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CAD/CAM

Dr. Ibrahim Al-Naimi

Chapter two

Computer Aided Design

(CAD)

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The information-processing cycle in a typical

manufacturing firm.

PRODUCT DESIGN AND CAD

Product design is a critical function in the production system. The quality of the product design (i.e., how well the design department

single most does its job) is probably the important factor in determining the commercial

. If the success and societal value of a productproduct design is poor, no matter how well it is manufactured. If the product design is good, there is still the question of whether the product can be produced at sufficiently low cost to contribute to the company's profits and success. Let us begin our discussion of product design by describing the general process of design.

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The Design Process

The process of designing something is

characterized as an interactive procedure, which

consists of six identifiable steps or phases:

1. Recognition of need.

2. Definition of problem.

3. Synthesis.

4. Analysis and optimization.

5. Evaluation.

6. Presentation.

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The Design Processinvolves the realization by Recognition of need

someone that a problem exists for which some

corrective action can be taken in the form of a

design solution. This recognition might mean

identifying some deficiency in a current machine

Problem definitiondesign by an engineer.

involves a thorough specification of the item to

be designed. This specification includes the

physical characteristics, function, cost, quality,

and operating performance.

The Design Process

are closely related and Synthesis and analysishighly interactive in the design process. Consider the development of a certain product design: Each of the subsystems of the product must be conceptualized by the designer, analyzed, improved through this analysis procedure, redesigned, analyzed again, and so on. The process is repeated until the design has been optimized within the constraints imposed on the designer. The individual components are then synthesized and analyzed into the final product in a similar manner.

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The Design Process

is concerned with measuring the Evaluationdesign against the specifications established in the problem definition phase. This evaluation often requires the fabrication and testing of a prototype model to assess operating performance, quality, reliability, and other criteria. The final phase in the design procedure

Presentationis the presentation of the design. is concerned with documenting the design by means of drawings, material specifications, assembly lists, and so on. In essence, documentation means that the design data base is created.

Application of Computers in Design

is defined aided design (CAD)-Computeras any design activity that involves the effective use of the computer to create, modify, analyze, or document an engineering design. CAD is most commonly associated with the use of an interactive computer graphics system, referred to as a CAD system.

There are several good reasons for using a CAD system to support the engineering design function:

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Fundamental reasons for implementing CAD

system:

1. To increase the productivity of the designer.

This is accomplished by helping the

designer to conceptualize the product and

its components. In turn, this helps reduce

the time required by the designer to

synthesize, analyze, and document the

design.



system:

2. To improve the quality of design.

The use of a CAD system with appropriate

hardware and software capabilities permits

the designer to do a more complete

engineering analysis and to consider a

larger number and variety of design

alternatives. The quality of the resulting

design is thereby improved.

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system:

3. To improve documentation

The graphical output of a CAD system

results in better documentation of the

design than what is practical with manual

drafting. The engineering drawings are

superior, and there is more standardization

among the drawings, fewer drafting errors,

and greater legibility.



system:

4. To create a data base for manufacturing.

In the process of creating the

documentation for the product design

(geometric specification of the product,

dimensions of the components, materials

specifications, bill of materials, etc.), much

of the required data base to manufacture

the product is also created.

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The design related tasks performed by CAD

system are:

1. Geometric modeling.

2. Engineering analysis.

3. Design review and evaluation.

4. Automated drafting.

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Geometric Modeling involves the use of a CAD Geometric modeling

system to develop a mathematical description of the geometry of an object. The mathematical description, called a geometric model, is contained in computer memory. This permits the user of the

an image of the model on a displayCAD system to graphics terminal and to perform certain operations

creating on the model. These operations include new geometric models from basic building blocks available in the system, moving the images around on the screen, zooming in on certain features of the

. These capabilities permit the image, and so forthdesigner to construct a model of a new product (or its components) or to modify an existing model.

Geometric Modeling There are various types of geometric models used in CAD.

