PREPARED BY: SAMERA BINTI SAMSUDDIN SAH SEM /2012 (NOV 2011)

INTRODUCTION ABOUT GEOMETRIC MODELING…  CAD tools have been defined as the melting pot of three disciplines: design, geometric modeling, and computer graphic.  A geometric model should be unique and complete to all engineering functions, from documentation to engineering analysis to manufacturing.

CURVES  Geometric description of curves defining an object can be tackled in several ways. coordinate data analytic equation  A curve can be described by arrays of coordinate data or by an analytic equation.  Majority of the curves were circles, but some were free-form.  Those are curves arising from applications such as ship hull design to architecture.

 When they had to be drawn exactly, the most common tool was a set of templates known as French curves.  These are carefully designed wooden curves and consist of pieces of conics and spirals.  a conic section is a curve obtained by intersecting a cone with a plane.  3 types of conic section are ellipse, parabola, hyperbola.

1. Ellipse  In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve.  Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis.  An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant.

Example of an ellipse:

2. Parabola  Parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface.  Given a point (the focus) and a corresponding line (the directrix) on the plane, the locus of points in that plane that are equidistant from them is a parabola.

Example of a parabola:

3. Hyperbola  a hyperbola is a smooth curve that lies in a plane, which can be defined either by its geometric properties.  A hyperbola has two pieces, called connected components or branches, which are mirror images of each other and resembling two infinite bows.  The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a cone.

Example of a hyperbola:

SPLINES  Another mechanical tool, called a spline was also used.  This was a flexible strip of wood that was held in place and shape by metal weights, known as ducks.  A spline “tries" to bend as little as possible, resulting in shapes which are both aesthetically pleasing and physically optimal.

 The mathematical counterpart to a mechanical spline is a spline curve, one of the most fundamental parametric curve forms.  Instead of referring to curves that minimize certain functional, spline curves are now mostly thought of as piecewise polynomial (or rational polynomial) curves with certain smoothness properties.

Example of Splines

SURFACES  Parametric surfaces were well understood after early work by Gauss and Euler.  They were immediately adopted in early CAD/CAM developments: A standard application is tracing a surface for plotting or for driving a milling tool.  Parametric surfaces are well-suited for both tasks.  The most popular of all surface methods was to become the tensor product surface.

Rectangular Surfaces  Rectangular surfaces are a map of a rectangular domain into 3D.  As a special case, we may map the domain to a 2D parametric surface, resulting in a distortion of the domain rectangle.  If we embed a curve in this domain rectangle, we will obtain a deformed curve.

Application of rectangular surfaces.

SOLID GEOMETRIC MODELING Solid model consist of volumetric information & surface info of an object Surface of the model represent boundary between outside & inside of the object Basic rule – all surfaces must touch another surface

Solid Modeling

3 different types of solid modeling – Primitive modeling – Constructive solid geometry (CSG) – Feature-based modeling (FBM)

1. Primitive Modeling Objects described using basic geometrical forms. Common geometric primitives.

Example:

2.Constructive Solid Geometry More flexible and powerful than primitive. Allow Boolean Operations: - union, difference & intersection

Boolean operation

Example using CSG:

Example using USG:

3. Feature-based Modeling 3D model is built using series of features, such as hole, slot, square block, etc. Each feature can be independent or linked to other feature. The geometry of each feature is controlled by modifiable constraints and dimensions.

FBM: 3D operations  Basic concept – 2D cross-section or profile is produced – Depth is given to the profile Generally 4 types** – Extrude – Revolve – Sweep – Blend **different terms might be used in different software/books

1. 3D Ops: Extrude  A linear sweep, where the profile is given a depth in straight line, perpendicular to the profile plane Cross-section is constant, start – end.

2. 3D Ops: Revolve  The profile is rotated around a defined axis, 0 – 360 degree Cross-section is constant

3. 3D Ops: Sweep  The new command and is similar to the EXTRUDE command, but it concentrates on using paths to define the direction of the extrusion.  This command SWEEP a 2D object along a path

3D Ops: Sweep & Blend

Steps in building 3D object

Examples of FBM + Boolean