1 5. GEOMETRIC MODELING Types of Curves and Their Mathematical Representation Types of Surfaces and Their Mathematical Representation Types of Solids and Their Mathematical Representation CAD/CAM Data Exchange
2 TYPES OF CURVES AND THEIR MATHEMATICAL REPRESENTATIONS Wireframe Model ( 2D in 1960s for drafting, 3D in 1970s) Wireframe Entities analytic entities (points, lines, arcs and circles, fillets and chamfers, and conics synthetic entities (splines and Bezier curves) methods of defining points: P(x,y,z), P(r,,z), P(x+x,y+y,z+z),, end points of existing entity, center point, intersection of two entities. methods of defining lines: between two points, parallel to axis, parallel or perpendicular to a line, tangent to entity methods of defining arcs and circles: center and radius, three points, center and a point, a radius and tangent to a line passing through a point. methods of defining ellipses and parabolas: ellipse (center and axes lengths, four points, two conjugate diameters), parabola (vertex and focus, three points). methods of defining synthetic curves: cubic spline (a set of data points and end slopes), Bezier curves (a set of data points), B-spline curves (interpolate a set of data points with local control possible).
3 Curve Representation: Two types of representation are parametric and non-parametric representation. In parametric representation all variables (i.e., coordinates) are expressed in terms of common parameters. For example, a point can be expressed with respect to a parameter as P( u) [ x( u), y( u), z( u)], u u u min max Non-parametric representation is the conventional representation as P [ x, y( x), z( x)] Ex. Non-parametric form of a circle: x^2+y^2=r^2, parametric form: P(u)=[r cos 2u, r sin 2u], 0u1. This form can be used to find slopes at a certain angle for example.
4 PARAMETRIC REPRESENTATION OF ANALYTIC CURVES The following list shows most of the analytic curve that are used in CAD/CAM system for part design and modeling. Lines Circles Ellipses Parabolas Hyperbolas Conic Curves
5 LINE AND CIRCLE A line between two points P 1 and P 2 can be expressed with respect to a parameter. P P u( P 1 ) 1 2 P A circle for a center and the radius can be written as x x y y z z c c c R cosu R sin u, 0 u 2
6 ELLIPSE An ellipse with a center and major and minor axes of 2A and 2B can be expressed as. x y z x y z c c c Acosu B sin u, 0 u 2
9 EXAMPLE 1
10 EXAMPLE 2
12 EXAMPLE 1
13 CONICS The most general form of planar quadratic curves is conic curves or conic sections that include the previously covered curves; lines, circles, ellipses, parabolas, and hyperbolas. The general implicit nonparametric quadratic equation that describes the planar conic curve has five coefficients and naturally needs five conditions to complete it. The conic parametric equation can be described if five conditions are specified appropriately. One case is specifying five points on the curve. L1 0, L2 0, L3 0, L4 0 L1 L2 0, L3L4 0 L L 1 2 al3 L4 0
14 EXAMPLE 1
15 PARAMETRIC REPRESENTATION OF SYNTHETIC CURVES Hermite Cubic Splines Bezier Curves: cubic curve with four control points. B-Splines: general case of Bezier s curve (non-uniform) Rational Curves (algebraic ratio of two polynomials) NURB (non-uniform rational B-spline) curve combines all features of previous curves
16 HERMITE CUBIC SPLINES
22 BEZIER CURVES Another alternative to create curves is to use approximation techniques which produce curves that do not pass through the given data points that are rather used to control the shape of the curves. Approximation techniques arc more often preferred over interpolation techniques in curve design due to the added flexibility and the additional intuitive feel. Bezier curves and surfaces are credited to P. Bezier of the French car firm Regie Renault who developed (about 1962) and used them in his software system called UNISURF which has been used by designers to define the outer panels of several Renault cars.
