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Module -II Introduction

GEOMETRIC MODELING “Geometric modeling is as important to CAD as governing
equilibrium equations to classical engineering fields as
& VISUAL REALISM mechanics and thermal fluids.”
Introduction to modeling and viewing – Wire frame modeling. intelligent decision on the types of entities necessary to use in a
Representation of curves – Hermite curve – Bezier curve – B– particular model to meet certain geometric requirements such as
spline curves. Techniques for surface modeling – Surface patch– slopes and/or curvatures.
Coons and bicubic patches– Bezier and B–spline surfaces. interpretation of unexpected results
Solid modeling techniques– Boundary representation (B–rep) and evaluations of CAD/CAM systems
Constructive Solid Geometry (CSG). Hidden – Line, surface, innovative use of the tools in particular applications.
solid removal algorithms, shading and coloring – computer creation of new attributes, or modify the obtained models to
animation. benefit new engineering applications.
understanding of terminology
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CAD Models CAD Models

• A CAD model is a computer representation of an • A 3D model is the most general model used in CAD
object or part software. This model is equivalent to an isometric
• It contains all of the design information including view
geometry, dimensions, tolerances, materials and • 2 basic types of 3D models are wire frame and
manufacturing information. surface models.
• CAD models replace the paper blueprints and • In a 3D wire frame model, only edges of the object
engineering drawings are represented.
• The simplest model used in CAD is a 2D model. This • A 3D surface model defines the object in terms of
model is essentially the computer graphics equivalent surfaces such as plates (flat) and shells (curved) in
to an orthographic projection addition to edges.

Why draw 3D Models?

• 3D models are easier to interpret.
• Simulation under real-life conditions.
• Less expensive than building a physical model.
• 3D models can be used to perform finite element
analysis (stress, deflection, thermal…..).
• 3D models can be used directly in manufacturing,
Computer Numerical Control (CNC).
• Can be used for presentations and marketing.
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Steps for Creating Geometric Model

• There are three steps in which the designer can create

geometric models by using CAD software, these are:
– 1) Creation of basic geometric objects: In the first step the
designer creates basic geometric elements by using
commands like points, lines, and circles.
– 2) Transformations of the elements: In the second step the
designer uses commands like achieve scaling, rotation and
other related transformations of the geometric elements.
– 3) Creation of the geometric model: During the final step
the designer uses various commands to that cause
integration of the objects or elements of the geometric
model to form the desired shape.

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Design Analysis Drafting

• Evaluation of areas and volumes. • Automatic planar cross sectioning.
• Evaluation of mass and inertia properties. • Automatic hidden line and surface removal.
• Interference checking in assemblies. • Automatic production of shaded images.
• Analysis of tolerance build-up in assemblies. • Automatic dimensioning.
• Analysis of kinematics — mechanics, robotics. •Automatic creation of exploded views for
• Automatic mesh generation for finite element technical illustrations.
analysis

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Manufacturing
Production Engineering
• Parts classification. – Bill of materials.
• Process planning. – Material requirement.
• Numerical control data generation and – Manufacturing resource requirement.
verification. – Scheduling.
• Robot program generation. Inspection and Quality Control
– Program generation for inspection machines.
– Comparison of produced part with design.

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3D Modelling Types of modelling for CAD

There are three basic types of three- dimensional computer
geometric modelling methods:
– Wireframe modelling
– Surface modelling
– Solid modelling

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WIREFRAME MODELING

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Wireframe Modeling Need ?

• A wireframe model of an object is the simplest • Geometry display by modeling systems
and represents mathematically in the computers. • Visualization of motion (simple animations)
• It is most commonly used technique and all • Modeling of geometries such as projected
commercial CAD/CAM systems are wire-frame profiles and revolutions.
based. • 2D drafting
• Basic wire-frame entities can be divided into
analytic and synthetic entities.

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Wireframe Modeling Wireframe Modeling

• Contains information about the locations of all the points
(vertices) and edges in space coordinates.
• Each vertex is defined by x, y, z coordinate.
• Edges are defined by a pair of vertices.
• Faces are defined as three or more edges.
• Wireframe is a collection of edges, there is no skin defining the
area between the edges.

