1. Lectures 1234567 Computer Aided Design
Hikmet Kocabaş, Prof., PhD.
Istanbul Technical University

2. Computer Aided Design Prof. Dr. Hikmet Kocabas
I.T.U. Faculty of Mechanical Engineering
office no: office phone: / 2468
web site:
Interests: Mechanical Design
Finite Element Analysis
Mechanism Simulation
CAD – CAE – CAM

3. Intro. to Computer Graphic Systems
Textbooks:
Computer Aided Engineering Design, Anupam Saxena, Birendra Sahay, Springer, 2005
Engineering Design A Systematic Approach, G. Pahl, W. Beitz, Springer-Verlag, 2007

4. Lectures 1234567 Lecture 1 Intro. to Computer Graphic Systems
Lecture 2 Geometry
Lecture 3 Vector Algebra
Lecture 4 Transformations
Lecture 5 Curves
Lecture 6 Surface Modeling
Lecture 7 Solid Modeling

5. Computer Aided Design

6. Geometry
solids
A typical solid model is defined by solids, surfaces, curves, and points.
Solids are bounded by surfaces. They represent solid objects.
Surfaces are bounded by lines. They represent surfaces of solid objects, or planar or shell objects.
Curves are bounded by points. They represent edges of objects.
Points are locations in 3-D space. They represent vertices of objects.
lines, curves, points
surfaces

7. geometric elements

8. Geometry There is a built-in hierarchy among solid model entities.
Points are the foundation entities.
Curves are built from the points,
Surfaces from curves,
and Solids from surfaces.
Points
Curves
Surfaces
Solids
Points
Curves
Surfaces
Solids
I’ll just
change
this line
Curves
Points
Surfaces
Solids
OOPs!

9. Line (Combinations of Points)
Let P1 and P2 be points in space.
if 0 ≤ t ≤ 1 then P is somewhere
on the line segment joining P1 and P2 .
We may utilize the following notation
P = P(t) = (1 - t) P1+ t P2
α2= t
α1= 1- t 1 P1 P2
Linear interpolation P1 P2
P1+ t (P2 - P1)
t (P2 - P1) P

10. Line (Combinations of Points)
We can then define a combination
of two points P1 and P2 to be
P = α1 P1+ α2 P2 where α1 + α2 = 1
derive the transformation by setting α2 = t
α2= t
α1= 1- t 1 P1 P2
Linear interpolation P1 P2
P1+ t (P2 - P1)
t (P2 - P1) P

11. Linear Surface We can generalize the line to define a combination of
an arbitrary number of points.
P = α1 P1+ α2 P2 + α3 P3
where α1 + α2 + α3 = 1
0 ≤ α1 , α2 , α3 ≤ 1
Illustration shows the
point P generated when
α1 = α2 = 1/4
α3 = 1/2 P2 P1
α3 (P3 - P1) P3 P
α2 (P2 - P1)

12. Lecture 5 Curves
Both Bézier curves and B-splines are polynomial parametric curves. Polynomial parametric forms can
not represent some simple curves such as circles.
Continuity

13. Curves
Bézier curves and B-splines are generalized to rational Bézier curves and Non-Uniform Rational B-splines, or NURBS for short. Rational Bézier curves are more powerful than Bézier curves since the former now can represent circles and ellipses. Similarly, NURBS are more powerful than B-splines. The relationship among these
NURBS
B-spline
Bezier
Rational Bezier

14. de Casteljau’s Algorithm
Curve Subdivision

15. Tangent, Normal, Curvature of a Curve

16. Degree Elevation of a Curve

17. Rational Bezier curve Effect of the weights of the control points
of a Rational Bezier curve

18. Bezier curves n=3 C(n,i) = n! / ( i! (n-i)!)
B(i,n,u) = C(n,i) ui (1-u)n-i
P(u) = ∑ B(i,n,u) Pi
i=0 n

19. Bezier curves
The order of the degree of curve is variable and related to the number of points defining it. n+1 points define an n'th degree curve which permit higher-order continuity.
Control points form the vertices of what is called characteristic polygon. Curve lies entirely within the convex hull defined by the polygon vertices.
Only the first and the last control points or vertices of the polygon actually lie on the curve. The other vertices define the order, derivatives, and the shape of the curve.
The curve is also always tangent to the first and last polygon segments. In addition, the curve shape tends to follow the polygon shape.

