Geomentric Modelling
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The document discusses different geometric modeling techniques. It describes Bezier curves, which interpolate control points and are tangent to the characteristic polygon. It also describes B-spline curves as another effective curve generation method. It discusses properties of B-splines and rational curves defined by ratios of polynomials. The document then summarizes wireframe modeling using lines, surface modeling using patches, and solid modeling techniques like CSG and boundary representation.
Geomentric Modelling
- 1. CHARACTERISTICS OF BEZIER CURVE The curve interpolates the first and last control points. The curve is tangent to the first and last segments of characteristics polygon. The curve shape can be modified by either changing one or more vertices of its polygon or by keeping the polygon fixed and specifying multiple coincident points at a vertex as shown in figure.
- 4. B-SPLINE CURVE B-spline curves provides another effective method, besides Bezier, of generating curves defined by data points. In fact, B-spline curves are the proper and powerful generalization of Bezier curves.
- 6. THE B-SPLINE FUNCTIONS HAVE FOLLOWING PROPERTIES
- 10. RATIONAL CURVES A rational curve is defined by the algebraic ratio of two polynomials while a nomenclature curve is defined by one polynomial. A rational B – spline curve defined by 𝑃 𝑢 = 𝑖=0 𝑛 𝑃𝑖 𝑅𝑖 𝑘 𝑢 , 0 ≤ 𝑢 ≤ 𝑢 𝑚𝑎𝑥 𝑅𝑖, 𝑘 𝑆 are the rational B – spline Basis function are given by 𝑅𝑖 𝑘 𝑢 = ℎ𝑖 𝑁𝑖 𝑘 (𝑢) 𝑖=0 𝑛 ℎ𝑖 𝑁𝑖 𝑘 𝑢
- 12. GEOMETRIC MODELING The mathematical description of the geometry of an object using a software is called as geometric modeling There are three basic methods Wire – Frame Modeling Surface modeling Solid modeling
- 13. WIRE FRAME MODELING This is one of the most popular and commonly used method of geometric modeling. In construction of wire frame model, the edges of an object are presented as lines. Wire frame model is used for following representations 2D Representation Orthographic views representation
- 14. 2D Wire Frame Model 3D Wire Frame Model The co-ordinate system is 2D co-ordinate system i.e. x and y co-ordinates only 3D co-ordinate system is used for representing objects; x, y and z coordinates are used 3 Dimensional wire frame system generation is difficult Both 2D and 3D wire frame generation is possible Hidden lines may not complicate the figure Difficult for the viewer to interpret the figure unless the hidden lines are removed Curved surfaces are indicated by circles, arcs and ellipses Curved surfaces are represented by spaced generators.
- 15. DRAWBACK OF WIRE FRAME MODEL Lack of clarity When part geometry is complex and in case of 3D wire frame systems, the hidden lines causes the image to be confusing to the viewer. During the surface definition, there might be confusing. By eliminating hidden lines, some clarity can be obtained.
- 16. SURFACE MODELING The techniques of representation of objects (or) components by surface is called surface modeling. Objects can be clearly interpreted by the user. Main draw back here is that, no data is available about the interior of solid. Application is modeling car bodies, ships, aerospace structure, dies, etc.
- 17. TECHNIQUES AVAILABLE FOR SURFACE MODELING Surface patch Coons patch Bicubic patch Bezier surfaces B-spline surfaces NURBS (Non-Uniform Rational B-Splines)
- 18. SURFACE PATCH The building blocks of the surfaces are known as surface patch. Generally u and v are two variables used for representing a patch. 0 ≤ 𝑈 ≤ 1 𝑎𝑛𝑑 0 ≤ 𝑉 ≤ 1
- 20. COONS PATCH Coons surface batch is a form of “transfinite interpolation”, which indicates that the coons scheme interpolates to an infinite number of data points. Coons patch is particularly useful in blending four prescribed intersecting curves which forms a closed boundary. In practice, coons surface can be used to create windshield designs for cars and skyline windows for houses.
- 22. BEZIER SURFACE Bezier surface is an extension of the Bezier curve in two parametric directions u and v. An orderly set of data or control points is used to build a topologically rectangular surface as shown in figure. The surface equation can be written as 𝑃 𝑢, 𝑣 = 𝑖=0 𝑛 𝑗=0 𝑚 𝑃𝑖𝑗 𝐵𝑖,𝑛 𝑢 𝐵𝑗,𝑚 𝑣 , 0 ≤ 𝑢 ≤ 1, 0 ≤ 𝑣 ≤ 1 where,P(u,v) is any point on the surface 𝑃𝑖𝑗 are the control points
- 24. B-SPLINE SURFACE B - Spline surface is an extension of the B - Spline curve. A rectangle set of data points creates the surface. A B – Spline surface can approximate or interpolate the vertices of the polyhedron as shown in figure. B – Spline surface equation is defined by 𝑃 𝑢, 𝑣 = 𝑖=0 𝑛 𝑗=0 𝑚 𝑃𝑖𝑗 𝑁𝑖,𝑘 𝑢 𝑁𝑗,𝑙 𝑣 , 0 ≤ 𝑢 ≤ 1, 0 ≤ 𝑣 ≤ 1 where,P(u,v) is any point on the surface 𝑃𝑖𝑗 are the control points
- 27. SOLID MODELING Solid modeling is one of the most effective geometric modeling method. In this approach, models are displayed as solids to viewer, there by eliminating any chance of misinterpretation. In solid modeling , to make the object more realistic.
- 32. SOLID MODELING TECHNIQUES Half – space Method Boundary representation Method Constructive Solid Geometry (CSG) Analytical Solid Geometry (ASM) Primitive instancing Sweep representation Spatial portioning representation
- 33. CONSTRUCTIVE SOLID GEOMETRY (CSG) This method is also known as C-rep. In this method, solid graphic primitives are employed for constructing the model. The solid primitives include cubes, spheres, cylinders, rectangle blocks and pyramids.
- 37. CSG MODEL The constructive solid model uses building block approach The physical objects can be divided into set of elements and combined in order to form an object.
- 38. ADVANTAGES OF CSG This requires less storage space This method is advantageous in the initial creation of solid models. Using Boolean operations, it is easy to construct solid models precisely. Less skill is enough CSG is more user friendly
- 39. DISADVANTAGES OF CSG This method involves more computational work for creating a solid model For complicated solid geometry, in this method is not appropriate The tree is not unique for the same part design
- 40. Boundary representation Method Boundary representation is one of the most popular and widely used schemes to create solid models of physical objects. In this method, front view, top view, bottom view, side view of an object is sketched and connected by means of lines to create a relationship.
- 41. PRIMITIVES OF B-rep MODEL Edge Vertex Face Loop Genus or Handle Body
- 42. POLYHEDRAL OBJECT Simple polyhedra Polyhedra with faces of inner loops Polyhedra with no through hole Polyhedra with through holes
- 44. EULER EQUATIONS For simple polyhedra V – E + F = 2 For closed polyhedra object F – E + V – L = 2 (B - G) For open polyhedra object F – E + V – L = B - G
- 47. ADVANTAGES OF B-rep This method is very powerful for creating complex shapes solid models. B-rep model can be easily converted into wire frame model system. B-rep system stores an explicit definition of the model boundaries. B-rep system is very much compatible with other systems.
- 48. DISADVANTAGES OF B-rep This requires more storage space. This concept cannot be applied for tool path generation.