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Wireframe Modeling Real objects & wire-frame models

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Non-sense object Ambiguous wireframe model

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Basic database information Basic database structures

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Basic database structures Basic Database Structures

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Model validity criteria

Wireframe Cylinder
(restrict to linear edges)

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Free Form Surfaces Free Form Surfaces

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Free Form Surfaces

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Free Form Surfaces

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Representation of curves

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Representation of curves

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Representation of curves

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Representation of curves

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Representation of curves

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Hermite Curve

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Hermite Curve contd… Hermite Curve contd…

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Hermite Curve contd… Hermite Curve contd…

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Advantages of parametric forms Definition- Bezier curve

• More degrees of freedom • A Bezier curve is a mathematically defined curve used in two dimensional
graphic applications.
• Directly transformable • The curve is defined by four points: the initial position and the terminating
position (which are called "anchors") and two separate middle points
• Dimension independent (which are called "handles").
• No infinite slope problems • The shape of a Bezier curve can be altered by moving the handles.
• The mathematical method for drawing curves was created by Pierre Bezier
• Separates dependent and independent variables in the late 1960's for the manufacturing of automobiles at Renault.

• Inherently bounded
• Easy to express in vector and matrix form
• Common form for many curves and surface
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Properties Properties

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Properties Properties

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Properties Properties

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Properties Properties

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Properties Design Technique

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Design Technique Design Technique

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Design Technique Design Technique

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Design Technique Application

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B-splines B-splines

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B-splines B-splines

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B-splines properties B-splines properties

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B-Splines Properties B-splines properties for CAD

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B-splines properties for CAD

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Surface Modeling Surface Modeling

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Surface Modeling Surface Modeling

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Surface Modeling Surface Modeling

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Surface Modeling Surface Modeling

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Surface Modeling Surface: definition

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Surface modeling Surface modeling

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Advantages of
Plane surface
Solid & Surface Modelling

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Plane surface Ruled Surface (lofted surface)

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Ruled surface Ruled surface

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Lofted Surface Revolved surface

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Revolution of Surface Tabulated cylinder

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Tabulated Cylinder Tabulated Cylinder

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Swept Surface Surface modeling

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TECHNIQUES IN
Surface Patch
SURFACE MODELLING
i. Surface Patch The patch is the fundamental building block for surfaces. The two
variables u and v vary across the patch; the patch may be termed
iparametric. The parametric variables often lie in the range 0 to 1.
ii. Coons Patch Fixing the value of one of the parametric variables results in a
curve on the patch in terms of the other variable (Isoperimetric
curve). Figure shows a surface with curves at intervals of u and v
iii. Bi-cubic Patch of 0 : 1.

iv. Be’zier Surface

v. B-Spline Surface
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Coons Patch Coons Patch

• The sculptured surface often involve interpolation
across an intersecting mesh of curves that in
effect comprise a rectangular grid of patches, each
bounded by four boundary curves.
• The linearly blended coons patch is the simplest
for interpolating between such boundary curves.
• This patch definition technique blends for four
boundary curves Ci(u) and Dj(v) and the corner
points pij of the patch with the linear blending
functions,
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Bi-cubic Patch Bezier Surface

• The bi-cubic patch is used for surface descriptions
defined in terms of point and tangent vector
information.
• The general form of the expressions for a bi-cubic
patch is given by:

• This is a vector equation with 16 unknown

parameters k ij which can be found by Lagrange
interpolation through 4 x 4 grid.
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B-Spline Surfaces B-Spline Surfaces

• The B-spline surface approximates a
characteristics polygon as shown and passes
through the corner points of the polygon, where
its edges are tangential to the edges of the
polygon
• This may not happen when the control polygon is
closed
• A control point of the surface influences the
surface only over a limited portion of the
parametric space of variables u and v.
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Surface Patch Surface Patch

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Surface Patch Analytical Surfaces

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Primitive Surfaces Primitive Surfaces