20. B-spline curves Piecewise Polynomials Approximating Splines Spline eq.
P(u) = ∑ N(i,k,u) Pi
i=0 n

21. B−spline Curves a B−spline curve is the union of a number of
Bezier curves joining
together with
C − continuity.

22. B-Spline Surfaces

23. B-spline curves Since a NURBS curve is rational, circles, ellipses and
N0, N1,0 N2,0 N3,0 N4,0 N5,0 N6,0 N7,0
N0, N1,1 N2,1 N3,1 N4,1 N5,1 N6,1
N0, N1,2 N2,2 N3,2 N4,2 N5,2
N0, N1,3 N2,3 N3,3 N4,3
N0, N1,4 N2,4 N3,4
B-spline curves
Since a NURBS curve
is rational, circles,
ellipses and
hyperbolas
can be represented
NURBS

24. De Boor’s Algorithm De Boor’s algorithm is implemented
by repeated knot
insertion.

25. Shape Editing
The local modification property guarantees that only part of the curve will be affected when a control point changes its position.
In fact, position change
is parallel to the
Movement of
the control point.

26. Weight Change Increasing weight pulls the curve toward the
selected control point.
Decreasing weight
pushes the curve
away from the

27. Surface Modeling Control points Bezier, B−spline and
NURBS surface is a
tensor product surface
and is the product
of two curves.
Surfaces are defined by a grid and have two sets of parameters, two sets of knots and so on.

28. Surface Modeling
Surface by Recursive Subdivision Algorithm

29. Surface Modeling
Surface subdivision

30. Surface Modeling Surface Reconstruction by Recursive Subdivision
Curve-based design
and
Surface deformation

31. Surface Modeling
Finite Element Simulation

32. Interpolation and Approximation

33. Surface Modeling Extension of curve modeling
Parametric representation:
p = p(u,v)
which is equivalent to
x = x(u,v)
y = y(u,v)
z = z(u,v)

34. Surface Modeling
Defining a surface with hole

35. Surface Patch Isoparametric curves can be used for tool
path generation.

36. Linearly Blended Coons Patch
Surface is defined by linearly interpolating between the boundary curves
Simple, but doesn’t allow adjacent patches to be joined smoothly P1
T(v)
R(v)
Q(u)
S(u)

37. Linearly Blended Coons Patch
T(v)
R(v)
Q(u)
S(u)
Linearly Blended Coons Patch
P1(u,v)
P2(u,v)
P3(u,v)
P(u,v) = + -

38. Bicubic Patch Extension of cubic curve
16 unknown coefficients – 16 boundary conditions
Tangents and “twists” at corners of patch can be used
Like Lagrange and Hermite curves, difficult to work with

39. Bezier Surfaces Bezier curves can be extended to surfaces
Same problems as for Bezier curves:
no local modification possible
smooth transition between adjacent patches difficult to achieve
Isoparametric curves used for tool path generation.

40. B-Spline Surfaces
As with curves, B-spline surfaces are a generalization of Bezier surfaces
The surface approximates a control polygon
Open and closed surfaces can be represented

41. B-Spline Surfaces

42. Surfaces from Curves Tabulated cylinder (extrusion)
Ruled surface (lofting or spined)
Surface of revolution
Swept surface
Sculptured surface

43. Tabulated Cylinder Project curve along a vector
In SolidWorks, created by extrusion
Generating curve C
P(u,v) = C(u)+ V(v) v u
V(v)
C(u)
Vector V

44. Ruled Surface Linear interpolation between two edge curves
Created by lofting through cross sections u v
C1(u)
C2(u)
P(u,v) = (1-v) C1(u)+ v (C2(u)
Edge curve 2
Edge curve 1
Linear interpolation

45. Surface of Revolution Revolve curve about an axis Axis C2(u) C1(u) v
P(u,v) = C1(u)+ v (C2(u) – C1(u))
C2(u)

46. Swept Surface Defining curve swept along an arbitrary spine curve
C2(v)
P(u,v) = C1(u)+ C2(v)
Spine

47. Lofted Surface Defining curve swept along Surface Pipe
an arbitrary spine curve
Defining curves
Spine

48. Spined Surface

49. Sculptured Surface General surface Composed of connected
surface patches

50. Fillets and Blends
Often necessary to create a blend between intersecting surfaces
Most common application is a fillet
Fillet required here

