GEOMETRIC MODELING

ME8691
COMPUTER AIDED DESIGN & MANUFACTURING
UNIT 2 – GEOMETRIC MODELING
S.BALAMURUGAN
ASSISTANT PROFESSOR
MECHANICAL ENGINEERING
AAA COLLEGE OF ENGINEEERING & TECHNOLOGY

GEOMETRIC MODELING
• It plays a crucial role in the overall application
of CAD-CAM-CAE system.
REQUIREMENTS OF GEOMETRIC MODELING
• Information entered through geometric
modeling is utilized in number of downstream
applications.
DESIGN ANALYSIS
• Evaluation of centroid, area (cross-sectional &
surface) & volume.
• Estimate the mass & Inertia properties.
• Interference checking in assemblies.
• Kinematic / Dynamic analysis & Simulation
• Finite element analysis for Stress, Vibration,
Thermal & Optimization
DRAFTING
• Automatic 2D view generation
• Automatic planar cross-sectioning
• Automatic dimensioning
ME 8691 COMPUTER AIDED DESIGN & MANUFACTURING S.BALAMURUGAN, AP/MECHANICAL, AAACET

MANUFACTURING
• NC Tool path generation & verification
• Manufacturing process simulation
• Part classification & Process planning
PRODUCT INDUSTRIAL & ENGINEERING
• Material Requirement Planning
• Scheduling
• Preparation of Bill of Material
• Marketing
GEOMETRIC MODELING

GEOMETRIC
MODELING
THREE DIMESIONAL 3 - D
WIREFRAME
MODELING
SURFACE
MODELING
SOLID
MODELING
TWO DIMENSIONAL 2 -D

TWO DIMESIONAL 2 – D
• To prepare manufacturing drawings
• Difficult to represent complex objects
THREE DIMENSIONAL 3 – D
• It provides all the information required for CAD-CAM-CAE applications
• Provide all details required from documentation to engineering analysis to
manufacturing.
GEOMETRIC MODELING

WIREFRAME MODELING
• In this method the complete object is represented by number of lines,
points, arcs & curves & their connectivity relationships.
ADVANTAGES
• The construction of a wireframe model is simple
• It does not require much computer time & memory
• It can be used for simple NC tool path generation
DISADVANTAGES
• It can not be used for calculation of mass, inertia properties
• The interpretation of wireframe models having many edges is very
difficult.

SURFACE MODELING
• The surface model is constructed essentially from surfaces such as planes, rotated
curved surfaces & even very complex synthetic surfaces.
• Surface creation on existing CAD system usually requires wireframe entities as a
start(Points & Curves)
• Surface & wireframe form the core of all existing CAD system
ADVANTAGES
• It is relatively more complete & less ambiguous representation than its wireframe
model
• This method is very much useful for specific non-analytic surfaces(Free form
surfaces) – Used in modeling automobile, airplane bodies & turbine blades etc.
• It is used in NC tool path generation, Sectioning & Interference detections
DISADVANTAGES
The calculations of mass & inertia properties would be difficult.

SOLID MODELING
• Solid modeling of an object is a more complete representation than surface
model, as all the information required for engineering analysis &
manufacturing can be obtained with this technique.
• It provides more Topology information in addition to the Geometric
information, helps to represent the object un ambiguously.
ADVANTAGES
• Solid modeling produces accurate design
• mass & inertia properties can be determined
• Provides complete 3D definition
• Improves the quality of design
• Improves Visualization
GEOMETRY
• Definition of the coordinates & dimensions of an object & its entities.
TOPOLOGY
• The connectivity & associativity of the object entities. It determines the
relationship between object entities.

ENGINEERING CURVES
ANALYTIC CURVES
• This curve described by analytic equations such as lines, circle, conics etc.,
• Provide very compact forms to represent shapes & simplify the computation
of related properties such as areas & volume.
• Analytic curves not sufficient to meet today’s geometric design requirements
of complex mechanical parts like automobile bodies, aero plane wings,
propeller blades etc.
• That require synthetic curves & surfaces (Free form surfaces)
SYNTHETIC CURVES
• This curves are defined by a set of data points(control points) such as
Splines, Bezier curve etc.
• Synthetic curves provide designers with great flexibility & control of a curve
shape by changing the positions of one or more data points.