-One classification distinguishes between twoD) -3dimensional (-D) and three-2dimensional (

dimensional models are best utilized for -Twomodels.design problems in two dimensions, such as flat objects and layouts of buildings. In the first CAD systems developed in the early 1970s, 2-D systems were used principally as automated drafting systems. They were often used for 3-D objects, and it was left to the designer or draftsman to properly construct the various views of the object. Three-dimensional CAD systems are capable of modeling an object in three dimensions. The operations and transformations on the model are done by the system in three dimensions according to user instructions. This is helpful in conceptualizing the object since the true 3-D model can be displayed in various views and from different angles.

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Geometric Modeling Geometric models in CAD can also be classified

frame models or solid -either wireas being uses frame model-wireA models.

interconnecting lines (straight line segments) to depict the object as illustrated in the following

of complicated frame models-WireFigure (a). geometries can become somewhat confusing because all of the lines depicting the shape of the object are usually shown, even the lines representing the other side of the object. Techniques are available for removing these so called hidden lines, but even with this improvement, wire-frame representation is still often inadequate.

Geometric Modeling are a more recent development in Solid models

, Figure solid modelinggeometric modeling. In (b), an object is modeled in solid three dimensions, providing the user with a vision of the object very much like it would be seen in real life. More important for engineering purposes, the geometric model is stored in the CAD system as a 3-D solid model, thus providing a more accurate representation of the object. This is useful for calculating mass properties, in assembly to perform interference checking between mating components, and in other engineering calculations.

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Engineering Analysis

After a particular design alternative has been developed, some form of engineering analysis often must be performed as part of the design process. The analysis

strain calculations, heat -stressmay take the form of The transfer analysis, or dynamic simulation.

computations are often complex and time consuming, and before the advent of the digital computer, these analyses were usually greatly simplified or even omitted

The availability of software for in the design procedure.engineering analysis on a CAD system greatly increases the designer's ability and willingness to perform a more thorough analysis of a proposed design. The term

is often used for aided engineering (CAE)-computerengineering analyses performed by computer. Examples of engineering analysis software in common use on CAD systems include:

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• Mass properties analysis, which

involves the computation of such features

of a solid object as its volume, surface

area, weight, and center of gravity. It is

especially applicable in mechanical

design.

• Interference checking

• Tolerance analysis

Engineering Analysis• Finite element analysis. Software for finite element

analysis (FEA), also known as finite element modeling(FEM), is available for use on CAD systems to aid in stress strain, heat transfer, fluid flow, and other engineering computations. Finite element analysis is a numerical analysis technique for determining approximate solutions to physical problems described by differential equations that are very difficult or impossible to solve. In FEA, the physical object is modeled by an assemblage of discrete interconnected nodes (finite elements), and the variable of interest (e.g., stress, strain, temperature) in each node can be described by relatively simple mathematical equations. By solving the equations for each node, the distribution of values of the variable throughout the physical object is determined.

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• Kinematic and dynamic analysis. Kinematic

analysis involves the study of the operation of

mechanical linkages to analyze their motions. A

typical kinematic analysis consists of specifying

the motion of one or more driving members of

the subject linkage, and the resulting motions of

the other links are determined by the analysis

package. Dynamic analysis extends kinematic

analysis by including the effects of the mass of

each linkage member and the resulting

acceleration forces as well as any externally

applied forces.

Design Evaluation and Review

Design evaluation and review procedures can be augmented by CAD. Some of the CAD features that are helpful in evaluating and reviewing a proposed design include:

• Automatic dimensioning routines that determine precise distance measures between surfaces on the geometric model identified by the user.

• Error checking. This term refers to CAD algorithms that are used to review the accuracy and consistency of dimensions and tolerances and to assess whether the proper design documentation format has been followed.

