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UNIT -IV 3-D OBJECT REPRESENTATION: spline representation, Hermite curve, Bezier curve and B-spline curve, Polygon surfaces, quadric surfaces, , Solid modeling Schalars – wire frame, CSG, B-rep. Bezier and B-spline surfaces, Basic illumination models, shading algorithms
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spline representation, Hermite curve, Bezier curve and B-spline curve,Polygon surfaces, quadric surfaces, , Solid modeling Schalars – wire frame, CSG, B-rep. Bezier and B-spline surfaces, Basic illumination models, shading algorithms
To obtain a display of a three dimensional scene that has been modeled in world coordinates, we must setup a co-ordinate reference for the ‘camera’. This coordinate reference defines the position and orientation for the plane of the camera film which is the plane we want to use to display a view of the objects in the scene. Object descriptions are then transferred to the camera reference coordinates and projected onto the selected display plane. The objects can be displayed in wire frame form, or we can apply lighting and surface rendering techniques to shade the visible surfaces.
Parallel Projection: Parallel projection is a method for generating a view of a solid object is to project points on the object surface along parallel lines onto the display plane. In parallel projection, parallel lines in the world coordinate scene project into parallel lines on the two dimensional display planes. This technique is used in engineering and architectural drawings to represent an object with a set of views that maintain relative proportions of the object. The appearance of the solid object can be reconstructed from the major views.
Perspective Projection: It is a method for generating a view of a three dimensional scene is to project points to the display plane alone converging paths. This makes objects further from the viewing position be displayed smaller than objects of the same size that are nearer to the viewing position. In a perspective projection, parallel lines in a scene that are not parallel to the display plane are projected into converging lines. Scenes displayed using perspective projections appear more realistic, since this is the way that our eyes and a camera lens form images.
Depth Cueing: Depth information is important to identify the viewing direction, which is the front and which is the back of displayed object. Depth cueing is a method for indicating depth with wire frame displays is to vary the intensity of objects according to their distance from the viewing position. Depth cueing is applied by choosing maximum and minimum intensity (or color) values and a range of distance over which the intensities are to vary.
Visible line and surface identification: A simplest way to identify the visible line is to highlight the visible lines or to display them in a different color. Another method is to display the non visible lines as dashed lines.
Surface Rendering: Surface rendering method is used to generate a degree of realism in a displayed scene. Realism is attained in displays by setting the surface intensity of objects according to the lighting conditions in the scene and surface characteristics. Lighting conditions include the intensity and positions of light sources and the background illumination. Surface characteristics include degree of transparency and how rough or smooth the surfaces are to be.
Exploded and Cutaway views: Exploded and cutaway views of objects can be to show the internal structure and relationship of the objects parts. An alternative to exploding an object into its component parts is the cut away view which removes part of the visible surfaces to show internal structure.
Three-dimensional and Stereoscopic Views: In Stereoscopic views, three dimensional views can be obtained by reflecting a raster image from a vibrating flexible mirror. The vibrations of the mirror are synchronized with the display of the scene on the CRT. As the mirror vibrates, the focal length varies so that each point in the scene is projected to a position corresponding to its depth. Stereoscopic devices present two views of a scene; one for the left eye and the other for the right eye. The two views are generated by selecting viewing positions that corresponds to the two eye positions of a single viewer. These two views can be displayed on alternate refresh cycles of a raster monitor, and viewed through glasses that alternately darken first one lens then the other in synchronization with the monitor refresh cycles.
2.2 Three Dimensional Object Representations Representation schemes for solid objects are divided into two categories as follows:
1. Boundary Representation ( B-reps) : It describes a three dimensional object as a set of surfaces that separate the object interior from the environment. Examples are polygon facets and spline patches. 2. Space Partitioning representation : It describes the interior properties, by partitioning the spatial region containing an object into a set of small, nonoverlapping, contiguous solids(usually cubes). Eg: Octree Representation
2.2.1 Polygon Surfaces Polygon surfaces are boundary representations for a 3D graphics object is a set of polygons that enclose the object interior. Polygon Tables The polygon surface is specified with a set of vertex coordinates and associated attribute parameters. For each polygon input, the data are placed into tables that are to be used in the subsequent processing. Polygon data tables can be organized into two groups: Geometric tables and attribute tables.