NEED OF SYNTHETIC CURVES
The need for synthetic curves in design arises on two occasions
• When a curve is represented by a collection of measured data points (in
case of reverse engineering)[graphical visualization of experimental data]
• When an existing curve must change to meet new design requirements
INTERPOLATION TECHNIQUE APPROXIMATION TECHNIQUE
• Curve resulting form this technique pass
through the given data points.
Ex- Hermite Cubic Spline
• Produce curves that do not pass through
the given data points.
• The control points are used to control the
shape of the resulting curves.
• Ex- Bezier Curve

NON-PARAMETRIC REPRESENTATION ANALYTICAL CURVES
aXY
b
Y
a
X
RYX
cmXY
4
1
2
2
2
2
2
222
=
=+
=+
+=Line
Circle
Ellipse
Parabola
• Non-parametric representations of curve equations are used in some cases,
they are not in general suitable for CAD because:
• If the slope of a curve at a point is vertical or near vertical, its value becomes
infinity or very large.
• Shapes of most engineering objects are intrinsically independent of any
coordinate system.
• If the curve is to be displayed as a series of point or straight-line segments,
the computations involved could be extensive.

PARAMETRIC REPRESENTATION OF
STRAIGHT LINE
Parametric equation of a straight line
P(u) = A + (B – A) u
• X = X1 + (X2 – X1) u
• Y = Y1 + (Y2 – Y1) u where, 0 ≤ u ≤ 1
• The point P on the line is changed from A to B, as the value of ‘u’ is varied
from 0 to 1.
Parametric equation of Circle
• X = r cos Ф
• Y = r sin Ф
where, 0 ≤ Ф ≤ 2π

PARAMETRIC REPRESENTATION OF CURVES
ADVANTAGES
• It can be easily expressed in terms of vectors & matrices which enables
the use of simple computation techniques to solve complex analytic
geometry problem.
• To check whether a given point lies on the curve or not, reduces to finding
the corresponding “u” values & checking whether that value les in the
stated “u” range.
BLENDING OF CURVES
• Blending is used to construct composite curve. Blending of two curves
implies the joining of two curves subjected to the satisfaction of continuity
equation.
• Various continuity requirements can be specified at data points to impose
various degrees of smoothness of the resulting curve.
• The order of continuity becomes important when a complex curve is
modeled by several curve segments pieced together end-to-end.

• Synthetic curves represent a curve-fitting problem to construct a smooth curve that
passes through given data points. Zero-order continuity C0 yields a position
continuous curve.
• First C1 and second C2 order continuities imply slope and curvature continuous
curves respectively. A C1 curve is the minimum acceptable curve for engineering
design.

HERMITE CUBIC SPLINE
• The parametric equation of a cubic spline segment is given by
• VECTOR FORM – P(u) =σ𝒊=𝟎
𝒏
𝑪𝒊 𝒖𝒊 u – parameter, 0 ≤ u ≤ 1
• P(u) = C0 + C1u + C2u2 + C3u3 Ci – Polynomial Coefficients
• P(u) = C3u3 + C2u2 + C1u + C0
• MATRIX FORM
• P(u) = [ C3 C2 C1 C0 ]
𝒖 𝟑
𝒖 𝟐
𝒖
𝟏
• Cubic polynomial has four coefficients & thus requires four
conditions to evaluate.
A cubic polynomial is the minimum-
order polynomial that can guarantee the
generation of C0, C1 or C2 curves.

HERMITE CUBIC CURVE
• This curve is defined by the two data points that lie at the beginning & at
the end of the curve, along with the slopes at these points.
• This curve is used to interpolate the given data points.
LIMITATIONS or DISADVANTAGES
• The curve cannot be modified locally, i.e., when a data point is moved, the
entire curve is affected, resulting in a global control.
• The order of the curve is always constant(cubic), regardless of the data
points.
• Increase in number of data points increase shape flexibility. This requires
more data points , creating more splines, that are joined together(only two
data points & slopes are utilized for each spline)

• Parametric equation of Hermite Cubic spline
• P(u) =σ𝒊=𝟎
𝒏
𝑪𝒊 𝒖𝒊 u – parameter, 0 ≤ u ≤ 1
• P(u) = C0 + C1u + C2u2 + C3u3 ……. Ci – Polynomial Coefficients
• Control point, P(u) = C3u3 + C2u2 + C1u + C0
• To define a tangent vector, differentiate the above equation
• P’(u) =σ𝒊=𝟎
𝒏
𝒊 𝑪𝒊 𝒖𝒊 _ 𝟏
• P’(u) = 3 C3u2 + 2 C2u + C1 ……. , u – parameter, 0 ≤ u ≤ 1
• To find the coefficients C3, C2, C1 & C0. Use Boundary conditions
• At P0 & P0’, u = 0, At P1 & P1’, u = 1
• Substitute in Equ. 1 & 2,
• P0 = C0
• P0’ = C1
• P1 = C3 + C2 + C1 + C0
• P1’ = 3 C3 + 2 C2 + C1 P1 = C3 + C2 + P0’ + P0 , P1’ = 3 C3 + 2 C2 + P0’
HERMITE CUBIC CURVE
1
2

