Curve and Surface

SPPU Solid Modeling and Drafting

1. Matoshri College of Engineering & Research Center
Course Name: Solid Modeling and Drafting
Course Code: 202042
Class: S.E. (Semester -I)
By Prof. H.B.Wagh
Matoshri College of Engineering and Research Centre, Nashik 1

2. Syllabus
Unit II Curves & Surfaces
Curves: Methods of defining Point, Line and Circle, Curve
representation -Cartesian and Parametric space, Analytical and
Synthetic curves, Parametric equation of line, circle, ellipse,Continuity
(C0, C1 & C2), Synthetic Curves -Hermit Cubic
Spline, Bezier, B-Spline Curve, Non-Uniform Rational B-Spline curves
(NURBS).
Surfaces: Surface representation, Types of Surfaces, Bezier,B-Spline,
NURBS Surface, Coonspatch surface, Surface Modeling.
Reverse Engineering: Introduction, Point Cloud Data (PCD), PCD file
formats, Quality issues in PCD, Requirements for conversion of surface
models into solid models, Applications of PCD.
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3. Curves and Surfaces
Types of Curves and Their Mathematical Representation
Types of Surfaces and Their Mathematical Representation
Reverse Engineering
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4. Objective:
To introduce the curves and surfaces and their implementation in
geometric modeling
Outcome: UTILIZE knowledge of curves and surfacing features and
methods to create complex solid geometry
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5. Geometric Modeling
Wireframe Modeling Surface Modeling
Solid Modeling
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6. Methods of defining points
1. Absolute Cartesian coordinates
2. Absolute cylindrical coordinates
3. Incremental Cartesian coordinates
4. Incremental cylindrical coordinates
5. Point of intersection
6. Defining middle or break point.
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8. Matoshri College of Engineering and Research Centre, Nashik 8

9. Methods of defining lines
1. Defining endpoints
2. Parallel or perpendicular to existing line
3. Vertical or horizontal line
4. Tangent to existing line.
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10. Defining points
Horizontal (Parallel to X or Y Axis)
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11. Parallel or perpendicular to existing line
Tangent to existing entities
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12. Methods of defining circles
1. Radius or Diameter and center.
2. Defining three points
3. Center and point on the circle
4. Tangent to a line, pass through a given point and with a radius
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13. 1.Radius or Diameter and center. 2. Defining three points
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14. 3. Center and point on the circle4. Tangent to a line, pass through a given point and with a radius
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15. Matoshri College of Engineering and Research Centre, Nashik 15

16. Curve Representation
All forms of geometric modeling require the ability to define curves.
A geometric model contains description of the modeled object’s
shape.
Since geometric shapes are described by surfaces, curves are used to
construct them.
Computer geometric modeling uses curves to control the object’s
surfaces as they are easy to manipulate.
The curves may be constructed using analytic functions, a set of
points, or other curves and surfaces.
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17. Curve Use in Design & Representation of “irregular surfaces”
Engineering design requires ability to express complex curve
shapes. –Some examples
•Turbine blades
•Ship hulls
•Auto industry (car body design)
•Artist’s representation
•Clay / wood models
•Surface modeling (“body in white”)
•Scaling and smoothening
•Tool and die Manufacturing
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18. Ideally, a curve should be:
Reproducible
–the representation should give the same curve every time
Computationally Quick
Easy to manipulate and edit
–especially important for design purposes.
Easy to combine with other curves
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19. Curve Representation
All forms of geometric modeling require the ability to
define curves.
A geometric model contains description of the modeled
object’s shape.
Since geometric shapes are described by surfaces, curves
are used to construct them.
Computer geometric modeling uses curves to control the
object’s surfaces as they are easy to manipulate.
The curves may be constructed using analytic functions, a
set of points, or other curves and surfaces.
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20. Representation of Curves
The curves can be mathematically represented by two
methods
1. Non-parametric Representation
2.Parametric Representation
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21. 1. Non-parametric Representation
Represented as relationship between x ,y and z.