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Standards for dimensioning

A drawing is expected to convey a complete description of every detail of a part. However, dimensioning is as important as the geometric information. According to the American National Standards Institute (ANSI) standards, the following are the basic rules that should be observed in dimensioning any drawing:

1. Show enough dimensions so that the intended sizes and shapes can be determined without calculating or assuming any distances.

2. State each dimension clearly, so that it can be interpreted in only one way.

3. Show the dimensions between points, lines, or surfaces that have a necessary and specific relation to each other or that control the location of other components or mating parts.

Standards for dimensioning

4. Select and arrange dimensions to avoid

accumulations of tolerances that may

permit various interpretations and cause

unsatisfactory mating of parts and

failure in use.

5. Show each dimension only once.

6. Where possible, dimension each feature

in the view where it appears in profile,

and where its true shape appears.

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Conventional tolerance

Since it is impossible to produce the exact dimension specified,

a tolerance is also used to show the acceptable variation in a

dimension. The higher the quality a product has, the smaller the

tolerance value specified. Tighter tolerances are translated into

more careful production procedures and more precise inspection.

There are two types of tolerances: bilateral tolerance and unilateral

tolerance (as shown in the following Figure). Unilateral tolerances,

such as , specify dimensional variation from the basic size

(i.e., decrease) in one direction in relation to the basic size; for

example,

The basic location where most dimension lines originate is the

reference location (datum). For machining, the reference location

provides the base from which all other measurements are taken. By

stating tolerance from a reference location, cumulative errors can be

eliminated.

00.0

05.000.1

00.195.000.1 00.0

05.0

Conventional tolerance

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Conventional tolerance- Most mechanical parts contain both working surfaces and nonworking surfaces. Working surfaces are those for items such as bearings, pistons, and gear teeth, for which optimum performance may require control of the surface characteristics. Nonworking surfaces, such as the exterior walls of an engine block, crankcase, or differential housings, seldom require surface control. For surfaces that require surface control, control surface symbols can be used.

- In the symbol, several surface characteristics are specified. The roughness height is the roughness value as normally related to the surface finish. It is the average amount of irregularity above or below an assumed centerline. It is expressed in micro inches or, in metric system, in micrometers.

Conventional toleranceSurface control symbols

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Dimensioning

TOLERANCE

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TOLERANCE

1. Check that the tolerance & dimension specifications are reasonable

for assembly.

2. Check there is no over or under specification.

TOLERANCE

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TOLERANCE

TOLERANCE GRAPH

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CAD Systems Architecture

Modeling objects

The model of an engineering object consists of geometry, topology, and auxiliary information. Geometry includes points, lines, circles, planes, cylinders and other surfaces. It defines the basic shape characteristics. Topology represents the relationships of the geometry of an object. In addition to its shape, an engineering object also possesses some other attributes. Dimensions, tolerances, and surface finish are some important attributes.

CAD Systems Architecture

Functions of CAD Systems

CAD is a tool not only to represent an

engineering model, but also to manipulate it. To

construct or display a model, geometric

transformations and view transformations are

needed.

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Modeling

Many properties of products have to be

modeled, including form, dimension,

tolerance and structure. In all of these areas

geometry, images and spatial manipulation are

very important. For this reason, CAD is

founded on computational geometry and

computer graphics.

Defining the ModelRepresentation of Models

There are two types of models:

Models of form typically represented by drawings of components and their arrangement in assemblies.

Models of structure normally represented by diagrams that show the components of a system and how they are connected.

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Defining the Model

The representation of form using drawings

The technique of representing three-dimensional forms in two-dimensional space by means of engineering drawings -on paper or on a computer screen- is formally known as descriptive geometry.

Defining the ModelThe representation of structure using diagrams

In engineering diagrams the logical or physical structure of a system, in terms of the assembly of the primitive parts and the relationship between these, is shown by a series of symbols joined by connections. The rules for the symbols, and for the connections, are governed by conventions that have been established in standards.

