which is

P1P2 (P1P2 P1P2 ).

Therefore, the rotation matrix should be

3D Rotation (cont.)

u

y

xz

P1

P2

P3

vw

21

21

312121

312121

3121

3121

)()(

PPPP

PPPPPPPPPPPP

PPPPPPPP

R

y

xz

P1P2

P3

u

v

Yaw, Pitch, and Roll

Imagine three lines running through an airplane and intersecting at right angles at the airplane’s centre of gravity.

Roll: rotation around the

front-to-back axis.

Pitch: rotation around the

side-to-side axis.

Yaw: rotation around the

vertical axis.

An Example of the Airplane

Consider the following example. An airplane is oriented such that its nose is pointing in the positive z direction, its right wing is pointing in the positive x direction, its cockpit is pointing in the positive y direction. We want to transform the airplane so that it heads in the direction given by the vector DOF (direction of flight), is centre at P, and is not banked.

Solution to the Airplane Example

First we are to rotate the positive zp direction into the direction of DOF, which gives us the third column of the rotation matrix: DOF / |DOF|. The xp axis must be transformed into a horizontal vector perpendicular to DOF – that is in the direction of yDOF. The yp direction is then given by xp zp = DOF (y DOF).

DOFDOF

DOFyDOFDOFyDOF

DOFyDOFyR

)()(

Inverses of (2D and) 3D Transformations

1. Translation:

2. Scaling:

3. Rotation:

4. Shear:

),,(),,(1 zyxzyx dddTdddT

)1,1,1(),,(1zyx

zyx sssSsssS

)()()(1 TRRR

),(),(1 yxyx shshSHshshSH

Transformation as a change of a coordinate System

Mij = the transformation that converts a representation of a point in coordinate system j into its representation in coordinate system i.

P(i) = the representation of a point in coordinate system i.

Let P(i) = Mij . P(j) and P(i) = Mjk . P(k)

Then, P(i) = Mij . Mjk . P(k) = Mik . P(i)

So, Mik = Mij . Mjk

Let, M12 = T(4,2), M23 = T(2,3). S(0.5,0.5) and M34 = T(6.7, 1.8). R(45o)

Then point P(1) = (10,8), P(2) = (6,6), P(3) = (8,6), and P(4) = (4,2) are in coordinate systems 1 through 4 respectively.

Mij = Mji-1, So, M21 = M12

-1 = T(-4,-2),