• P1 = C3 + C2 + P0’ + P0 …...... P1’ = 3 C3 + 2 C2 + P0’ ……
• Two equation & Two Unknowns, Solve this 2 equations
• C2 = 3(P1 – P0) – (2P0’ + P1’)
• C3 = 2(P0 – P1) + P0’ + P1’
• Substitute C0, C1, C2 & C3 values in equation
• P(u) = (2P0 – 2P1 + P0’ + P1’) u3 + (3P1 – P0 – 2P0’ – P1’) u2 + P0’u + P0
• P(u) = (2u3 – 3u2 + 1) P0 + ( - 2u3 + 3u2) P1 + (u3 – 2u2 + u)P0’ + (u3 – u2) P1’
• In matrix form,
• P(u) = [ P0 P1 P0
’ P1
’ ]
𝟐𝒖 𝟑 − 𝟑𝒖 𝟐 + 𝟏
− 𝟐𝒖 𝟑 + 𝟑𝒖 𝟐
𝒖 𝟑 − 𝟐𝒖 𝟐 + 𝒖
𝒖 𝟑 − 𝒖 𝟐
• P(u) = [ P0 P1 P0
’ P1
’ ]
𝟐 −𝟑 𝟎 𝟏
−𝟐 𝟑 𝟎 𝟎
𝟏 −𝟐 𝟏 𝟎
𝟏 −𝟏 𝟎 𝟎
𝒖 𝟑
𝒖 𝟐
𝒖
𝟏
HERMITE CUBIC CURVE
3 4
1

• Find the parametric equation of the Hermite Cubic Spline with the
end point 𝐏0 (1,1) & 𝐏 𝟏 (7,4) whose tangent vector for end points are
given as 𝐏2 (5,6) & 𝐏3 (10,7). Evaluate the value of u = 0.2, 0.4, 0.6,
0.8 & 1.
• 𝐏0 (1,1) 𝐏 𝟏 (7,4) 𝐏2 (5,6) 𝐏3 (10,7)
• X – CO-ORDINATES
• P0
’ = Point 𝐏0 & Point 𝐏2 P1
’ = Point 𝐏 𝟏 & Point 𝐏3
• P0x = 1 P1x = 7 P0x
’ = 5 – 1 = 4 P1x
’ = 10 – 7 = 3
• Y – CO-ORDINATES
• P0Y = 1 P1Y = 4 P0Y
’ = 6 – 1 = 5 P1Y
’ = 7 – 4 = 3
• P(u) = [ P0 P1 P0
’ P1
’ ]
𝟐 −𝟑 𝟎 𝟏
−𝟐 𝟑 𝟎 𝟎
𝟏 −𝟐 𝟏 𝟎
𝟏 −𝟏 𝟎 𝟎
𝒖 𝟑
𝒖 𝟐
𝒖
𝟏
• Px(u) = - 5u3 + 7u2 + 4u + 1 Py(u) = 2u3 - 4u2 + 5u + 1
HERMITE CUBIC CURVE
U 0 0.2 0.4 0.6 0.8 1
Px(u) 1 2.04 3.4 4.84 6.12 7
Py(u) 1 1.85 2.48 2.99 3.46 4

BEZIER CURVE
• Based on Approximation techniques
• Developed by P.Bezier, Designer of French car Frim Regie Renault(1962).
• Used in his software system(UNISURF) to define the outer panels of
several Renault cars.
• Bezier curve uses the vertices of Control Polygon as control points for
approximating the generated curve.
• The curve will pass through the first & last point with all other points acting
as control points.
• The curve always tangent to the first & last polygon segment.
• The degree of Bezier curve is related to the number of data points.
• If Number of data points is 4 (n+1 = 4), then n=3, degree of curve = 3 Cubic
• This curve used for the design
of aesthetic surfaces.
• The flexibility of the curve
becomes more with more
control points