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Defining the ModelExamples of Electrical and Fluid Power Symbols

Defining the ModelBlock Diagrams

• At an early stage in the design process it may only be possible to define overall relationships between parts of a system, and a block diagrammay be most appropriate.

• As a design is prepared for construction and manufacture, detailed wiring or piping diagrams are required.

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Defining the ModelBlock diagram of injection system

Defining the ModelTop-Down Design

By exploiting representations such as block diagrams, the designer is able to subdivide a design problem into smaller elements. These in turn may be subdivided, such that a hierarchical decomposition of the problem is obtained. This

technique is known as "top-down" design.

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Defining the ModelExample: Top-Down Design “A hierarchical

arrangement of diagrams”

Defining the Model

computer representation of drawings and diagrams

Defining the graphic elements

The user has a variety of different ways to call a

particular graphic element and position it on the

geometric model. There are several ways of

defining points, lines, arcs, and other components

of geometry through interaction with the ICG

(interactive computer graphics) system. These

components are maintained in the database in

mathematical form and referenced to a 3D

coordinate system.

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Defining the ModelBasic geometry

A component must be modeled before it can be drawn.

Defining the ModelMethods of defining elements in interactive computer

graphics

Points

Methods of defining points in computer graphics include:

1. Pointing to the location on the screen by means of

cursor control.

2. Entering the coordinates via the alphanumeric

keyboard.

3. Entering the offset (distance in x, y, and z) from a

previously defined point.

4. The intersection of two lines.

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Defining the Model

Defining the ModelLines

Methods of defining lines include:

1. Using two previously defined points.

2. Using one point and specifying the angle of the line with the horizontal.

3. Using a point and making the line either normal or tangent to a curve.

4. Using a point and making the line either parallel or perpendicular to another line.

5. Making the line tangent to two curves.

6. Making the line tangent to a curve and parallel or perpendicular to a line.

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Defining the Model

Defining the ModelArcs and circles

Methods of defining arcs and circles include:

1. Specifying the center and the radius.

2. Specifying the center and a point on the circle.

3. Making the curve pass through three previously defined

points.

4. Making the curve tangent to two lines.

5. Specifying the radius and making the curve tangent to

two lines or curves.

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Defining the Model

Defining the Model

The curves and the surfaces should be

discussed at notebook

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Fundamentals of Solid Modeling


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Constructive Solid Geometry

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Boundary Representations

Objects are rep. by a collection of bounding faces plus topological information, which defines relationship:

Between faces, edges and vertices

Hierarchy:

Faces are composed of edges

Edges are composed of vertices

B-Reps are difficult to create but provide easy graphics interaction and display.

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Boundary Representation

A solid composed of faces, edges and vertices

E1F3

E2

E3

E4

E5E6

E7E8

V1

V2

V3V4

F1

F2

F4

F5

Fundamentals of Solid ModelingB.Rep. Models

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Fundamentals of Solid ModelingB.Rep. Model of Tetrahedron


Validity of an Engineering Part or Object

Polyhedron: a part which has flat or planar polygonal surfaces

only.

For the validity test of solids, Euler’s formula can be used.

For Polyhedrons without holes:

(Number of faces) + (Number of vertices) = Number of edges +2

F+V = E+2,

where F, E and V are number of faces, edges and vertices.

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For Polyhedrons with through – holes:

F+V = E+2+R-2H,

where R is the number of disconnected interior edge

rings in faces,

H is the number of holes in the body



Example: Euler’s formula

F+V = E+2,

F = 6, V = 8, E = 12

6 + 8 = 12 + 2

14 = 14 (valid object)

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Example: Object with through-hole

F+V = E+2+R-2H,

F = 10(6 plus additional 4)

V = 16, E = 24

R = 2 (as its through hole)

H = 1

10 + 16 = 24 +2 +2 –2(1)

26 = 26



Example: Part with blind hole

Formula check: F+V = E+2+R

F = 6+5 = 11

V = 16, E = 24

R = 1(as its blind hole)

