Manufacturing Technology (ME461)

Instructor: Shantanu Bhattacharya

Review of previous lecture

• Hermite cubic fit on two and three dimensional plane and space curves.

• Bezier curves theory.

• Properties of a Bezier curve.

Bezier Curves

• As in the previous section, we consider here one segment of the curve.

• For n+1 control points, the Bezier curve is defined by a polynomial of degree n as follows:

Bezier Curves

Here V0, V1, ……. Vn are position vectors of n+1 points (V0, V1, ……. Vn in Figure 3.19) that form the so-called characteristic polygon of the curve segment.

Properties of Bezier Curves

• The curves pass through the first and last control points (V0 and Vn from the preceding function) at parameter values 0 and 1. In the figure on the left the starting point V1’ of the second line and the endpoint V4 of the first line have the same position.

• The tangents at the first and last points are in the directions of the first and last segments of the characteristic polygon. This can easily be seen:

V’(0) = n *V(1)-V(0)+, V’(1) = n *V(n)-V(n-1)]

Where [V(1) – V(0)] and [V(n)-V(n-1)] define the first and last segments of the curve polygon. This implies that by aligning the last control point of the first Bezier curve segment, the connection point, and the first control point of the next curve segment will result in C1 continuity between the two curve segments.

Proof of the V’(0) and V’(1) values• The Bezier curve has the convex hull property. By convex hull property we mean that the entire curve lies within the characteristic polygon. This property is useful when curve intersection and spatial bounds on the curve segments are calculated.

Example problem

• Develop the equation of a Bezier curve, find the points on the curve for t = 0, ¼, ½, ¾, and 1, and plot the curve for the following data. The coordinates of the four control points are given by:

• V0 = [0,0,0], V1= [0,2,0], V2= [4,2,0], V3= [4,0,0]

Solution

B-Spline, Rational B-Spline, and Non uniform rational B-Spline curves

• The B-Spline is considered a generalization of the Bezier curve.

• Local control is an interesting feature of B-Spline curves which implies that any change in the local control point affects only part of the curve.

• Rational B-splines are generalizations of B-Splines. Interestingly, an RBS has an added parameter (called a weight) associated with each control point to control the behavior of the curve.

Surface Modeling

• In wireframe modeling, we take advantage of the simplicity of certain surfaces. For example a plane is represented by its boundaries. We say nothing about the middle of the plane, which is fine because we known that the middle of a plane is till a plane.

• It is common knowledge that the shapes of the cars, aircrafts and ships are very complex and do not consist of simple and regular geometric shapes. So, it is not easy to represent them by wireframe models and so surface modeling is prefered.

• Surface modeling system contains definition of surfaces, edges and vertices. It contains all the information that a wireframe does and in addition it also contains the information of how the two surfaces connect to each other.

Surface entitiesPlane surface:

A plane surface is the simplest surface that is defined by three non coincident points or its variation.

Ruled Surface:

A ruled surface can be defined as the linear interpolation between two general curves. Informally the straight lines connecting the two rails form the surface.

Surface of revolution: This surface is generated by rotating a planar curve in space about an axis at a certain angle.Tabulated cylinder: This surface is generated by sweeping a planar curve in space in a certain direction at a certain distance. In the example above a straight line sweeps about a path that is circle and forms the surface. The straight line is called the generatrix and the circle is the directrix.Bezier surface and B-Spline surface: Bezier and B-spline surfaces are both synthetic surfaces. Like synthetic curves a synthetic surface approximates the given input data. The Bezier and B-Spline surfaces are also formed in the same manner as the 2D curves.

Surface representations• In case of curves we have seen the representation of curves by

implicit/explicit equations.

• Implicit equation to describe a surface:

F(x, y, z) = 0, Its geometric meaning is that the locus of points that satisfy the constraint equation defines the surface.

• Explicit equation to describe a surface:

V = [x,y,z]T = [x,y,f(x,y)] T where V is the position of a variable point on the surface. In this equation, we directly define the variable oint coordinates x,y,z. The z-coordinates of the position vector of the variable points are defined by x,y through a suitable function f(x,y) as shown in the figure below.

•Comparing the equations for a 2-D curve and a 3-D surface the only difference between a space curve and a surface mathematically is that points on a space curve are defined by a single degree of freedom by that on a surface have two degrees of freedom.•Usually an arbitrary surface is defined in x,y with a functional relation f(x,y) by an x-y grid with P+1.Q+1 points.

Parametric Equation/ Representation of a Surface

• There are no extra parameters in equations represented earlier and as such these are called non-parametric representation of equations. The corresponding equations that utilize parameters are called parametric equations and have two degrees of freedom and are represented as :

V(s,t) = [x,y,z]T = [X(s,t), Y(s,t), Z(s,t)]T , smin< s< smax , tmin< t< tmax

Where x,y and z are functions of two parameters ‘s’ and ‘t’.

Example problem for Hermitian cubic surface:

Given a set of four space points and the tangent vectors at those points , find the equation of the hermitian surface patch. The four corner points are A(0,0,1), B(0,2,2), C(4,2,3), D(4,0,4). The tangent vectors of the corner points are 1 with a magnitude of 1. The cross vector slopes are assumed to be ‘0’. Find the basic equation from the transformation and formulate the surface.