• For (n+1) control points, the
Bezier curve is defined by
polynomial of degree n:
• The parametric equation of
Bezier curve
VECTOR FORM
• P(u) =σ𝐢=𝟎
𝐧
𝐏𝐢 𝐁𝐢, 𝐧(𝐮)
BEZIER CURVES
• P(u) is a point on the curve, Pi is a control point
• 𝑩𝒊, 𝒏(𝒖) – Berntein polynomials
• 𝑩𝒊, 𝒏(𝒖) = C(n, i) ui (1 – u )n – i
• P(u) = σ𝒊=𝟎
𝒏
𝑷𝒊 C(n, i) ui (1 – u )n – 𝒊
• P(u) = 𝐏0 C(n, 0) u0 (1 – u )n – 𝟎 + 𝐏1 C(n, 1) u1 (1 – u )n – 1 + 𝐏2 C(n, 2) u2 (1 – u )n – 2 +
……………. + 𝐏n C(n, n) un (1 – u )n – n
u – parameter, 0 ≤ u ≤ 1

• P(u) = 𝐏0 C(n, 0) u0 (1 – u )n – 𝟎 + 𝐏1 C(n, 1) u1 (1 – u )n – 1 +
𝐏2 C(n, 2) u2 (1 – u )n – 2 +….. + 𝐏n C(n, n) un (1 – u )n – n
• P(u) = 𝐏0 (1 – u )n + 𝐏1 C(n, 1) u1 (1 – u )n – 1 +
𝐏2 C(n, 2) u2 (1 – u )n – 2 + ……………. + 𝐏n un
• Four control points, then n = 3
• P(u) = 𝐏0 (1 – u )3 + 𝟑 𝐏1 u (1 – u )2 + 3 𝐏2 u2 (1 – u ) + 𝐏3 u3
BEZIER CURVES C(n,i) =
𝒏!
𝒊 ! 𝒏 − 𝒊 !
Four control points, then
n = 3
C(3,0) =
𝟑!
𝟎 ! 𝟑 −𝟎 !
= 1
C(3,1) =
𝟑!
𝟏 ! 𝟑 −𝟏 !
= 𝟑
C(3,2) =
𝟑!
𝟐 ! 𝟑 −𝟐 !
= 𝟑
C(3,3) =
𝟑!
𝟑 ! 𝟑 −𝟑 !
= 𝟏
P(u) = 𝐏0 (1 – u )3 + 𝟑 𝐏1 u (1 – u )2 + 3 𝐏2 u2 (1 – u ) + 𝐏3 u3
= 𝐏0 (1 – u3 − 3u + 3u2 ) + 𝟑 𝐏1 u (1 – 2u + u2 ) + 3 𝐏2 u2 (1 – u ) + 𝐏3 u3

• P(u)= 𝐏0 (– u3 + 3u2 − 3u + 1 ) + 𝐏1 (3u3 – 6u + 3u) + 𝐏2 (– 3u3 + 3u2) + 𝐏3 u3
• P(u) = [ P0 P1 P2 P3 ]
– u3 + 3u2 − 3u + 1
3u3 – 6u + 3u
– 3u3 +3u2
𝒖 𝟑
• P(u) = [ P0 P1 P2 P3 ]
−𝟏 𝟑 −𝟑 𝟏
𝟑 −𝟔 𝟑 𝟎
−𝟑 𝟑 𝟎 𝟎
𝟏 𝟎 𝟎 𝟎
𝒖 𝟑
𝒖 𝟐
𝒖
𝟏
BEZIER CURVES
• Find the parametric equation of the Bezier curve whose end points are 𝐏0 (0,0)
& 𝐏3 (7,0). The other control points are 𝐏1 (7,0) & 𝐏2 (7,6). Evaluate the value of
u = 0.2, 0.4, 0.6, 0.8 & 1.
ANSWER - Px(u) = 7u3 – 21u2 + 21u Py(u) = 18u2 – 18u3
• Find the equation of a Bezier curve which is defined by four control points as
(80,30,0), (100,100,0), (200,100,0) & (250,30,0). Evaluate the value of u = 0.2, 0.4,
0.6, 0.8 & 1.
ANSWER - Px(u) = - 130 u3 +240u2 + 60u + 80 Py(u) = - 210u2 + 210u + 30