Listing the geometric data in three tables provides a convenient reference to the individual components (vertices, edges and polygons) of each object. The object can be displayed efficiently by using data from the edge table to draw the component lines. Extra information can be added to the data tables for faster information extraction. For instance, edge table can be expanded to include forward points into the polygon table so that common edges between polygons can be identified more rapidly.
This is useful for the rendering procedure that must vary surface shading smoothly across the edges from one polygon to the next. Similarly, the vertex table can be expanded so that vertices are cross-referenced to corresponding edges. Additional geometric information that is stored in the data tables includes the slope for each edge and the coordinate extends for each polygon. As vertices are input, we can calculate edge slopes and we can scan the coordinate values to identify the minimum and maximum x, y and z values for individual polygons. The more information included in the data tables will be easier to check for errors. Some of the tests that could be performed by a graphics package are:
Spline Representations A Spline is a flexible strip used to produce a smooth curve through a designated set of points. Several small weights are distributed along the length of the strip to hold it in position on the drafting table as the curve is drawn. The Spline curve refers to any sections curve formed with polynomial sections satisfying specified continuity conditions at the boundary of the pieces. A Spline surface can be described with two sets of orthogonal spline curves. Splines are used in graphics applications to design curve and surface shapes, to digitize drawings for computer storage, and to specify animation paths for the objects or the camera in the scene. CAD applications for splines include the design of automobiles bodies, aircraft and spacecraft surfaces, and ship hulls.
Spline curve can be specified by a set of coordinate positions called control points which indicates the general shape of the curve. These control points are fitted with piecewise continuous parametric polynomial functions in one of the two ways.
A set of six control points interpolated with piecewise continuous polynomial sections
A set of six control points approximated with piecewise continuous polynomial sections
Interpolation curves are used to digitize drawings or to specify animation paths. Approximation curves are used as design tools to structure object surfaces. A spline curve is designed , modified and manipulated with operations on the control points.The curve can be translated, rotated or scaled with transformation applied to the control points. The convex polygon boundary that encloses a set of control points is called the convex hull. The shape of the convex hull is to imagine a rubber band stretched around the position of the control points so that each control point is either on the perimeter of the hull or inside it.
UNIT – IV Part - B Geometric modeling techniques like wire frame, surface and solid modeling have totally changed not only the drawing office practices but also have helped to integrate design with analysis, simulation and optimization as well as to seamlessly integrate design with downstream manufacturing applications. Data created in geometric models can thus be directly passed on to all the application software packages like finite element analysis, mechanism analysis, CNC programming, inspection etc. Geometric modeling has therefore paved the way for CIM. The salient features of the different modeling techniques are discussed in this chapter. The starting point of new product development is conceptual design. The designer has to develop the shape of the product which in turn has to accommodate the functional parts inside. Whether it is a consumer durable like a camera, and an electric iron, a washing machine, an automobile, an entertainment electronic item like television or a sports item like a golf club, shape design is a critical activity in product design. This chapter also discusses conceptual design techniques and transfer of data to modeling software.
Product development activity starts with the design of the product. As mentioned in Chapter 2 this is a very critical activity which will influence the cost, performance, service life, quality, manufacturability, maintainability etc. The challenges before the product designers today are listed below:
The machine operator is provided with the drawing of the finished part and an operation sheet which gives step by step instructions to produce the part. Drawings are also required for process planning, tool design, production planning, and CNC programming, inspection, assembly, costing and vendor development. Thus, drawings are essential documents for product development as well as for regular production. However, with the introduction of computer integrated manufacturing the importance of drawing as the basic document for product development and manufacture has diminished. In addition to production drawings of components, the design department has to create layout drawings, assembly drawings, and tool drawings (Jigs, fixtures, templates, special tools, inspection fixtures etc). The number of drawings required for a product varies with the complexity of the product. In the case of the development of a centre lathe, it may be necessary to create about 400-500 drawings. For an aircraft, the number of drawings will be of the order of 30,000 to 60,000. In addition to component drawings, it is necessary to create hundreds of tool drawings and jig and fixture drawings for manufacture, assembly and inspection.