H = 0

11 +16 = 24 +2 +1 – 2(0)

27 = 27

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Example: Part with Projection

F + V = E +2 +R-2H

F =11(6 + 4 +1)

V = 16, E = 24, H = 0

R = 1 (at base of projection)

F + V = E + 2 +R – 2H

11 +16 = 24 +2 +1-2(0)

27 = 27

For 2 projections on a part,

F=16, V=24, E=36, R=2, H=0

16+24 = 36 +2+2

40 = 40



Example: Projection and Blind Hole

F + V = E + 2 +R –2H

F=5+11 (from previous slide) =16

V=8+16=24

E=12+24=36

R=1+1 (at base of projection and top

of hole)

F+V = E+2+R-2H

16+24 = 36+2+2-2(0)

40 = 40

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Example: Projection and Through Hole

F + V = E + 2 +R –2H

F=4+11 (from previous slide) =15

V=8+16=24

E=12+24=36

R=1+2 (at base of projection and top

of hole)

F+V = E+2+R-2H

15+24 = 36+2+3-2(1)

39 =39


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Entry Manipulation and Data Storage

Manipulation of the Model

Manipulation Modification of drawings, erase unwanted parts,

move some geometry around the drawing, or to copy some repeated

detail.

The facilities that typically provided for manipulation of the

model: Four groups of functions:


1. Those that apply the transformations of translation, rotation and

scaling to elements of the model (moving the geometry, copying

the geometry to create one or more duplicate sets of entities in

the data structure.

2. Those that allow the user to make changes to individual

geometric elements to trim or extend them to their intersections

with other elements.

3. Functions for the temporary or permanent deletion of entities

from the model.

4. Miscellaneous functions that, for example, allow entities to be

grouped together.

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Transformations

1. Object transformations

Object transformations mathematical operations of

the manipulation.

When the entities of a CAD model are manipulated by

moving them around, or by taking one or more

copies at different locations and orientations, we

image the coordinate system to be stationary, and

the object to move.


Transformations

2. Coordinate system transformations

we image the object to be stationary, and the coordinate

system to move.

Coordinate system transformations = Viewing

transformation.

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TransformationsThe main task:

Define the new object (Transformed)

How?

where,

is the object new coordinates matrix (new object)

is the object original coordinates matrix, or points matrix

(original object)

is the transformation matrix

]][[*][ TPP

*][P

][P

][T

Transformations

TRANSFORMATIONS• The aim of these lectures and notes is to give an

understanding of what is happening within CAD

systems.

• By understanding how something works allows us

to use it more effectively.

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Transformations

TRANSFORMATIONSCAD and Geometry

The simplest CAD systems are 2D or 3D drafting tools. They

allow geometry to be created, stored and manipulated.

Example: A line might be stored as two points:

L→P1P2→(x1,y1; x2y2)

Or in matrix notation:

Where:

The graphical representation:

22

11

2

1

yx

yx

P

PL

222111 , yxPyxP

Transformations


Example: Representation of a Triangle: (in 2D ordinary

coordinates)

Graphical representation:

33

22

11

3

2

1

yx

yx

yx

P

P

P

P

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Transformations


In this format it is not easy to do matrix manipulation in 2D or 3D

(which is what we want to do). Thus we want homogeneous

coordinates.

Homogeneous Coordinates:

Presents a unified approach to describing geometric

transformations.

:

TransformationsHomogeneous Coordinates

Assume a 2D point lies in 3D space.

Any 2D point can be represented in such a 3D space as:

P(x, y, z) = P(hx1, hy1, hz)

That is, along a ray from the origin (called homogeneous space).

:

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TransformationsHomogeneous Coordinates

For instance, consider point P(2, 4) in ordinary coordinates. This

can be considered as:

P(4, 8, 2), where h=2; or P(6, 12, 3), where h=3; or P(2, 4, 1),

where h=1 in homogeneous space.