26 B-SPLINES In contrast to Bezier curves, the theory of B-spline curves separates the degree of the resulting curve from the number of the given control points. The B-spline curve P(u) for the degree k defined by n + 1 control points n P( u) B ( u) P, 0 u u j0 j0 partition of unity n k k j B j ( u) 1 j max
27 otherwise otherwise u u u n j for otherwise u u u u B j j j j j, 0, 1,, 0, 1, ) ( the recursive property 1 ), ( ) ( ) ( k u B u u u u u B u u u u u B k j j k j k j k j j k j j k j open parametric knots otherwise k j n j k n k j u j, 1, 1 0, for 0 j n + k + 1
28 Knot Values number of knots = number of points + degree + 1 (n k = n+1+k+1 must be satisfied) The individual knot values are not meaningful by themselves; only the ratios of the difference between the knot values matter. To be able to use the parametric knots produced by the coordinates, the values must be in increasing order. Thus, if the curve intersects itself, this cannot be used. The number of duplicate values is limited to no more than the degree. Duplicate knot values make the b-spline curve less smooth. At the extreme, a full multiplicity knot in the middle of the knot list means there is a place on the b-spline curve that can be bent into a sharp kink.
29 Summary of B-spline Basis Function B j k a polynomial function of degree k nonnegative for all j and k (nonnegativity) non-zero only on [u j,u j+k+1 ] (local support) at most k+1 basis functions of degree k are non-zero on knot span [u j,u j+1 ], namely, B j-kk, B j-k+1k,, B j k sum of all degree k basis functions on knot span [u j,u j+1 ] is one (partition of unity) at a knot of multiplicity m, basis function B j k is C k-m continuous. a composite curve of degree k polynomials with joining knots in [u j,u j+k+1 ] n+1 control points, P 0,P 1,,P n n-2 basis functions (i.e., cubic polynomial curve segments, Q 3,Q 4,,Q n ) n-1 knot points, t 3,t 4,,t n+1 (any additional ones needed for computation are set to same values as the nearest) B j k is - defined over a knot interval [u j,u j+1 ] - defined by four of the control points, P j-3,,p j
30 Notable benefits of b-spline include: Independent degree: the degree is set by user and it has nothing to do with number of data points a single piecewise curve of a particular degree: there is no need to stitch together separate curves as in interpolation splines such as natural cubic spline
36 MCAD PROGRAMMING
40 UNIFORM QUADRATIC B-SPLINES
41 UNIFORM CUBIC B-SPLINES
42 RATIONAL CURVES
45 RATIONAL FORMS OF COMMON CURVES
48 NURBS Remember that the b-spline curve is weighted sum of its control points. n k P( u) P B ( u), 0 u u j0 j j max Since the weights depend on the knot vector only, it is useful to add another weight w j to every control point that can be set independently. The weights can be added as n w j B ( u) P( u) N ( u) P, N ( u), 0 u u j0 k j j k j n j0 w j k j B Here, N j k is NURBS basis functions. Thus, the NURBS curve of degree k is defined as weighted sum of control points with NURBS basis functions. k j ( u) max
49 Some of Valuable Properties of NURBS Curves and Surfaces invariant under affine as well as perspective transformations (affine invariance and projective invariance). single mathematical form for both analytical and free-form shapes. flexibility to design variety of shapes. less memory is used when storing shapes in comparison to other methods. numerical algorithms can evaluate them easily and quickly.
50 EXAMPLE Solution. For three control points n = 2 and for the second degree k=2. The uniform knots can be computed by the program in the B-spline example. The following shows result in MathCad with modified program for B jk. It now uses the knot vector as an input.
52 Shown below are various curves that vary with different weights. Note that the last shows violation of convex hull property with negative weight for midpoint.
53 CURVE MANIPULATIONS The cases that require manipulation of curves include: Displaying Evaluating Points on Curves Blending Segmentation Trimming Intersection Transformation
58 DESIGN APPLICATIONS Example 1. For the given state of stresses (a) determine the principal stresses and (b) state of stresses on a plane a-a. A Example 2. For a plane motion of a bar sliding down the step, find the locus of the point C that is located at one-third length from B. C B
59 TYPES OF SURFACES AND THEIR MATHEMATICAL REPRESENTATIONS Surface model is an extension of wireframe but has advantages: less ambiguous, provide realism for display with hidden lines, mesh, and shading. Surface Entities Plane surface Ruled (lofted) surface (surface created by two curves being blended) Surface of Revolution
60 Tabulated Cylinder (surface created by a curve and a vector) Bezier surface: only approximates the given data points permitting only global control B-spline: surface that can approximate and interpolate permitting local control Coons Patch: used to create a surface using curves that form closed boundaries in contrast to the above surfaces that use open boundaries or set of points. Fillet surface: B-spline surface that blends two surfaces together Free-form surface: formed by free-form curves that are extensions of Bezier, B-spline, and NURB curves.
