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University of Illinois-Chicago. Chapter 3 Transformation and Manipulation of Objects. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche University of Illinois-Chicago. 3.1 Introduction.

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slide1

University of Illinois-Chicago

Chapter 3

Transformation and Manipulation of Objects

Principles of

Computer-Aided

Design and

Manufacturing

Second Edition 2004

ISBN 0-13-064631-8

Author: Prof. Farid. Amirouche

University of Illinois-Chicago

3 1 introduction
3.1 Introduction

Motorcycle Engine Design

3 2 transformation matrix
3.2 Transformation Matrix

3.3 2D Transformation

slide7

Where

= Shear along x-direction.

= Shear along y-direction

slide9

3.4 Arbitrary Rotation about the Origin

Counterclockwise rotation of x and y to obtain and

slide10

x1

Where

is the rotation matrix.

3 5 rotation by different angles
3.5Rotation by Different Angles

Arbitrary rotation of axes x and y

3 6 concatenation
3.6 Concatenation

3.7 2D Translation

slide13

R=

R=

Reverse the order of the 2 matrices

3 9 overall scaling
3.9 Overall Scaling

R=

.

=

An example for overall scaling of an 2D object

3 10 rotation about an arbitrary point
3.10 Rotation about an Arbitrary Point
  • Example

Rotation of an Object about an Arbitrary Point in 2D

Let C describe an object or configuration of some geometry, where C is an array of data-point coordinates.

solution
Solution:

+

-

-

[R] =

Rotation about arbitrary point.

example
Example

Uniform Scaling in 2D

Find the transformation matrix that would produce rotation of the geometry about point A, s shown in Figure 3.11(a), followed by a uniform scaling of the geometry down to half its original size.

solution19
Solution:

Step 1: Place the points into a matrix.

Step 2: Translate point A to the origin, that is, -2- along the x-axis and -10 along the y

Axis, as shown in Figure 3.11(b).

Step 3: Rotate the object 30 degrees about the z-axis, as shown is Figure 3.12(c).

Step 4: Translate point A to its original position as shown in Figure 3.12(d).

Step 5: Scale the object to half its original size, as shown is Figure 3.13(e).

3 11 2d reflection
3.11 2D Reflection

R=

Reflection about y-axis

slide24

R=

Reflection about x-axis

slide27

Reflection about arbitrary axis:

a) Coordinate transformation to move the line so it passes through o.

T1=

b) Rotation to make the x-axis align with the given line

T2=

slide28

c) Reflection about the x-axis

R=

d) Rotation back by an angle 

T3=

The concatenated matrix expressing the above steps is defined by

3 12 3d transformation
3.12 3D TRANSFORMATION

A trailer with a lower-attachment

An energy-fuel vehicle

3 13 3d scaling
3.13 3D Scaling

(a) Local Scaling: 

(b) Overall Scaling : Overall scaling can be achieved by the following transformation matrix where the final coordinates need to be normalized

where

slide33

C =

Let

R =

Then

3 14 3d rotation of objects
3.14 3D Rotation of objects

Rx

Rotation about x-axis

slide38

Ry =

Rz =

Rotation about y-axis

Rotation about z-axis

example 3 5 rotation in 3d space
Example 3.5: Rotation in 3D Space
  • The box shown in Figure 3.26(A) will demonstrate rotation about an axis in 3D space. The box shown in the figure is at the initial starting point for all three rotations. The labeled points of the box listed in matrix format (see Sec. 3.3) are used with the transformation rotation matrices, equations (3.37), (3.39), and (3.40), to obtain the new coordinates after rotation (rotations are in a counterclockwise direction in this example)
slide40

Solution:

[C]=

Rotation about the x Axis:

slide41

[C*]=

=

Rotation about x-axis for 30 degrees.

slide42

Rotation about the y Axis:

[C*]=

=

Rotation about y-axis for 30 degrees

slide43

Rotation about the z Axis:

[C*] =

=

Rotation about z-axis for 30 degrees

slide45

Reflection about the x-y plane is given:

Reflection about the y-z plane is given:

Reflection about the x-z plane is given:

example building of a block
Example : Building of a Block
  • Symmetry is the similarity between two objects with respect to a point or a line or a plane. Dimensions of the object with measured from the symmetric plane will be equal for both the object. One object look similar to the mirror image of the other assuming that the central plane acts as a mirror. This concept of symmetry and mirroring are widely used in design and modeling field to reduce model creation time. Use reflection to simplify the creation of the block shown in
solution48
Solution:

=

Step 1: Establish the transformation matrix to reflect the quarter block about the x-y plane

[C*] = CR1 = C

slide50

Step 2: Reflect the half portion of the block about the y-z plane

Reflection of half portion of the block about yz plane.

slide53

Example : Translation of a Block in 3D

Using the same box of Figure in Example 3.5, translate the box 2 units in the x direction, 1 unit in the y direction, and 1 unit in the z direction.

slide54

Solution:

Using previous equation, we substitute the numerical values into the translation matrix and apply equation to find the new coordinates of the points after translation. We know x=2, y=1, and z=1. The new coordinates of the box are

3 17 3d rotation about an arbitrary axis
3.17 3D ROTATION ABOUT AN ARBITRARY AXIS

Transformation matrix could be achieved through a procedure as described below:

1. The object is translated such that the origin of coordinates passes through the line

2. Rotation is accomplished

3. The object is translated back to its origin

rotation about an arbitrary axis can be classified into 3 types
Rotation about an arbitrary axis can be classified into 3 types
  • 1) Axis of rotation parallel to any one of the coordinate axes.

