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 Motorcycle Engine Design
3.2 Transformation Matrix 3.3 2D Transformation
Where = Shear along x-direction. = Shear along y-direction
3.4 Arbitrary Rotation about the Origin Counterclockwise rotation of x and y to obtain and
x1 Where is the rotation matrix.
3.5Rotation by Different Angles Arbitrary rotation of axes x and y
3.6 Concatenation 3.7 2D Translation
R= R= Reverse the order of the 2 matrices
3.8 Projection onto a 2D Plane R1 = where X*=x, y*=y
3.9 Overall Scaling R= . = An example for overall scaling of an 2D object
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: + - - [R] = Rotation about arbitrary point.
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.
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 R= Reflection about y-axis
R= Reflection about x-axis
Reflection about any arbitrary Point T= R= T1=
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=
c) Reflection about the x-axis R= d) Rotation back by an angle T3= The concatenated matrix expressing the above steps is defined by
Reflection about an arbitrary axis y=mx+c Reflection of the object
3.12 3D TRANSFORMATION A trailer with a lower-attachment An energy-fuel vehicle
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
C = Let R = Then
3.14 3D Rotation of objects Rx Rotation about x-axis
R-1 = = RT Or
Ry = Rz = Rotation about y-axis Rotation about z-axis
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)
Solution: [C]= Rotation about the x Axis:
[C*]= = Rotation about x-axis for 30 degrees.
Rotation about the y Axis: [C*]= = Rotation about y-axis for 30 degrees
Rotation about the z Axis: [C*] = = Rotation about z-axis for 30 degrees
3.15 3D Reflection and mirror imaging An example for symmetry
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 • 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
Solution: = Step 1: Establish the transformation matrix to reflect the quarter block about the x-y plane [C*] = CR1 = C
Step 2: Reflect the half portion of the block about the y-z plane Reflection of half portion of the block about yz plane.