67 BILINEAR SURFACE Bilinear surface is a linear interpolation of four corners in two different directions (u, v). P P u,0 u,1 ( u) P( u, v) ( u) (1 u) P (1 u) P (1 v) P 0,0 0,1 u,0 up up vp 1,0 1,1 u,1
68 RULED SURFACE
69 REVOLVED SURFACE
70 TABULATED CYLINDER
71 PARAMETRIC REPRESENTATION OF SYNTHETIC SURFACES
72 HERMITE BICUBIC SURFACES
76 BILINEAR SURFACE Bilinear surface is a linear interpolation of four corners in two different directions (u, v). P P u,0 u,1 ( u) P( u, v) ( u) (1 u) P (1 u) P (1 v) P 0,0 0,1 u,0 up up vp 1,0 1,1 u,1
77 COON S PATCH Bilinear surface generated by four corners has straight sides and produce quite flat surfaces. In contrast, Coon s surface uses four side curves. ), ( ), ( ), ( ) ( ) ( ) (1 ), ( ) ( ) ( ) (1 ), ( v u v u v u u v u v v u v u v u v u BT LR T B BT R L LR P P P P P P P P P The surface obtained as above does not produce the end curves. Thus, evaluating the surface on boundaries and forcing it to confirm to the boundary curves yields extra-terms that must be subtracted.
78 Evaluating the surface along edges one finds extra terms that must be subtraceted. P P LR BT ( u, v) ( u, v) P(0, v) (1 u) P (1 v) P P P P LR L L L B ( v) up ( u) vp (0, v) P BT ( v) (1 v) P ( v) ( u) (0, v) B R T (0) vp (0) T must ( v) (1 v) P0,0 vp0,1 PL ( v) On the left edge, the extra terms are identified as ( 1 v) P vp 0,0 0,1
79 Similarly, the other extra terms are found along the other edges and all surplus terms are surplus (1 v) P (1 v) P (1 u) P (1 u) P 0,0 1,0 0,0 0,1 vp vp up 0,1 1,1 up 1,0 1,1 P( u, v) P ( u, v) P ( u, v) surplus LR BT Advantages: Follows boundary curves Limitation: not able to control internal shape
80 EXAMPLE For given data along four edges, use cubic Bezier curves along the edges to create Coon s patch.
81 First find the curves along the edges. Their three components in (u,v) are L,R,B, and T refer to left, right, bottom, and top.
82 The plot is shown below. L,R,B, and T refer to left, right, bottom, and top.
83 The Coon s patch is then obtained as below and the plot is shown with data points.
84 BEZIER SURFACE
87 EXAMPLE [PROGRAMMING]
89 B-SPLINE AND NURBS SURFACES
90 EXAMPLE [PROGRAMMING] xp yp zp ,,, wt
96 OTHER SURFACES
97 SURFACE MANIPULATIONS As in the curve manipulations, the surfaces have to be manipulated for various reasons in the CAD designs. Some of the applications are
99 DESIGN AND ENGINEERING APPLICATIONS
100 SOLIDS AND THEIR REPRESENTATIONS Solid Model is based on informationally complete (or spatial addressability), valid, and unambiguous representation of objects and stores geometric data as well as topological information of associated objects. This representation permits automation and integration of tasks such as interference analysis, mass property calculation, finite element modeling, CAPP (computer-aided process planning), machine vision, and NC machining. It is very easy to define an object with a solid model than other two previous modeling techniques (curves and surfaces) because solid models do not need individual locations as with wireframe models.
101 GEOMETRY AND TOPOLOGY The above figure illustrates the difference between geometry and topology. The geometry that defines the object is the lengths of lines, areas of surfaces, the angles between the lines, and the radius and the center of the cylinder and the height. On the other hand, topology (sometimes called combinatorial structure), is the connectivity and associativity of the object entities. It has to do with the notion of neighborhood and determines the relational information between object entities. From a user point of view, geometry is visible and topology is considered to be nongraphical relational information that is stored in solid model databases and are not visible to users.