Express in the form f(x,y,z)=0
E.g. y=mx+ c
f(x,y,z) = 0
x
2
+ y
2
–R
2
= 0
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22. Non Parametric Representation
1.Explicit Non Parametric Representation
2. Implicit Non Parametric Representation
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23. 1.Explicit Non Parametric
Representation
The equation of line
y = mx+c
y= f(x)= mx+c
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24. 2. Implicit Non Parametric Representation
When we have any equation of the form:
f (x, y, z, …. a, b, c,) = 0
We say that the equation is implicit.
A good example of an implicit equation is the equation of the circle:
??????
2
+y
2
-c
2
= 0
Straight Line: ax+ by+ c = 0
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25. Limitation of Non parametric Representation
Due to one to one relation between coordinates curves can not
use for closed curves like circles and ellipses, parabolas,
hyperbolas.
If straight line is vertical or near vertical its slope (m) is infinity
or very large value. Such values are difficult to handle
In Implicit form curves requires to solve simultaneous equation,
which is highly inconvenient and lengthy.
Implicit form represents unbounded geometry.
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26. Parametric Representation
Each point on curve is expressed as a function of parameter
u/s/t/ѳ
X = x(u)
Y = y(u) u min ≤ u ≤ u max
Z = z(u)
The parametric equations for a circle are:
x=r cos(ѳ) y=r sin(ѳ)
They are modeled as piecewise polynomial and have to
aspects
Interpolation & Approximation.
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27. Types of Curves
1. Analytic Curve 2. Synthetic Curve
1.Analytic Curve-defined by analytical equations.
E.g. Lines, Circles, ellipses, parabolas and hyperbolas.
2.Synthetic Curve-defined by set of data point.
Curve represented by collection of data points.
E.g. Various types of splinesand Bezier curves
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28. Analytic Curves
Line
Two cases
1.Line connecting Two End Points
2.Line starting from given point of Given Length and direction.
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29. 1.Line connecting Two End Points
Straight line connecting two end points P
1and P
2. The parameter
‘u’ such that its value are 0 and 1 at point P
1and P
2respectively.
1.Parametric equation of a straight line is given as,
P = P
1+ u(P
2-P
1) 0≤ u ≤1
In scalar form,
x = x
1+ u(x
2-x
1)
y = y
1+ u(y
2-y
1)0≤ u ≤1
z = z
1+ u(z
2-z
1)
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31. Line starting from given point of given length and
direction
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32. Problem No.1
Write equation of line having end points P1(3,5,8) and
P2(6,4,3). Find the tangent vector and coordinates of
point on line at u= 0.25,0.5,0.75
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34. Numerical No.2
A Line is represented by end points P(5,7,2) and Q(-4,6,3). If
‘u’ at P and Q is 0 and 1 resp. Determine its length. Also
determine the coordinates of points represented by u=
0.4,u=-0.25 and u = 1.5
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37. Circles
Parametric equation of circle
The general form of equation
x= x
c
+R cos(u)
y= y
c
+ R sin(u)
z= z
c
u= angle measured from X-axis to
any point P on the circle
0≤u≤2π
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38. Incremental form of equation
Coordinates of point P
n
on circle
x
n
= x
c
+R cos(u)
y
n
= y
c
+ R Sin(u)
z
n
= z
c
Coordinates of next point P
n+1
on circle with increment of ∆u
x
n+1
= x
c
+R cos(u+∆u)
y
n+1
= y
c
+ R Sin(u+∆u)
z
n+1
= z
c
Hence,
x
n+1
= x
c
+R cos(u)cos(∆u)-R sin(u)sin(∆u)
y
n+1
= y
c
+ R Sin(u)cos(∆u)+ R cos(u)sin(∆u)
z
n+1
= z
c
Matoshri College of Engineering and Research Centre, Nashik 38

39. Recursive relationship for coordinates of point on circle:
Substitute value of Rcos(u) and Rsin(u) as (x
n
-x
c
) and (y
n
-y
c
)
x
n+1
= x
c
+(x
n
-x
c
) cos(∆u)-(y
n
-y
c
)sin(∆u)
y
n+1
= y
c
+ (y
n
-y
c
)cos(∆u)+ (x
n
-x
c
) sin(∆u)
z
n+1
= z
c
The circle can start from any point and successive points with equal
spacing can calculated.