In general, P(m, n, h) in homogeneous space is P(m/h, n/h, 1) in

ordinary coordinates.

Thus, the triangle in 2d space can be represented in

homogeneous coordinates as:

Why? To help with transformations.

1

1

1

33

22

11

yx

yx

yx

P

TransformationsTRANSFORMATIONS

Transformation is the backbone of computer graphics, enabling

us to manipulate the shape, size, and location of the

object.

It can be used to effect the following changes in a geometric

object:

1. Change the location

2. Change the shape

3. Change the size

4. Rotate

5. Copy

6. Generate a surface from a line

7. Generate a solid from a surface

8. Animate the object

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Transformations

Types of transformations1. Modeling Transformation/ Object Transformation

This transformation alters the coordinate values of the object. Basic

operations are scaling, translation, rotation and combination of one

or more of these basic transformations.

Object transformation = Move (transform) an object in the 3D space.

2. Visual/ Viewing Transformation (Coordinate System

Transformation)

In this transformation there is no change in either the geometry or the

coordinates of the object. A copy of the object is placed at the

desired sight, without changing the coordinate values of the object.

Coordinate system transformation = Move (transform) the coordinate

system. View the objects from the new coordinate system.

Transformations

Examples

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Transformations

Examples

Transformations

Examples

Coordinate System Transformation

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Transformations

Basic Modeling/Object Transformations

Scaling, translation, and Rotation.

Other transformations, which are modification or

combination of any of the basic transformations,

are Shearing, Mirroring, Copy, etc.

Transformation can be expressed as:

where, is the new coordinates matrix

is the original coordinates matrix, or points matrix

is the transformation matrix

]][[*][ TPP

*][P

][P

][T

TransformationsScaling

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Transformations

Scaling

Or in matrix form:

This is object scaling about the origin.

If sx = sy → → uniform scaling → → Magnify command

yx ysyxsx

yxPyxP

*,*

*)*,(*),(

1100

00

00

11*** yxy

x

ysxss

s

yxyxP


1. Uniform Scaling

For uniform scaling, the scaling transformation matrix is given as:

• In ordinary 3D coordinate system:

Here, s is the scale factor

• In homogeneous 3D coordinates:

s

s

s

Ts

00

00

00

][

1000

000

000

000

][s

s

s

Ts

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2. Non-Uniform Scaling

• Scaling transformation matrix in 3d ordinary coordinates:

• In 3d Homogeneous Coordinates:

where, , are the scale factors for the x, y, and z coordinates

of the object.

z

y

x

s

s

s

s

T

00

00

00

][

1000

000

000

000

][z

y

x

ss

s

s

T

zyx , sss ,

Transformations

Example: If the triangle A(1, 1), B(2, 1), C(1, 3) is

scaled by a factor 2, find the new coordinates of

the triangle.

Solution:

Writing the points (original) matrix in homogeneous 3D

coordinates, we have

1031

1012

1011

][P

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TransformationsThe scaling matrix is:

The new points matrix can be evaluated by the equation:

1000

0200

0020

0002

][ sT

]][[*][ TPP

1062

1024

1022

*][P

Transformations

Translation Transformation

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Transformations


Or in matrix form (homogeneous coordinates):

You can now see that homogeneous coordinates are needed

for translation transformation.

This is what the Move command does in CAD systems.

yyy

xxx

*

*

1

010

001

11***

yx

yxyxP

Transformations


In translation, every point on an object translates exactly the

same distance. The effect of translation transformation is

that the original coordinate values increase or decrease by

the amount of the translation along the x, y, and z-axes.

The translation transformation matrix has the form:

In 3D Homogeneous Coordinates:

where are the values of translation in the x, y, and z direction,

respectively.

For translation transformation, the matrix equation is:

1

0100

0010

0001

][

zyx

Tt

]][[*][ tTPP

zyx ,,

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Transformations

Translation TransformationExample: Translate the rectangle (2, 2), (2, 8), (10, 8), (10, 2) 2 units

along x-axis and 3 units along y-axis.