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Planar Surface Offset Surface

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Tabulated Cylinder Surface of Revolution

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Surface of Revolution Swept Surface

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Rulled Surface Rulled Surface

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Synthetic Surfaces Linearly Blended Coons Surface

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Linearly Blended Coons Surface Linearly Blended Coons Surface

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Linearly Blended Coons Surface Bicubic Patch

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Bicubic Patch Bicubic Patch

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Bicubic Patch Bicubic Patch

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Bicubic Patch Bicubic Patch

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Bicubic Patch Example Bezier Surfaces

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Bezier Surfaces Bezier Surfaces

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Bezier Surfaces Bezier Surfaces

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Bezier Surfaces B-Spline Surfaces

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B-Spline Surfaces B-Spline Surfaces

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B-Spline Surfaces

SOLID MODELLING

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Why solid modeling? Solid model

• Recall weakness of wireframe and surface • Solid modeling is based on complete, valid
modeling and unambiguous geometric representation
– Ambiguous geometric description
of physical object.
– Complete points in space can be classified.
– incomplete geometric description (inside/ outside)
– lack topological information – Valid vertices, edges, faces are connected
– Tedious modeling process properly.
– Awkward user interface – Unambiguous there can only be one
interpretation of object

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Solid model representation Constructive solid geometry (CSG)

schemes
• Objects are represented as a combination
1. Constructive solid geometry (CSG) of simpler solid objects (primitives).
• The primitives are such as cube, cylinder,
2. Boundary representation (B-rep) cone, torus, sphere etc.
3. Spatial enumeration • Copies or “instances” of these primitive shapes
are created and positioned.
4. Instantiation.
• A complete solid model is constructed by
combining these “instances” using set
specific, logic operations (Boolean)

Constructive solid geometry (CSG) Constructive solid geometry (CSG)-

boolean operation
• Boolean operation
– each primitive solid is assumed to be a set of • Union
points, a boolean operation is performed on point – The sum of all points in each of two defined
sets and the result is a solid model. sets. (logical “OR”)
– Boolean operation
– Also referred to as Add, Combine, Join, Merge
– union, intersection and difference
– The relative location and orientation of the two A B
primitives have to be defined before the boolean AB
operation can be performed.
– Boolean operation can be applied to two solids
other than the primitives.

Constructive solid geometry (CSG)- Constructive solid geometry

boolean operation
(CSG)- boolean operation
• Difference • intersection
– The points in a source set minus the points – Those points common to each of two defined
common to a second set. (logical “NOT”) sets (logical “AND”)
– Set must share common volume – Set must share common volume
– Also referred to as subtraction, remove, cut – Also referred to as common, conjoin
A B A B
A-B AB

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Constructive solid geometry (CSG)- Constructive solid geometry

boolean operation (CSG)- boolean operation
• When using boolean operation, be careful to • Boolean operation
avoid situation that do not result in a valid – Are intuitive to user
solid
– Are easy to use and understand
– Provide for the rapid manipulation of large
amounts of data.
A B
AB • Because of this, many non-CSG systems
also use Boolean operations

Constructive solid geometry Constructive solid geometry

(CSG)- data structure (CSG)- CSG tree
• Data structure does not define model shape
explicitly but rather implies the geometric shape • CSG tree stores the history of applying
through a procedural description boolean operations on the primitives.
– E.g: object is not defined as a set of edges & faces but – Stores in a binary tree format
by the instruction : union primitive1 with primitive 2 – The outer leaf nodes of tree represent the
• This procedural data is stored in a data structure primitives
referred to as a CSG tree – The interior nodes represent the boolean
• The data structure is simple and stores compact operations performed.
data (easy to manage)

Constructive solid geometry (CSG)- Constructive solid geometry

CSG tree (CSG)- not unique
• More than one procedure (and hence database) can
be used to arrive at the same geometry.


+
-
-

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Constructive solid geometry (CSG) Constructive solid geometry (CSG) -

representation advantage

• CSG representation is unevaluated • CSG is powerful with high level command.

