UNIT 2-Geometric Modeling.pptx

Geometric Modeling

1. Unit 2 GEOMETRIC MODELING Representation of curves- Hermite curve- Bezier curve- B-spline curves-rational curves-Techniques for surface modeling – surface patch- Coons and bicubic patches- Bezier and B-spline surfaces. Solid modeling techniques- CSG and B-rep

2. REPRESENTATION OF CURVES Mathematically , curve is a continuous map from one-dimensional space to n dimensional space. A curve is an infinitely large set of points . The points in a curve have a property and any point has two neighbors except for a small number of points which have one neighbor (these are the endpoints). Some curves have no endpoints either because they are infinite (similar to a line) or they are closed (loop around and connect to themselves)

3. REPRESENTATION OF CURVES The problem that we need to address is how to describe a curve or to give " names " or representations to all curves so that we can represent them on a computer. known shapes such as line segments, circles, elliptical arcs, etc. A general curve which does not have a " named ” shape is sometimes called a free-form curve because a free-form curve can take on just about any shape and it is much harder to describe.

4. Mathematical Representation of Curves Explicit: F ree parameter to the set of points on the curve. The explicit form of a curve in two dimensions gives the value of one variable. It may be a dependent variable in terms of the other or independent variable. For cach value of x, only a single value of y is normally computed by the function. y = f(x )

5. Mathematical Representation of Curves (b) Implicit I mplicit curve in two dimension is defined by an implicit function f (x, y) = 0 A common example is the circle whose implicit representation.

6. Mathematical Representation of Curves (c) Parametric curve: The parametric form of a curve defines a function that assigns positions to values of the free parameter. A two- dimensional or three dimensional parametric curve may be represented x= X(u); y= Y(u); z=Z(u )

7. Synthetic Curves The analytical curves are not sufficient to fully some of the geometries that are encountered in real life . Ex: Aircraft bodies, automobile body, moulds and die,Horse saddles, Ship hulls etc . For this purpose, free-form and synthetic curves are developed . These synthetic curves have many curve segments .

8. Synthetic Curves The properties of these curves are : 1. Easy to enter the data and control the continuty the curves . 2. Require lesser computer storage for the data representing the curves 3. Have no computational problems and faster in computing time. Since synthetic curve have different types of curve segments , they need to maintain certain continuity requirements.

9. Synthetic Curves Three types of continuity are possible : Zero order continuity(Co) - simply that the curves meet First order continuity (C¹) - first parametric derivatives of the coordinate functions for two successive curve sections are equal at their joining points. Second order continuity (C²) - both first and second parametric derivatives of two curve sections are the same at the intersection

10. Various order of continuity

11. Interpolation and Approximation Modeling Interpolation Modeling: P olynomial sections are fitted so that the curve passes through each control point. R esulting curve is said to interpolate the set of control points. Therefore , the interpolation essentially tries to pass a curve on a surface called interpolation .

12. Approximation Modeling P olynomials are fitted to the general control-point path without necessarily passing through any control points. R esulting curve is said to approximate the set of control points . Therefore, an approximation tries to fit a smooth curve on the surface which may be closer to three points but it may not actually pass through each of them.

13. HERMITE CURVE (OR) HERMITE CUBIC SPLINE Curves draw their name from the traditional drafting tool called ‘ French Curves ’ (or) splines . Cubic splines use cubic polynomial has four coefficients and thus requires four conditions to evaluate. A cubic spline uses four data points . The Hermite cubic spline uses two data points at its ends and two tangent vectors at these points

14. U ses: The cubic splines in design applications is not very popular due to the need for tangent vectors or slopes to define the curve . T he control of the curve is not very obvious from the input data due to its global control characteristics. Example: 1.Changing the position of a data point 2.An end slope changes the entire shape of the spline.

15. Bezier curve was developed by Pierre Bezier at French car company “ Renault Automobile Company ”. Bezier and B-spline curves are created on the basis of approximation techniques . Bezier curve have a number of properties which make them highly useful and convenient for curve and surface design . Bezier curves are widely available in various CAD systems . Most graphics software includes a pen tool for drawing paths with Bezier curves BEZIER CURVE

16. Mathematical Formulation of Bezier Curves Bezier uses : control polygon for curves in place of points and tangent vectors as in the case of cubic splines. The Bezier curve is defined in terms of locations with n+1 points which are called control points. Bezier curve section can be fitted to any number of control points. The number of control points to be approximated and their relative position determine the degree of the Bezier polynomial .