Solution: Using the matrix equation for translation, we have:

Substituting the numbers, we get

Note that the resultant coordinates are equal to the original x and y

values plus the 2 and 3 units added to these values, respectively.

]][[*][ tTPP

10512

101112

10114

1054

1032

0100

0010

0001

10210

10810

1082

1022

*][P

TransformationsRotation

• We will first consider rotation about the z-axis, which passes through the

origin (0, 0, 0), since it is the simplest transformation for understanding the

rotation transformation. Rotation about an arbitrary axis, other than an

axis passing through the origin, requires a combination of three or more

transformations.

• When an object is rotated about the z-axis, all the points on the object

rotate in circular arc, and the center of the arc lies at the origin. Similarly,

rotation of an object about an arbitrary axis has the same relationship with

the axis, i.e., all the points on the object rotate in circular arc, and the

center of rotation lies at the given point through which the axis is passing.

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Derivation of the Rotation Transformation Matrix

Original coordinates of point P:

sin ,cos ryrx



The new coordinates:

Using the trigonometric relations, we get:

We get:

In matrix form:

)sin(* ),cos(* ryrx

cossin)cossinsin(cos*

sincos)sinsincos(cos*

yxry

yxrx

cossin

sincos** yxyx

sincoscossin)sin(

sinsincoscos)cos(

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In general, the points matrix and the transformation matrix are re-written

as (For 2D objects): [In Homogeneous Coordinates]

OR:

1000

0100

00cossin

00sincos

1010**

yxyx

100

0cossin

0sincos

11**

yxyx



• For 3D geometry: Rotation about z-axis

1000

0100

00cossin

00sincos

11***

zyxzyx

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• Transformation matrix for rotation about y-axis:

1000

0cos0sin

0010

0sin0cos

yRT



• Translation matrix for rotation about x-axis:

1000

0cossin0

0sincos0

0001

xRT

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• For use with 2D geometry:

• For use with 3D geometry:

This is what the Rotate command does in CAD system.

10***][ ,10][ yxPyxP

1****][ ,1][ zyxPzyxP

TransformationsRotation of an Object about an Arbitrary Axis

Rotation of a geometric model about an arbitrary axis, other than

any of the coordinate axes, involves several rotational and

translational transformations. When we rotate an object

about the origin (in 2D), we in fact rotate it about z-axis.

Every point on the object rotates along a circular path, with

the center of rotation at the origin. If we wish to rotate an

object about an arbitrary axis, which is perpendicular to the

xy-plane, we will have to first translate the axis to the origin

and then rotate the model, and finally, translate so that the

axis of rotation is restored to its initial position.

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Thus, the rotation of an object about an arbitrary axis, involves

three steps:

Step 1: Translate the fixed axis so that it coincides with the z-axis

Step 2: Rotate the object about the axis

Step 3: Translate the fixed axis back to the original position

(reverse translation)

Note: When the fixed axis is translated, the object is also

translated. The axis and the object go through all the

transformations simultaneously.


Example:

Rotate the rectangle (0, 0), (2, 0), (2, 2), (0, 2) shown

below, 30o ccw about its centroid and find the

new coordinates of the rectangle.

68

Rotation of an Object about an Arbitrary Axis


69


Transformations

Rotation about an Arbitrary Point (in xy-plane)In order to rotate an object about a fixed point, the point is first moved

(translated) to the origin. Then, the object is rotated around the

origin. Finally, it is translated back so that the fixed point is

restored to its original position. For rotation of an object about an

arbitrary point, the sequence of the required transformation

matrices and the condensed matrix is given as:

OR:

where is the angle of rotation and the point (x, y) lies in the xy-plane.