– Faces, edges, vertices not defined in explicit • Easy to construct a solid model – minimum
• CSG model are always valid step.
– Since built from solid elements. • CSG modeling techniques lead to a concise
database less storage.
• CSG models are complete and – Complete history of model is retained and can
unambiguous be altered at any point.
• Can be converted to the corresponding
boundary representation.

CSG -disadvantage Solution

• Only boolean operations are allowed in the
modeling process with boolean operation alone, • CSG representation tends to accompany the
the range of shapes to be modeled is severely corresponding boundary representation
restricted not possible to construct unusual shape. hybrid representation
• Requires a great deal of computation to derive the
• Maintaining consistency between the two
information on the boundary, faces and edges which is
important for the interactive display/ manipulation of
representations is very important.
solid.

Boundary representation Boundary representation

(B-Rep) (B-Rep)
• Why B-Rep includes such topological
• Solid model is defined by their enclosing information?
surfaces or boundaries.
- A solid is represented as a closed space in
• This technique consists of the geometric 3D space (surface connect without gaps)
information about the faces, edges and vertices
of an object with the topological data on how - The boundary of a solid separates points
these are connected. inside from points outside solid.

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B-Rep vs Surface modeling B-Rep data structure

• Surface model
– A collection of surface entities which simply enclose • B-Rep graph store face, edge and vertices
a volume lacks the connective data to define a solid as nodes, with pointers, or branches
(i.e topology). between the nodes to indicate connectivity.
• B- Rep model
– Technique guarantees that surfaces definitively divide
model space into solid and void, even after model
modification commands.

B-Rep data structure Boundary representation-validity

v5

f3 f2 E3
E1 v4
E4
E7
v3
• System must validate topology of created
f4 E2
f5 E6 solid solid.
E8 f1
v1 E5 v2
face1 face2 face3 face4 face5 • B-Rep has to fulfill certain conditions to
Combinatorial disallow self-intersecting and open objects
edge1 edge2 edge3 edge4 edge5 edge6 edge7 edge8 structure /
topology
• This condition include
– Each edge should adjoin exactly two faces and
vertex1 vertex2 vertex3 vertex4 vertex5 have a vertex at each end.
Metric information/ – Vertices are geometrically described by point
(x, y, z) geometry coordinates

Boundary representation-validity Boundary representation-validity

• Validity also checked through mathematical
• This condition include (cont)
evaluation
– At least three edges must meet at each vertex.
– Evaluation is based upon Euler’s Law (valid for
– Faces are described by surface equations
simple polyhedra – no hole)
– The set of faces forms a complete skin of the solid
with no missing parts. –V–E+F=2 V-vertices E- edges F- face loops
– Each face is bordered by an ordered set of edges
forming a closed loop. v5 V = 5, E = 8, F = 5
– Faces must only intersect at common edges or f3 E4 f2 E3
vertices. E1 v4 E7 5–8+5=2
f4 f5E2 E6 v3
– The boundaries of faces do not intersect themselves E8 f1
v1 E5 v2

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Boundary representation-
Boundary representation-validity
ambiguity and uniqueness
• Expanded Euler’s law for complex polyhedrons
(with holes) • Valid B-Reps are unambiguos
• Euler-Poincare Law: • Not fully unique, but much more so than
– V-E+F-H=2(B-P) CSG
– H – number of holes in face, P- number of passages or through • Potential difference exists in division of
holes, B- number of separate bodies.
V = 24, E=36, F=15, H=3, – Surfaces into faces.
P=1,B=1 – Curves into edges

Boundary representation- Boundary representation-

advantages disadvantages
• Capability to construct unusual shapes that • Requires more storage
would not be possible with the available • More prone to validity failure than CSG
CSG aircraft fuselages, swing shapes
• Model display limited to planar faces and
• Less computational time to reconstruct the linear edges
image - complex curve and surfaces only approximated

Visual Realism Introduction Continued..