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40. CIRCULAR ARCS
x= x
c
+R cos(u)
y= y
c
+ R Sin(u) u
s
≤u≤u
e
z= z
c
where(x
c
, y
c
,z
c
)= center of arc
R=radius of arc
u
s
=starting angle of arc
u
e
=ending angle of arc
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41. Problem No.01
Write a parametric equation of a circle with center at
point (5,5,0) and with radius 5 units. Calculate
coordinates of the four quadrant points circles.
Solution-P
c (x
c,y
c,z
c)=(5,5,0) R=5
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42. 1.Parametric Equation of Circle:
x= x
c+R cos(u)
y= y
c+ R Sin(u) 0≤u≤2π
z= z
c
x= 5+R cos(u)
y= 5 + R Sin(u) 0≤u≤2π
z= z
c
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43. Coordinate of Points on circle
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44. Problem No.02
Write a parametric equation of a circle with center at point (3,3,0)
and with radius 3 units. Calculate coordinates of circles.
If it is divided in eight parts.
Solution-P
c (x
c,y
c,z
c)=(3,3,0) R=3
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45. 1.Parametric Equation of Circle:
x= x
c+R cos(u)
y= y
c+ R Sin(u) 0≤u≤2π
z= z
c
x= 3+R cos(u)
y= 3 + R Sin(u) 0≤u≤2π
z= z
c
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46. Coordinate of Points on circle
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47. Circle is represented by center point (5,5) and radius 6 units. Find
the parametric equation of circle and determine the various points on
the circle in first quadrant if increment of angle is 45
0
and 90
0
.
Solution-P
c (x
c,y
c,z
c)=(5,5,0) R=6
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48. 1.Parametric Equation of Circle:
x= x
c+R cos(u)
y= y
c+ R sin(u) 0≤u≤2π
z= z
c
x= 5+6 cos(u)
y= 5 + 6sin(u) 0≤u≤2π
z= z
c
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49. Coordinate of Points on circle
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50. Surfaces
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51. WireframeModeling
Wire-framemodeling usespointsandcurves(i.e.lines,
circles, arcs) to defineobjects.
The user uses edges and vertices of the part to form a 3-D object.
Wireframemodel
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52. Wireframe modeling -Advantages
Simple to construct
Less computer memory for storage compare to surface and solid
models.
Form the basis for surface models.
Can quickly and efficiently convey informationthan multi view
drawings
Can be used for finite element analysis.
Contain most of the information needed to create surface, solid and
higher order models
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53. Wireframe modeling -Disadvantages
Ambiguities present in the wireframemodel
Volume or surfaces of object notdefined
Forcomplexitems,theresultcanbeajumbleoflines
thatis impossible todetermine
Limited ability for checking interference between matingparts.
No ability to determine computationally information such as the line
of intersect between two faces of intersectingmodels.
Cannot be used to calculate dynamicproperties
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54. SurfaceModeling
“Asurfacemodelrepresentstheskinofanobject,
these skins have no thickness or material type”
Surfacemodelingismoresophisticatedthanwireframe
modelinginthatitdefinesnotonlytheedgesofa3Dobject,
butalsoitssurfaces.
Insurfacemodeling,objectsaredefinedbytheirbounding
faces.
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55. Surface modeling -Advantages
Eliminatesambiguityandnon-uniquenesspresentinwireframe
modelsbyhidinglinesnotseen.
Rendersthemodelforbettervisualizationandpresentation,
objectsappearmorerealistic.
ProvidesthesurfacegeometryforCNCmachining.
Providesthegeometryneededformoldanddiedesign.
Canbeusedtodesignandanalyzecomplexfree-formed
surfaces(carbodies)
Surfacepropertiessuchasroughness,colorandreflectivitycan
beassignedanddemonstrated.
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56. Surface modeling -Disadvantages
Surfacemodelsprovidenoinformationabouttheinsideofan
object.
Cannotbeusedtocalculatedynamicproperties.
Requiresmoretrainingandmathematicalbackgroundonthe
partofuser.