B-SPLINE CURVES• Single piecewise parametric polynomial curve through any
number of control points with the degree of polynomial
selected by designer.
• It provides the ability to add control points without increasing
the degree of the curve.
• B-Spline exhibit a local control of the curve shape. i.e.
Whenever a single vertex is moved, only those vertices
around that will be affected while rest remains the same.
• In contrast to Bezier curve, the theory of B-Spline curve
separates the degree of resulting curve from the number of
the given control points.
• Four control points can always produce a cubic Bezier curve
but four control points can produce linear, Quadratic or Cubic
B-Spline curve.
• A B-spline is a piecewise polynomial, and its knots are the
points where the pieces meet.
• A knot would have the same type as the argument to the
polynomials

HERMITE CUBIC SPLINE BEZIER CURVE B-SPLINE CURVE
• It is represented by the
polynomial of degree 3
• Curve with (n+1) data
points are represented
by the polynomial of nth
degree.
• Curve with (n+1) data
points are represented
by the polynomial of nth
degree.
• To draw the curve, it
needs two data points &
two tangent vector
• To draw Bezier curve, it
require two data points &
one or more control
points in between is
required.
• To draw Bezier curve, it
require two data points &
one or more control
points in between is
required.
• Degree of polynomial is
independent of data
points.
depends on the number
of data points.
depends on the number
of data points.
• The shape of the curve
depends on the tangent
vectors at the end.
• The shape is controlled
by the control points.
• The shape is controlled
by the control points.
• It is not convenient to
control the shape of the
curve.
• The curve is affected
globally by the
movement of the control
points
• It affects the curve locally
by the movement of the
control points

SURFACE MODELING

SURFACE MODELING – ANALYTICAL SURFACE
PLANE SURFACE
• It is the simplest surface which requires three
non-coincident points to define a plane.
• The plane surface can be used to generate
cross-sectional view by interesting a surface
model with it.
RULED SURFACE
• It is a surface constructed by transitioning
between two or more curves by using linear
blending between each section of the surfaces.
• It interpolates linearly between two boundary
curves that define the surface.
LOFTED SURFACE
• It is a surface constructed by transitioning
between two or more curves by a smooth i.e.
higher order blending between each section of
the surface.
• Used for modeling engine manifolds, turbine
blades etc.

SURFACE OF REVOLUTION
• It is an Axi-Symmetric surface that
can model axi-symmetric objects,
• It is generated by a rotating a
planar wireframe entity in space
about the axis of symmetry to the
required angle.
TABULATED CYLINDRICAL SURFACE
• It is a surface generated by
translating a planar curve a certain
distance along a specified
direction.
• Plane of the curve is perpendicular
to the axis of the cylinder.
• It is used to generate surfaces that
have identical curved cross
sections.
SURFACE MODELING – ANALYTICAL SURFACE

RULED SURFACE
BEZIER SURFACE B-SPLINE SURFACE

SURFACE MODELING – SYNTHETICAL SURFACE
BI - CUBIC PATCHES
• It is generated by the four boundary curves
connects four corner data points & utilizes a
bi-cubic equation.
• bicubic interpolation is an extension of cubic
interpolation for interpolating data points on a
two-dimensional regular grid
• The patch is defined by the 16 control points
i.e. 4 control points on each curve.
B - SPLINE SURFACES
• The surface is formed by using B-Spline
curve.
• It is a synthetic surface that can either
approximate or interpolate given input data.
• Its not necessary to pass the surface from all
control points
• Permits local control of the surface.

COONS PATCH or COON SURFACE
• A linear interpolation between four bounded
curves is used to generate a coons surface.
• It is easy to create, so many 2-D cad packages
utilize this option for generating models.
• The surface is inflexible & cannot create very
smooth surface.
• The single patch can be extended in both the
directions by adding further patches.
BEZIER SURFACE
• It is a synthetic surface that approximates given
input data, i.e. it does not pass all given data
points
• Allows only global control of the surface.
• The surface is contained in the convex hull of
the polygon set.
• The degree of the surface in each polynomial
direction is one less than the number of
defining polygon vertices in that direction
SURFACE MODELING – SYNTHETICAL SURFACE

SOLID MODELING
• Solid modeling techniques provide the user with the means to create, store,
and manipulate complete representations of solid objects with the potential
for integration and improved automation.
SOLID REPRESENTATION
• Several representation schemes are available for the creation of solid
models. Some of the most popular are given:
• Constructive Solid Geometry (CSG).
• Boundary Representation (B-Rep).
• Sweeping
CYLINDER
E – Edges
F – Faces
V – Vertices

CONSTRUCTIVE SOLID GEOMETRY
• A CSG model is based on the topological
notation that a physical object can be divided
into a set of primitives (basic elements or
shapes).
• This primitives can be combined in a certain
order following a set of rules (Boolean
operations) to form the object.
• The available operators are Union ( U or +),
Intersection (∩ or I) and difference ( - ).
• The Union operator (U or +): is used to
combine or add together two objects or
primitives.
• The Intersection operator (∩ or I):
intersecting two primitives gives a shape equal
to their common volume.
• The Difference operator (-): is used to
subtract one object from the other and results
in a shape equal to the difference in their
volumes.