Considerable manpower and time will be required to create such a large volume of drawings and the
time required for this task represents a significant portion of the lead time required for product
development.
Computer aided design and drafting (CADD) is a powerful technique to create the drawings.
Traditionally, the components and assemblies are represented in drawings with the help of elevation,
plan, and end views and cross sectional views. In the early stages of development of CADD, several
software packages were developed to create such drawings using computers. Figure shows four
views (plan, elevation, end view and isometric view) of a part. Since any entity in this type of
representation requires only two co-ordinates (X and Y) such software packages were called two-
dimensional (2-D) drafting packages. With the evolution of CAD, most of these packages have been
upgraded to enable 3-D representation.
Computer representation of the geometry of a component using software is called a geometric
model. Geometric modeling is done in three principal ways. They are:
i. Wire frame modeling ii. Surface modeling iii. Solid modeling
These modeling methods have distinct features and applications.
In wire frame modeling the object is represented by its edges. In the initial stages of CAD,
wire frame models were in 2-D. Subsequently 3-D wire frame modeling software was introduced.
The wire frame model of a box is shown in Fig. (a). The object appears as if it is made out of thin
wires. Fig. (b), (c) and 6.2(d) show three objects which can have the same wire frame model of the
box. Thus in the case of complex parts wire frame models can be confusing. Some clarity can
Apart from standard surface types available for surface modeling (box, pyramid, wedge, dome, sphere, cone, torus, dish and mesh) techniques are available for interactive modeling and editing of curved surface geometry. Surfaces can be created through an assembly of polygonal meshes or using advanced curve and surface modeling techniques like B-splines or NURBS (Non- Uniform Rational B-splines). Standard primitives used in a typical surface modeling software are shown in Fig.. Tabulated surfaces, ruled surfaces and edge surfaces and revolved are simple ways in which curved geometry could be created and edited. Surface modeling is discussed in detail later in
this chapter.
The representation of solid models uses the fundamental idea that a physical object divides the 3-D Euclidean space into two regions, one exterior and one interior, separated by the boundary of the solid. Solid models are:
i. Spatial Enumeration: In this simplest form of 3D volumetric raster model, a section of 3D space is
described by a matrix of evenly spaced cubic volume elements called voxels.
ii. Cell Decomposition: This is a hierarchical adaptation of spatial enumeration. 3D space is sub-divided
into cells. Cells could be of different sizes. These simple cells are glued together to describe a solid object.
iii. Boundary Representation: The solid is represented by its boundary which consists of a set of faces, a set
of edges and a set of vertices as well as their topological relations.
iv. Sweep Methods: In this technique a planar shape is moved along a curve. Translational sweep can be
used to create prismatic objects and rotational sweep could be used for axisymmetric components.
v. Primitive Instancing: This modeling scheme provides a set of possible object shapes which are described
by a set of parameters. Instances of object shape can be created by varying these parameters.
vi. Constructive Solid Geometry (CSG): Primitive instances are combined using Boolean set operations to
create complex objects.
In most of the modeling packages, the approach used for modeling uses any one of the following three
techniques:
i. Constructive solid geometry (CSG or C-Rep) ii. Boundary representation (B-Rep) iii. Hybrid method which is a combination of B-Rep and CSG.
Constructive Solid Geometry (CSG)
In a CSG model, physical objects are created by combining basic elementary shapes known
as primitives like blocks, cylinders, cones, pyramids and spheres. The Boolean operations like union
(∪), difference (–) and intersection ∩ are used to carry out this task. For example, let us assume
that we are using two primitives, a block and a cylinder which are located in space as shown in Fig.
A “union” operation (A∪B) will combine the two to convert them into a new solid. (Fig.(c)) The
difference operation (A – B) will create a block with a hole (Fig. (D)). an intersection operation (A ∩B)
will yield the portion common to the two primitives. (Fig. (E)).