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57. Types of SurfaceEntities
Analytic entities include
1.Planesurface
2.Ruledsurface
3.Surface of revolution
4.Tabulatedcylinder
Synthetic entitiesinclude
1.Hermite Cubic Spline
surface
2.B-Splinesurface
3.Bezier surface
4.Coonspatches.
5.Fillet Surface
6.Offset Surface
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58. 1.Plane Surface
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59. 2.Ruled (lofted) surface.
This is a linear surface.
It interpolates linearly between two boundary curves that define
thesurface.
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60. 3.Surface of revolution
This is an axisymmetric surface that can model axisymmetric
objects.
It is generated by rotating a planar wireframe entity in space
about the axis of symmetry a certain angle.
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61. 4.Tabulated cylinder.
This is a surface generated by translating a planar curve a certain distance
along a specified direction (axis of thecylinder).
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62. Synthetic Surfaces
Defined by set of data points.
Needed when a surface is represented by a collection of data
points.
Represented by polynomials.
Examples
Bezier Surface, B-spline Surface , Coons patch , fillet surface,
and offset surface.
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63. 1.Beziersurface
This is a surface that approximates given input data.
Similarly to the Bezier curve, it does not pass through all given
data points.
It is a general surface that permits, twists, and kinks.
The Bezier surface allows only global control of thesurface
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64. 2.B-Spline surface
This is a surface that can approximate or interpolate given input
data.
It is a general surface like the Bezier surface but with the
advantage of permitting local control of thesurface.
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66. 3.Cone path
Interpolation with four curve
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67. 4.Filletsurface
5.Offsetsurface
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68. Solid Modeling
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69. Solid Model contains geometrical and topological information of
object.
It converted into wireframe models.
Used to generates orthographic views.
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70. Methods of Creating SolidModels
Constructive Solid Geometry (CSG)
CADpackages; Unigraphics, AutoCAD –3Dmodeler.
Boundary Representation (B-rep), mostly used in finite element
programs.
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71. Geometry and Topology
Geomerty
Actual dimensions that define the entities of object.
Topology
Different entities of the object are connected together
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72. Sr. No WireframeModeling Solid Modeling
1
Objectis create by using2D geometrical
entities such as Points, Line, curve,
Polygons
Object is created by using 3Dgeometrical entities known as
primitives
2 Containonly geometrical data. Contains both geometric data and topological information.
3
Not possible to calculateobject properties
such as mass, volume, moment of inertia.
Possible to calculateobject properties such as mass, volume,
moment of inertia.
4 Can notconverted to solid model Can be converted to wire-frame model.
5
Can not be usedto fully integrate and
automate the design and manufacturing.
Can be usedto fully integrate and automate the design and
manufacturing.
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73. Solid Entities(Primitive)
Solid Entities(Primitive)
Object is created by using the
3D geometric entities.
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74. Basic Primitive Solid:
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75. Method of Solid Modeling
1.Constructive Solid Geometry(CSG or C-REP)
Combiningbasicandgenerated(usingextrusionandsweeping
operation)solidshapes.
Objectsarerepresentedasacombinationofsimplersolidobjects
(primitives).
CSGusesBooleanoperationstoconstructamodel.
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76. ThreebasicBooleanoperations:
Union(Unite,join)-theoperationcombinestwovolumes
includedinthedifferentsolidsintoasinglesolid.
Subtract(cut)-theoperationsubtractsthevolumeofone
solidfromtheothersolidobject.
Intersection-theoperationkeepsonlythevolume
commontobothsolids
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77. Union
The sum of all points in each of two defined sets. (logical
“OR”)
Also referred to as Add, Combine, Join, Merge
Boolean Operation
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78. Difference
–The points in a source set minus the points common to a second set.
(logical “NOT”)
–Set must share common volume
–Also referred to as subtraction, remove, cut
Boolean Operation
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79. Intersection
–Those points common to each of two defined sets (logical
“AND”)
–Set must share common volume
–Also referred to as common
Boolean Operation
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80. Union
Subtract
Intersection
Boolean Operation
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81. Union
Solid Modeling Example Using
CSG
Cut
Cut
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82. CSG -Advantage
CSGispowerfulwithhighlevelcommand.