CSG PRIMITIVES
• Primitives are usually translated and/or rotated to position and orient
them properly applying Boolean operations.
• Following are the most commonly used primitives:

BOOLEAN OPERATIONS
Figure below shows Boolean operations of a clock P and Solid Q

• Data structures for the CSG
representation are based on
the binary tree structure.
• The CSG tree is a binary tree
with leaf nodes as primitives
and interior nodes as Boolean
operations
LIMITATION or DISADVANTAGES
• Inconvenient for the designer to determine simultaneously a sequence of
feature creation for all design iterations
• The use of machining volume may be too restrictive
• Problem of non-unique trees. A feature can be constructed in multiple ways
• Tree complexity
• Surface finish and tolerance may be a problem

• The CSG tree is organized upside down, with the root representing the composite
solid at the top & primitives called as leaves at the bottom

• The creation of a model in CSG can be simplified by the use of a table
summarizing the operations to be performed. The following example
illustrates the process of model creation used in the CSG
representation.

BOUNDARY REPRESENTATION (B-REP)
• A B-Rep model or boundary model is based on the topological notation that a
physical object is bounded by a set of Faces.
• Each face is bounded by edges and each edge is bounded by vertices.
• These faces are regions or subsets of closed and orientable surfaces.
• A closed surface is one that doesn’t have a boundary or end, such as a
sphere, cube pyramid & cone etc. The surface is closed if it has a definite
inside & outside. There is no way to get from the inside to the outside
surface without passing through the surface.
• An orientable surface is one in which it is possible to distinguish two sides
by using the direction of the surface normal to a point inside or outside of
the solid model.

BOUNDARY REPRESENTATION (B-REP) DATA STRUCTURE
• A general data structure for a
boundary model should have both
topological and geometrical
information.
• Geometry relates to the information
containing shape defining
parameters, such as the
coordinates of the vertices.
• Topology describes the
connectivity among the various
geometric components, that is, the
relational information between the
different parts of an object
Topology Geometry
Object
Body
Genus
Face
Loop
Edge
Vertex
Surface
Curve
Point

B-REP GEOMETRY VS TOPOLOGY
Same geometry but different topology
Same topology but different geometry

B-REP ENTITIES DEFINITION
• Vertex is a unique point in space
• An Edge is a finite, non-self-intersecting,
directed space curve bounded by two
vertices
• A Face is defined as a finite connected, non-
self-intersecting, region of a closed oriented
surface bounded by one or more loops.
• A Handle (Genus or Through hole) is
defined as a passageway that passes
through the object completely.
• A Body (Shell) is a set of faces that bound a
single connected closed volume. Thus a
body is an entity that has faces, edges, and
vertices.
• A Loop is an ordered alternating sequence of
vertices and edges. A loop defines a non-
self-intersecting, piecewise, closed space
curve which, in turn, may be a boundary of a
face.

• To ensure topological validation of the boundary model, special operators
are used to create and manipulate the topological entities. These are
called Euler Operators.
• The Euler’s Law gives a quantitative relationship among faces, edges,
vertices, loops, bodies or genus in solids
• EULER LAW
F = number of faces, E = number of edges, V = number of vertices
L = Faces inner loops, B = number of bodies, G = number of genus (handles)
)(2 GBLVEF −=−+−
F – E + V = 2
6 – 12 + 8 = 2
F – E + V = 2
10 – 24 + 16 = 2

SWEEP REPRESENTATION
• Solids that have a uniform thickness in a particular direction & axisymmetric
solids can be created by Transitional (Extrusion) or Rotational (Revolution)
Sweeping.
• Sweeping requires two elements – a surface to be moved and a trajectory,
analytically defined, along which the movement should occur.
EXTRUSION
TRANSITIONAL SWEEPING.
REVOLUTION
ROTATIONAL SWEEPING.

GEOMETRIC MODELING

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