]][][[][ cond trt TTTT

10

0100

0010

0001

1000

0100

00cossin

00sincos

10

0100

0010

0001

][ cond

yxyx

T

70

Transformations

Rotation about an Arbitrary Point (in xy-plane)Solution: We first translate the point (3, 2) to the origin, then rotate the rectangle

about the origin, and finally, translate back so that the original point is

restores to its original position (3, 2). The new coordinates of the rectangle

are found as follows:

These are the new coordinates of the rectangle after the rotation.

1063.063.2

1037.263.1

1087.177.0

1013.077.1

1023

0100

0010

0001

1000

0100

00866.05.0

005.0866.0

1023

0100

0010

0001

1031

1032

1012

1011

]][][][[*][ trt TTTPP

Transformations

Mirroring (Reflection)

• In modeling operations, one frequently used operation is mirroring an

object. Mirroring is a convenient method used for copying an object

while preserving its features. The mirror transformation is a special

case of a negative scaling, as will be explained below.

• Let us say, we want to mirror the point A(2, 2) about the x-axis (i.e.,

xz-plane). The point matrix [P*]=[2 -2] can be obtained with the

matrix transformation given below:

71

Transformations


The transformation matrix above is a special case of non-uniform scaling

with sx=1 and sy=-1.

1022

1000

0100

0010

0001

1022*][

P

Transformations


Transformation Matrix for Mirroring about x-axis:

Transformation Matrix for Mirroring about y-axis:

1000

0100

0010

0001

xmT

1000

0100

0010

0001

ymT

72

Transformations

Mirroring about an Arbitrary Plane

If mirroring is required about an arbitrary plane, other

than one defined by the coordinate axes, translation

and/or rotation can be used to align the given plane

with one of the coordinate planes. After mirroring,

translation or rotation must be done in reverse order to

restore the original geometry of the axis.

We will use the figure shown below, to illustrate the

procedure for mirroring an object about an arbitrary

plane. We will mirror the given rectangle about a plane

passing through the line AB and perpendicular to xy-

plane.

TransformationsMirroring about an Arbitrary Plane

73

TransformationsMirroring about an Arbitrary Plane

It should be noted that in each of the

transformations, the plane and the rectangle

have a fixed relationship, i.e., when we move

the plane (or line AB), the rectangle also moves

with it.

Note: We are using line AB to represent the

plane, which passes through it. Mirroring can be

done only about a plane, and not about a line.

Transformations

Mirroring about an Arbitrary Plane

Procedure for mirroring the rectangle about the plane:

Step 1: Translate the line AB (i.e., the plane) such that it passes through

the origin, as shown by the dashed line.

Step 2: Next, rotate the line about the origin (or the z-axis) such that it

coincides with x or y axes (we will use the x-axis).

Step 3: Mirror the rectangle about the x-axis.

Step 4: Rotate the line back to its original orientation.

Step 5: Translate the line back to its original position

The new points matrix, in terms of the original points matrix and the five

transformation matrices is given as:

[P*] = [P][Tt][Tr][Tm][T-r][T-t]

Where, the subscripts t, r, and m represent the translation, rotation, and

mirror operations, respectively.

Note: A negative sign is used in the subscripts to indicate a reverse

transformation.

74

Transformations


Coordinate frame moves to a new location.

Transformations


The origin has been translated (moved) from (0, 0, 0) to (a, b, c)

Or:

For coordinate system:

For the object:

zyxP

czbyax , ,

czbyax , ,

]][[*][ TPP

1

0100

0010

0001

cba

Tt

75

Transformations


Note: The sign in the T matrix need to be changed

1*][ czbyaxP

Transformations

Example: Coordinate Transformation

If the coordinate system has been rotated about z-axis by -30o, then

translated to [a b c], what is the coordinates of the point

[x y z] in the new coordinate system?

76

Transformations

Example: Coordinate Transformation

Solution:

1***

1

0100

0010

0001

1000

0100

0030cos30sin

0030sin30cos

1*][

zyx

cba

zyxPoo

oo

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