Easytoconstructasolidmodel–minimumstep.
Completehistoryofmodelisretainedandcanbealteredat
anypoint.
Canbeconvertedtothecorrespondingboundary
representation.
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83. Boundary Representation(B-rep)
Thistechniqueconsistsofthegeometricinformationaboutthe
faces,edgesandverticesofanobjectwiththetopologicaldata
onhowtheseareconnected.
B-repmodeliscreatedusingEuleroperation
Datastructure:
B-Repgraphstoreface,edgeandverticesasnodes,withpointers,or
branchesbetweenthenodestoindicateconnectivity.
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85. Building block of Boundary Represention(B-rep)
1.Vertex-Point in 3 dimensional space. point of intersection of two or
more edges.
2.Edge-Curve or line bounded by two vertices.
1.Intersection of two points.
3.Face-Closed surface bounded by three or more than three edges.
4.Loop-Hole in face.
1.Two dimensional entities.
5.Handle or Genus-Through hole in body or solid.
1.3D entities.
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86. Verification of Topological Validity of B-Rep Model-Euler’s
Equation
Euler’s equation for general
3D object
F-E+V-L = 2(B-G)
F=No of faces
E=No of edges
V= No of vertices
L=No of loop
G=No of genus or handles
B= No of bodies
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87. Verification of Topological Validity of B-Rep Model-
Euler’s Equation cont….
Euler’s equation for Simple 3D object
•L=0 G=0 and B=1
Simplified equation
•F-E+V = 2
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88. Verification of Topological Validity of B-Rep Model-
Euler’s Equation cont….
Euler’s equation for 2D object
•F-E+V-L = B-G
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89. Sweeping
Solid Model of an object is created by moving a surface along
given path.
Types of Sweeps
1.Linear Sweep
•Translation Sweep
•Rotational Sweep
2.Non-Linear Sweep
3.Hybrid Sweep
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90. 1.Linear Sweep
Surface moves in linear path.
Translation sweep-When
the planar domain is
translated
Rotational Sweep-when the
planar region is revolved.
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91. 2.Non-linear Sweep
Surface moves along the curved path.
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92. Feature Based Modeling
Most widely used method of creating solid models.
Feature-combination of shape and operation to build the part.
Shape-2D sketch.
Example-rectangle, square, triangle ,circle ,ellipse etc
Operation-converts sketch into 3D shape.
Example-extrude, revolve, sweep, fillet, chamfer etc
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93. Steps in Feature based
Modeling
1.Create Shapes(Sketch)
2.Create features
3.Combine Feature
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94. Creating Features fromSketches
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95. AppliedFeature
•Applied feature does not require asketch.
•They are applied directly to themodel.
•Fillets and chamfers are very common appliedfeatures.
Fillet
Chamfer
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96. Geometric Features
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103. Constraint Based Modeling
Sr.No Constraints Without constraints With Constraint
1 Coincident
2
Collinear
3 Concentric
4 Horizontal
5
Vertical
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104. Sr.No Constraints Without constraints With Constraint
6 Midpoint
7 Parallel
8 Perpendicular
9 Tangent
10 Symmetry
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105. Solid modeling -Advantages
Surfacemodels(uniqueness,non-ambiguous,realistic,surface
profile)plusvolumetricinformation.
Allowsthedesignertocreatemultipleoptionsforadesign.
2Dstandarddrawings,assemblydrawingandexplodedviewsare
generatedformthe3Dmodel.
CaneasilybeexportedtodifferentFiniteElementMethods
programsforanalysis.
Massandvolumetricpropertiesofanobjectcanbeeasilyobtained;
totalmass,masscenter,areaandmassmomentofinertia,volume,
radiusofgyration.
Matoshri College of Engineering and Research Centre, Nashik 105

106. Solid modeling -Disadvantages
More intensive computation than wireframe and surface modeling.
Requires more powerful computers (faster with morememory and
goodgraphics)
Matoshri College of Engineering and Research Centre, Nashik 106