Parametric representation of surface

Parametric representation of a surface

Solution

Bezier surface patch• Mathematically, the only difference between a

Hermite surface patch and a Bezier surface patch is that different basis functions are used.

• As with the Bezier curve, the Bernstien basis function is used for Bezier surface patch. The transformation matrix M is the only thing which changes.

Surface ManipulationsVarious surface manipulation techniques are available in CAD/CAM systems.

Displaying: The simplest and most widely used method is to display a surface by a mesh of curves. This is usually called mesh in a CAD software. By holding one parameter constant at a time, a mesh of curves is generated to represent a surface which are displayed. Shading of a surface are also available in many CAD/CAM systems, which gives the displayed objects a realistic image.

Trimming and segmentation: Segmentation and trimming is essentially a problem of representing an entity by its particular portion as required. For example, if we want to show a particular partition of a surface on a screen or get a portrait on a printer or for editing purposes, the surface is split first and then the required part is displayed and the rest is blanked.

Intersection: This function is usually provided in CAD/CAM systems. Various curves may result when two general surface intersect. This also serves as a way to define a curve.

Projection: Projecting an entity onto a plane or surface is basically the problem of projecting a point to plane or surface. When a curve or surface is projected, the point projections are performed repeatedly. This function is used in determining shadows of entities.

Transformation: As with the curve transformation we can translate, rotate, mirror and scale a surface in most CAD/CAM systems. To transform a surface, the control points of the surface are evaluated and then transformed to a new position or orientation and the new object is reconstructed using the transformed position coordinates.

CAD/ CAM data exchange

• It is common knowledge that the primary cause of data sharing problems between two or more systems is software incompatibility.

• This is due to the fact that vendors of computer applications design different proprietary formats to store the data required and produced by their systems.

• There are two solutions one of which is to develop the systems in a way that they are compatible with each other, in which case the two commercial products are integrated and the second is to have a neutral data exchange format for the purposes of data sharing.

• By neutral file, we mean that the file has a format that can be utilized by various systems.

Translation Strategies• In a direct translation strategy, one translator is

capable only of translating the information of one pair of systems.

• If we have N systems, every system has to have N-1 translators installed in order to input the database of any other system created.

• The total no. of translators needed would then be N(N-1).

• Moreover, the strategy is so complicated that it is almost impossible for a system to transfer data to all other systems.

Translation Strategies• In the indirect strategy, a neutral database structure is created. “Neutral” means that

the file format is independent of different formats utilized by the various CAD/CAM system vendors.

• With this strategy, each system needs only to have a preprocessor and a post processor to transfer the database universally. The function of a preprocessor is to translate the neutral file format database to the system’s own database format when the system is reading the database .

• The function of a post processor is to translate the given database format to neutral file format when output is made. In this strategy, N CAD/CAM systems will need only 2N translators. However, indirect translating is usually slower than direct translating.

• A successful data exchange format or standard or neutral file must meet a minimum set of requirements. The standard must cover the common entities, such as wireframe and surface entities, used in various modeling systems.

• The standard format has to be compact, which may help achieve faster storage and retrieval; i.e., higher speed in converting data to and from the neutral format and smaller size resulting neutral file.

• Among many standards available today, the IGES and PDES standards are the most widely used.

IGES (Initial graphics exchange specification)

• This was first published in 1980 and later updated in 1983, 86, 88 and 1990. It is the first widely accepted standard exchange format used to communicate a modeling database among dissimilar CAD/CAM systems.

• In fact IGES has been used for supplying data between suppliers and customers, as a means to create model of objects etc.

• The basic elements of IGES are entities we have discussed in previous sections. Therefore, the format of the data is basically a description of entities. In IGES, each entity is assigned a number.

• Numbers 1 through 599 and 700 through 5000 are allotted for specific assignments, and 600 through 699 and 10,000 through 99,999 are for user defined entities.

• Entities are classified as geometric entities (such as shape, curve, surface) and non geometric entities (such as relation between various entities).

• The geometric entities are described via two distinct but related Cartesian coordinate systems, namely, the model space coordinate system (MCS) or local coordinate system (W-workingCS).

PDES (Product data exchange standard)

• The international organization for standardization (ISO) is currently involved in developing an international standard called STEP (Standard for transfer and exchange of product model data).

• STEP is a step for global standardization of exchange of information related to automated manufacturing and results in the PDES.

• The fundamental differences between IGES and PDES is that the IGES utilizes the basic entities as elements of the design whereas PDES data exchange is done in terms of applications.

• This philosophy is also called mental models and is reflected in the PDES three layer architecture: Application layer-Interface between user and PDES in which the application model is explicitly expressed and the description and information are expressed, Logical layer which is used to provide a consistent and computer independent description, Physical layer which takes care of the structure of the data and its format of the exchange file.

Drawing Exchange File format

• DXF has been developed by Auto desk inc. to assist in interchange of drawings between AutoCAD and other programs.

• The overall organization of DXF is as follows:

1. Header section: general information about the drawing.

2. Tables section: Definitions of named items.

3. Blocks section: Describe the entities constituting each block in the drawing.

4. Entities section: The drawing entities, including any block references.

5. END OF FILE