 What is realistic image?
 Visual Realism is a method for interpreting picture data fed into a
 A picture that capture many of the effect of light interacting
computer and for creating pictures from difficult
with real physical object.
multidimensional data sets.
 the ultimate goal of picture is to convey information, then a picture
 Visualization in geometric modeling is related to displaying that is free of the complications of the shadows and reflections.
geometric models of object

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Continued.. Application of realistic pictures

Creating a realistic pictures involves following stages: 1. Simulation

1) Model of the object. 2. Design of 3D object such as automobiles, airplanes etc.

2) Viewing specifications and lighting conditions, 3. Entertainment and advertisement

3) Visible surface determination. 4. Research and education

4) Color determination of each pixel 5. Command and control

5) Animated sequence

Classification of Visualization
Difficulties in visual realism
 Total visual realism is the complexity of the real world.
 Visualization in geometric modeling- Shading the parts
Achieved sub goal of realism: to provide sufficient
with various shadows, colors and transparency information to let the viewer understand the 3D spatial
 Visualization in scientific computing- It changes the data relationships among several objects.
in numerical form into picture display,  Most display devices are 2D therefore, 3D objects must be
projected into 2D, with considerable attendant loss of
information which can sometimes create ambiguities in the
image.

Existing visualization methods

1) Parallel projections

2) Perspective projection.

3) Hidden line removal Hidden Line Removal

4) Hidden surface removal

5) Hidden solid removal

6) Shaded models

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HLR - Introduction HLR - Introduction

• Hidden line removal (HLR) is an extension of wireframe model
rendering where lines (or segments of lines) covered by
surfaces are not drawn.
• Hidden line removal (HLR) is the method of computing
which edges are not hidden by the faces of parts for a
specified view and the display of parts in the projection of a
model into a 2D plane.
• It is considered that information openly exists to define
a 2D wireframe model as well as the 3D topological
information.

HLR - Introduction Visibility Techniques

Algorithms might be slow in calculation and storing data.
These techniques will rectify the problem.

• Minimax Test
• Containment Test
• Surface Test
• Computing Silhouettes
• Edge Intersection
• Segment Comparison
• Homogeneity Test

Minimax Test/ Bounding Box Test Containment Test

• Determines the given point is located in or out of the
• To check whether a point is in a given bounded
polygon and to check the vertices of one polygon for
surface or volume and checks if two polygons
containment in the other.
overlap.
• Types of Method
Conditions
– Intersection Method
– Angle Method

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Containment Test Surface Test/ Back Face/Depth Test

Containment proceeds: • The process used to determine which surfaces and
(a)If the sum of the angles is equal to zero, point P is outside the parts of surfaces are not visible from a certain
polygon. viewpoint.
(b) If the sum is equal to 360°, point P is inside the polygon.

Condition :
– Faces whose surface normal are
positive in Z direction for visible
surfaces
– Faces whose surface normal are
negative in Z direction for non visible
surfaces

Computing Silhouettes Computing Silhouettes

• A set of edges that separates visible faces from invisible faces
of an object with respect to a given viewing direction is called
silhouette edges (or silhouettes).
• An edge that is part of the silhouette is characterized as the
intersection of one visible face and one invisible face.

Edge Intersection Segment Comparison (Scan Line)

• To determine the visibility of partially invisible edges. • Utilizes the image’s raster scan.
• Calculate line intersections to determine edge • Segment comparison are performed in x & z plane.
visibility. • Visibility techniques are further performed for each span by
comparing depth and edge segments.

x = x1 + (yboundary - y1) / m
y = y1 + m (xboundary - x1)

m = (y2 - y1) / (x2 - x1)

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Homogeneity Test Hidden Line Algorithms

• Utilizes neighbourhood points to test its visibility. • Priority Algorithm
• A Point can be projected onto neighbourhood of • Area Oriented Algorithm
projection of points, then the neighbourhood of point P
is decided to be visible or invisible. • Overlay Algorithm