51. Face Blend Fillet

52. Fillet between two selected surfaces
Spined fillet with drive curve

53. Surface functions Curve in surface intersection Offset Surface Planar
development
of surfaces

54. Fillet between three surfaces
Corner fillet

55. Fillet between three curves
Triangular Surface
by Coon patch

56. Sheet metal bending
Unfolding

57. Sheet metal bending

58. Offset curves on Surface

59. Molding and silhouettes

60. Lecture 7 Solid Modeling
Data storage structures
Vertex# Location Edge# Vertices Polygon# Edges
V ,5, E V1,V P E1,E3,E2
V ,15, E V1,V P E3,E4,E5
V ,10, E V2,V4
V ,0, E V2,V3
E V3,V4 E1 E3 E2 E4 E5 V3 V1 V4 V2 P1 P2 V3 V1 V4 V2 P1 P2
Vertex# Location Polygon# Vertices
V ,5, P V1,V2,V4
V ,15, P V2,V3,V4
V ,10,0
V ,0,0

61. Lecture 5 Solid Modeling
Data storage structure tree
????
Feature
Material = 1040
Volume
BOX
Boundary
grind
face1
face2
face6
Element
line1
line2
line4
???
Length tol. = 0.002

62. Solid modeling approaches
Boundary Representation
(B-rep)
Constructive Solid Geometry
(CSG)
Sweep

63. Solid modeling approaches
Hybrid
(Feature based modellers)
Analitical Solid Modeling
(ASM)

64. Solid Modeling B-rep modeling data structure
Octree solid representation

65. Manifold and non-manifold modeling

66. Constructive Solid Geometry (CSG)
Based on simple geometric primitives
cube, parallelepiped, prism, pyramid, cone, sphere, torus, cylinder, solid by points etc.
Primitives are positioned and combined using boolean operations
union (addition)
difference (subtraction)
Intersection
Represented as a boolean tree

67. CSG Primitives

68. Half Spaces used in CSG modeling
Surface descriptions

69. CSG modeling by Half Spaces

70. CSG boolean tree Examples
A-B
(A-B) È C
union (addition)
difference (subtraction) A B

71. Boundary Representation (B-Rep)
Solids represented by faces, edges and vertices
Topological rules must be satisfied to ensure valid objects
faces bounded by loop of edges
each edge shared by exactly two faces
each edge has a vertex at each end
at least 3 edges meet at each vertex

72. Boundary Representation (B-Rep)
Euler’s rule applies:
V – E + F = 2
where
V = number of vertices,
E = number of edges,
F = number of faces.
Euler-Poincare topological equation for solid with hole:
V - E + F - (L - F) -2 (S - G) = 0
where L = number of edge loops,
S = number of shells,
G = genus of solid (holes).
Surface must be closed

73. Boundary Representation (B-Rep)
Euler’s rule
V-E+F=2
Euler-Poincare rule
V-E+F-(L-F)-2(S-G)=0

74. Solid B-Rep Example
Vertices
Edges
Faces

75. Solid B-Rep Example Topology Same Geometry, Different Topology
Different Geometry,
Same Topology L2 L1 L3 C1 P1 R

76. Winged Edge Data Structure
Topology and geometry F1 F2 E
predecessor 2
successor 2
successor 1
predecessor 1 V2 V1
Vertex
Topology
Object
Genus
Face
Loop
Edge
Geometry
Underlying Surface equation
Curve equation
Point coordinates
Body

77. CSG vs. B-Rep CSG Simple representation Limited to simple objects
Stored as binary tree
Difficult to calculate
Rarely used anymore
B-Rep
Flexible and powerful representation
Stored explicitly
Can be generated from CSG representation
Used in current CAD systems

78. B-Rep of cylinder and circle
Boundary Model of Cylinder and Sphere
Limb E1 F1 V1
(silhouette edge) E3 F3 F2 V2 E2 V F
Limb

79. References
Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, Fourth Edition, Gerald Farin, September 1996
Geometric Modeling, Second Edition, Michael E. Mortenson, John Wiley & Sons, January 1997
The NURBS Book, Second Edition, Les A. Piegl and W. Tiller, Springer Verlay, January 1997
CAD/CAM Theory and Practice , Ibrahim Zeid,, McGraw Hill , 1991
Solid Modelling with DESIGNBASE, Hiroaki Chiyokura, Addison-Wesley Publishing Company, 1988.
Mathematical Elements for Computer Graphics, Rogers, D.F., Adams, J.A., McGraw Hill, 1990.