computer-aided-design
TRANSCRIPT
GEOMETRIC MODEL
Wireframe Model Surface Model Solid Model
Wireframe Model
A wireframe model of an object is the simplest, geometric model that can be used to represent it mathematically in the computer .It is sometimes referred to as a stick figure or an edge representation of the object.
A wire frame model is a visual presentation of a three dimensional or physical object used in 3D computer graphics It is created by specifying each edge of the physical object where two mathematically continuous smooth surfaces meet, or by connecting an object's constituent vertices using straight lines or curves. The object is projected onto the computer screen by drawing lines at the location of each edge.
Using a wire frame model allows visualization of the underlying design structure of a 3D model. Traditional 2-dimensional views and drawings can be created by appropriate rotation of the object and selection of hidden line removal via cutting planes.
Wireframe models
Wireframe Entities
Points Lines Arcs Circles Fillets Chamfers Ellipse Parabolas Hyperbolas Cubic spline B-spline Bezier curves
Lines
A line connecting two points p1 and p2 .A parameter u such that at p1 u has the value 0 and at p2 ,u value is 1.
Utilizing the triangle OPP1 equation can be written as: P=P1+(P-P1)
The vector (P-P1) is proportional to the vector P2-P1 P-P1=u(P2-P1) Equation of line becomes P=P1+u(P2-P1), 0<u<1
Scalar form ,this equation can be written
x=x1+u(x2-x1)
y=y1+u(y2-y1) 0 < u < 1
z=z1+u(z2-z1)
yP1 P P2P2-P1
U=0 U=1
x
z
o
Surface Model
A surface model of an object is a more complete and less ambiguous representation than its wireframe model . It is also richer in its associated geometric contents which makes it more suitable for engineering and design applications. A surface model can be used , for example , to drive the cutter of a machine tool while a wireframe model cannot
Surface modeling has been developing rapidly due to the shortcomings and in conviences of wireframes modelling . The former is considered an extension of the latter. In general a wireframe model can be extracted from a surface model by deleting or blanking all surface entities .
Surface Entities
Two Types
Analytic Surface Entities
Synthetic Surface Entities
Analytic Entities
Plane Surface It is the simplest surface .It requires three non coincident points to define an
infinite plane
Ruled (lofted) Surface Linear surface. It interpolates linearly between two boundary curves that
define the surface rails
Surface of Revolution This is an axisymmetric surface that can model axisymmetric object
Tabulated Cylinder Is a surface generated by translating a planar curve a certain distance along a
specified direction .The plane of the curve is perpendicular to the axis of the cylinder.
Synthetic Entities
Bezier surface 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 the surface.
B- spline surface This is a surface that can approximate or interpolate given input
data. General surface like the Bezier surface but with advantage of permitting local control of the surface.
Coons patch The above surfaces are used with either open boundaries or given
data points. The Coons patch is used to create a surface curves that form closed boundaries.
Fillet Surface This is a B-spline surface that blends two surface together The two orginal
surfaces may or may not be trimmed.
Offset Surface Existing surfaces can be offset to create new ones identical in shape but may
have different dimensions . It is a useful surface to use to speed up surface
construction.
Solid Model
Solid modelling has been acknowledged as the technological solution to automation and integrating design and manufacturing functions.Indeed the complete definition of part shape (geometry and topology) through solids models has been called a key to CIM
A solid model of an object is a more complete representation than its surface model .It is unique from the latter in the topological information it stores which potentially permits functional automation and integration
Defining an object with a solid model is the easiest of the available three modeling techniques (curves ,surfaces and solids). Solid models can be quickly created without having to define individual locations as with wireframes.In many cases, solid models are easier to build than wireframe or surface models .For example , representing the intersection of two cylinders using wireframe, modelling is not possible.
Primary uses of solid modeling are for CAD, engineering analysis, computer graphics and animation, rapid prototyping, medical testing, product visualization and visualization of scientific research.
Solid modeling software originally used either constructive solid geometry (CSG) or Boundary representation (B-REP) techniques to define solid shapes. Beginning in the late 1980s, software developers began applying higher-levels of abstraction to solid modeling construction techiques. The first of these techniques, called parametric feature-based solid-modelling, was introduced in commercial software by Parametric Technology Corporation in September 1987. These approaches made solid-modeling software easier to use and increased its acceptance among mechanical engineers
B-rep
B-rep is used to create solid models of physical objects .A B-rep model or boundary model is based on the topological notion that a physical object is bounded by a set of faces .These faces are regions or subsets of closed and orient able surfaces. A closed surface is one that is continuous with out breaks. Type of operation used in B-rep is Euler’s operation. Euler operators provide designers with drafting functionality.
Euler operations are used to create, manipulate, and edit the
faces, edges, and vertices of a boundary model.
CONSTRUCTIVE SOLID GEOMETRY
CSG is most popular because they are best understood representation scheme.
Easy to create and store , easy to check for validity A CSG model is based on the topological notion that a
physical object can be divided into a set of primitives that can be combined in a certain order following a set of rules (Boolean operations) to form the object .
The main building operations in CSG schemes are achieved by set operators
Operators are union , intersection , and differerence
SPLINE
The term "spline" is used to refer to a wide class of functions that are used in applications requiring data interpolation and/or smoothing. The data may be either one-dimensional or multi-dimensional. Spline functions for interpolation are normally determined as the minimizes of suitable measures of roughness (for example integral squared curvature) subject to the interpolation constraints.
Smoothing splines may be viewed as generalizations of interpolation splines where the functions are determined to minimize a weighted combination of the average squared approximation error over observed data and the roughness measure.
Contd……
Splines are popular curves in these subfields because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through curve fitting and interactive curve design.
B-SPLINE
In the mathematical subfield of numerical analysis, a B-spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. A fundamental theorem states that every spline function of a given degree, smoothness, and domain partition, can be represented as a linear combination of B-splines of that same degree and smoothness, and over that same partition. The term B-spline was coined by Isaac Jacob Schoenberg and is short for basis spline. B-splines can be evaluated in a numerically stable way by the de Boor algorithm.