Priority algorithm Priority algorithm

• This algorithm is also known as depth or Z
algorithm. Face Priority

• Imagines that objects are modelled with lines ABCD 1

and lines are generated where surfaces join. If ADFG 1

only the visible surfaces are created then the DCEF 1

invisible lines are automatically removed by ABHG 2

this algorithm. EFGH 2

BCEH 2

Priority algorithm Area Oriented algorithm

• ABCD, ADFG, DCEF are given higher priority-1. • Identify Silhouette Polygons
Hence, all lines in this faces are visible, that is, • Quantitative Hiding (QH)
AB, BC, CD, DA, AD, DF, FG, AG, DC, CE, EF and • Visibility of Silhouette segment
DF are visible. • Intersect the internal edges
• AGHB, EFGH, BCEH are given lower priority-2. • Display the edges
Hence, all lines in this faces other than
priority-1 are invisible, that is BH, EH and GH.
These lines must be eliminated.

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Overlay algorithm
• Applicable for curved surfaces by approximating
them as planar surfaces.
• u-v grid is used to make grid surface by making it as
straight edges.
Hidden Solid Removal
• STEPS:
– To Calculate uv grid using surface
equation.
– Grid Surface to Linear Edge
creation.
– Using proper visibility techniques
for required output.

Ray Tracing or Ray Casting

Introduction
Algorithm
• The process of displaying the solid model with • Ray Tracing is the process of tracking and plotting the
visible hidden line or surfaces removed. path taken by the rays of light starting at light source
to the center of projection.
• It is a method for creating an image by tracing the
path of light via pixels in an image plane and
reproducing the effects of its meets with virtual
objects.
• Ray tracing is simulating a wide range of optical
effects, such as scattering, reflection and refraction.

Ray Tracing or Ray Casting

Algorithm Advantages
• For each pixel • Flexibility to handle both flat and curved
– Trace Primary Eye
Ray, find intersection surfaces.
– Trace Secondary • Easily modified to provide realistic shading.
Shadow Ray(s) to all
light(s) • Computational expenses can be minimized by the
– Color the pixel for use of bounding box technique.
Visible Surface if it is
not an Illumination • A realistic simulation of lighting over other rendering.
Model
– Trace Reflected Ray • An effect such as reflections and shadows is easy and
– Color += (Reflectivity * effective.
Color of reflected ray)
• Simple to implement yet yielding impressive visual
results.

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Limitations Shading
• Scan line algorithms use data consistency to divide  Shading defines to describe depth perception in three
computations between pixels, while ray tracing normally
dimensioning models by different levels of darkness.
begins the process a new, treating every eye ray
separately. So the Computation Time is long.  Shading is applied in drawing for describes levels of
• Low Performance. darkness on paper by adding media heavy densely shade
for darker regions, and less densely for lighter regions.

Continued…

 The input to a shading model is

1. Intensity and color of light source

2. surface characteristics

3. Position & orientation of surface and source

 Shading surface algorithms

1. Constant shading
2. Gourand shading
3. Phong shading (a) Point light source (b) Ambient light source

Color of light

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1. Constant shading 2. Gourand shading

•  In this method a single intensity is calculated for each  Linearly interpolating the intensity values across the surface of

polygon and all points over the surface are then displayed with each polygon which are matched with the values of adjacent
the same intensity value. polygons along the common edges.

3. Phong shading
 Each vertex has its own surface normal and normal vectors are

interpolated

Transparency and reflection

 Now we consider transparent surface.  Improved Camera Models

 Simple models of transparency do not include the refraction

 So far we have consider camera model with a pinhole
(bending) of light through a transparent solid. lens and an infinitely fast shutter: all objects are in sharp
 More complex models include refraction, diffuse focus and represent the world at one instance in time.
transparency and the attenuation of light with respect to
 It is possible to model more accurately the way that we
distance. Also consider diffuse and specular reflection .
and cameras see the world.
 It requires knowledge of other surfaces besides the surface

being shaded.

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Other ways of realism

 Dynamics: We mean changes that spread across a sequence of

THANK
pictures, including changes in position ,size, material, properties
,lighting and viewing specification

 Most popular kind of dynamics is motion ranging from

simple transformation to animation.

 If a series of projections of same object, from a slightly different

YOU
viewpoint around the object, is displayed in rapid succession
then the object appears to rotate.
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