Rotation about a parallel axis

Translation of axis to coordinate axis

2 axis passing through origin and not parallel with any coordinate axis
2. Axis passing through origin and not parallel with any coordinate axis.

Rotation about an axis passing through origin

3 arbitrary line not passing through the origin and not parallel to any of the coordinate axis
3. Arbitrary line not passing through the origin and not parallel to any of the coordinate axis.

Rotation about an axis not passing through origin

example rotation of a box in 3d space
Example : Rotation of a Box in 3D Space

Using the box of Figure in Example 3.5, find the new coordinates of the box if it is rotated 30 degrees about the x-axis, 60 degrees about the y-axis, and 90 degrees about the z-axis. (Rotations are in the counterclockwise direction.) The rotations of the coordinate reference frames are illustrated in Figure 3.21. x’’’, y’’’, and z’’’ indicate the new coordinate system where the box resides [C*].

slide62

Solution:

where

And substituting =30, =60, and =90

example rotation and translation of a cube in 3d space
Example : Rotation and Translation of a Cube in 3D Space

Given the unit cube shown as follows, find the transformation matrix required for the display of the cube

Initial position of the cube

slide67

Step 3: Rotate the cube +90 degrees about the y-axis

By combining the transformation matrices, we have

The final answer is

example pyramid rotation and translation
Example : Pyramid Rotation and Translation

Give the concatenated transformation matrix that would generate the new position of the object shown in Figure 3.41. (Face A given by points ABCD lies in the x-z plane with its center along the x-axis.)

Initial position of the pyramid

slide69

Solution:

Step1: Determine the matrix to rotate the pyramid along the x-axis by 90 degrees

Rotation about the x-axis for 90 degrees

slide70

Step 2: Determine the matrix to translate the object –h units along the x-axis

Translation along the x-axis for –h units

slide71

Step 3: Rotate the object 90 degrees about the z-axis

Rotation about the z-axis for 90 degrees

slide72

3.18 3D VISUALIZATION

3.19 TRIMETRIC PROJECTION

(For z=0)

slide73

If we were to project the object onto x=0 or x=r plane, the projection matrix takes the following form:

(For x=0)

(For x=r)

In a similar fashion, the projection onto the y=0 or y=s plane is

(For y=0)

(For y=s)

slide74

Consider the following transformation :

defines the equation of a plane

slide75

In the case where q = 0 and p = 0, the equation becomes rx +1 = 0

and the distance from the origin is D = 1/ r. Therefore for a projection

onto a plane defined by as x = a, the projection matrix is

slide76

The equation of the plane x=a can also be written as

In order to normalize the representation of C matrix and have the last element equal to 1 we need to substitute the above

( 0 y z -x/r+1) by moving the geometry such that all coordinates have x=r and y and z are kept unchanged. Therefore,

example projection on a plane
Example : Projection on a Plane

Determine the projection of box in (a) x=6, (b) y=6, and (c) z=6.

Solution:

slide78

 (a) The projection of the box on x=6 plane (see Figure 3.46) has the following transformation matrix:

slide79

(b) The projection of the box on the y=6 plane has the following transformation matrix:

Therefore, the coordinates for the projection are

slide80

(c) The projection of the box on the z=6 plane has the following transformation matrix:

Therefore, the coordinates for the projection are

Projection on the plane z=6

3 20 isometric projection
3.20 ISOMETRIC PROJECTION

Combined rotations followed by projection from infinity form the bases for generating all axonometric projections. We perform the following: 

1. Rotate about the y-axis

2. Rotate about the x-axis

3. Project about the z=0 plane

4. Apply the final transformation conditions of foreshortening all axes equally

5. Get the final transformation matrix to yield the isometric view

slide83

Consider a point P given by (x y z 1). Let us find the isometric projection of this point while using the previous definitions. Operating on P by  and , we get

where [x* y* z*] represents the coordinates of the rotated point P about the y and x axes. The concatenated transformation matrix is given by

slide84

Suppose point P denotes different unit vectors along the x, y, and z-axes. Hence alone x, we have [1 0 0 1],

where

If we consider the unit vector along the y-axis, it transforms into

where

slide86

Using trigonometric relationships and the method of substitution, we can solve for  and  which yield =35.26, =45. We can then conclude that given geometry in 3D represented by [C}, its isometric projection is obtained by premultiplying it by R with  and  being 35.26 and 45 respectively. The resulting [C*] represents the projection for which we are looking.