17. Mathematical Formulation of Bezier Curves Bezier curve which has four control points (P1, P2, P3 and P4). Dashed lines connect the control point positions which forms the characteristics polygon . Only, first and last control points or vertices of the characteristics polygon actually lie on the curve . The other two vertices define the order derivates and shape of the curve. The curve is always tangent to first and last polygon segments . Examples: Two-dimensional Bezier curves generated from three, four, and five control points are

18. Mathematically, a parametric Bezier curve for n+1 control points is defined by is the Bernstein function are given by Where,  

19. Characteristics of Bezier Curve A Bezier curve is defined on n+1 points and is represented as a parametric polynomial curve of degree n. It always passes through the first and last control points . The Bezier curve is tangent to first and last segments of the characteristics polygon. The curve generally follows the shape of characteristics polygon. The degree of polynomial defining the curve segments is one less that the number defines the polygon points . Bezier curve exhibit a symmetry property. Each control point is weighted by its blending function for each u value. The curve lies entirely within the convex hull formed by four control points  

20. B-SPLINE CURVE L imitations: To reduce the degree of the curve is to reduce the number of vertices and vice versa . It increases the complexity of the curve and its calculation . It means that the value of blending function is non zero for all parametric values over the entire curve.

21. B-SPLINE CURVE It provide another effective method of generating curve defined polygons . These curves are widely used of approximation splines.  

22. B-SPLINE CURVE Where i- th normalized B-spline basis function of order k (degree k-1 ). Bi,k (u) are defined by Cox- DeBoor recursion formula. k subintervals of the total range of u u is referred as a knot vector. ui are elements of knot vector satisfying the relation is ui ≤ ui+1. P arameters varies from umin to umax along the curve P(u ).

23. B-spline functions Partition of unity- relationship between the curve and its defining control points Positivity- curve segment lies completely within the convex Local support- control point affects only k curve s Continuity-degree k-1 and CK-2 continuity over the range of u

24. Characteristics of the B-spline curves U sing multiple control points by placing several points at the same location.B -spline curve by moving the control points. F irst and last control points except when the linear blending functions are used .

25. Characteristics of the B-spline curves B-splines allow to vary the number of control points used to design a curve without changing the degree of the polynomial .

26. Characteristics of the B-spline curves A non-periodic B-spline curve pass through the first and last control points As the number of degree of curve increases, the curve tightens A second-degree curve (k = 3) is always tangent to the midpoints. B-spline curve becomes a Bezier curve if k equals the number of control points (n+1)

27. Characteristics of the B-spline curves Multiple control points results the regions of high curvature of a B-spline curve. As the degree of curve increases, it will be more difficult to control and calculate accurately .

28. RATIONAL CURVES A rational curve is defined by the algebraic ratio of two polynomials where as non-rational curve is defined by one polynomial. The most widely used rational curves are non-uniform rational b-splines (NURBS). A rational B – spline curve defined by are the rational B – spline Basis function are given by  

29. 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.

30. TECHNIQUES AVAILABLE FOR SURFACE MODELING Surface patch Coons patch Bicubic patch Hermite surfaces Bezier surfaces B-spline surfaces

31. SURFACE PATCH A surface patch is defined in terms of point data will usually be based on a rectangular array data points. In computer graphics, the parametric surface are sometimes called patches, curved surfaces or just surface. The building blocks of the surfaces are known as surface patch Generally u and v are two variables used for representing a patch.  

32. Cont …,

33. COONS PATCH A linear interpolation between four bounded curve is used to generate a coons surface , which is also called coons patch . The coons formulations interpolate to an infinite number of control points to generate the surface and it is referred as a form of transfinite interpolation .  

34. Cont …,

35. BICUBIC SURFACE Bicubic patch or surface is generated by four boundary curves which are parametric Bicubic polynomials. Bicubic parametric patches are defined over rectangular domain in uv - space and the boundary curves of patch are themselves cubic polynomial curves. The following are the major types of parametric bi-cubic surfaces used in CAD Hermite surface Bezier surface B-Spline surface

36. 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 where, P( u,v ) is any point on the surface are the control points  

37. 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 where, P( u,v ) is any point on the surface are the control points  

38. 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. The solid modeling is used to make the object more realistic.

39. Solid mod eli ng entities Which are building blocks which are also called primitives. Most of the commercially available solid modeling packages such as AutoCAD, SolidWorks , Unigraphics , IDEAS, Pro/Engineer etc. H ave certain set of solid primitives which can be combined by a mathematical set of Boolean operations to create the solid model .

40. Solid modeling entities

41. Solid Modeling Approaches Two different types of solid modeling approaches: primitive based modeling-create complex solids feature based modeling-more complex shape and it elaborates solids more readily Boolean operations are union or combination (U or +), intersection (∩or I) and difference or subtraction (-).

42. Solid Modeling Approaches

43. Solid Modeling Approaches

44. SOLID MODELING TECHNIQUES Constructive Solid Geometry (CSG ) Boundary representation Method(B-REP)

45. 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.

46. 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.

47. 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

48. 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

49. 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.

50. PRIMITIVES OF B-rep MODEL Edge Vertex Face Loop Genus or Handle Body

51. 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.

52. DISADVANTAGES OF B-rep This requires more storage space. This concept cannot be applied for tool path generation.