102 SOLID ENTITIES There are various basic building blocks, so called, primitives that can be combined in certain boolean operations to construct complex models. They include: block cylinder cone sphere wedge Torus In the previous figure, a block and a cylinder were combined with union (i.e., addition) and difference (i.e., subtraction). One more available Boolean operation is intersection.
103 Example Explain how to construct the solid model of the bearing support with primitives and boolean operations. EXAMPLE
104 SOLID REPRESENTATION Underlying fundamentals of solid modeling theory are geometry, topology, geometric closure, set theory, regularization of set operations, set membership classification, and neighborhood. Solid representation is based on the notion that a physical object divides an n-dimensional space, En, into two regions: interior and exterior separated by the boundaries. A region is a portion of space En and the boundary of a region is a closed surface. The set in the set theory is a collection of objects and the operations include complement, union, and intersection. Further, regularized set operations are used to avoid irregular object created by boolean operations. Regular set is a geometrically closed set. The set membership classification determines if some objects intersect with a given object.
105 Half-spaces are unbounded geometric entities; each one of them divides the representation space into two infinite portions, one filled with material and the other empty. By combining half-spaces in a building block fashion, various solids can be constructed. Thus, a solid model of an object can be defined as a point set S in threedimensional Euclidean space E3. If three sets for a region is denoted by Si (interior set), Se (exterior set), and Sb (boundary set), then the set for the solid model can be expressed as
106 The mathematical properties that the solid model should capture can be stated as: Rigidity Homogeneous three-dimensionality (ie. no dangling boundaries) Finiteness and finite describability Closure under rigid motion and regularized boolean operations Boundary determinism (ie. Boundary must contain the solid) The mathematical implication of the above properties suggests that valid solid models are bounded, closed, regular, and semi-analytic subsets of E3. These subsets are called r-sets (ie. regularized sets), which are curved polyhedra with well-behaved boundaries. Here, regular means no dangling portion and semi-analytic means no oscillation in value.
107 FUNDAMENTALS OF SOLID MODELING Various solid representation schemes require several underlying fundamentals of solid modeling theory. They include geometry, topology, geometric closure, set theory, regularization of set operations, set membership classification, and neighborhood. For detailed discussion of these topics, the readers are encouraged to refer to advanced topics on solid modeling. In the following some of most popular solid modeling techniques are discussed.
108 MODELING TECHNIQUES Boundary Representation (B-rep): A B-rep model is one of the two most popular and widely used schemes and is based on the topological notion that a physical object is bounded by a set of faces. The following shows various faceted B-rep solids.
109 Constructive Solid Geometry (CSG): This 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 (i.e. Boolean operations) to form the object. The basic elements are block, cylinder, cone, sphere, wedge, and torus and building operations are Boolean operations. The following bearing support was constructed with various primitives in a certain sequence by CSG technique.
110 Sweep Representation: This is especially useful for two-and-half dimensional objects used most frequently for extruded solids and revolved solids. This is based on sweeping of a section along a path that may be linear, nonlinear, and hybrid operations. When the path is straight, it s a simple extrusion and when it is axisymmetric, it becomes the revolution. For nonlinear path, the sweeping is done along a nonlinear curve in space. Cutting tool path simulation is one good applications of this technique. The section may vary along the sweeping path. The following shows such an example with variable section.
111 Analytic Solid Modeling (ASM): Historically it is closely related to three dimensional isoparametric formulation of finite element analysis for 8- to 20-node hexahedral elements. This arose from the need to model complex objects for finite element analysis. ASM uses the parametric representation of an object in three-dimensional space that is a mapping of a cubical parametric domain (so called, master domain) into a solid described by the global coordinates (MCS).
112 Other Representations: primitive instancing: is based on notion of families of objects or family of parts. cell decomposition scheme: an object can be represented as the sum of cells. spatial enumeration scheme: a solid is represented by the sum of spatial cells that it occupies. The cells have fixed size. octree encoding scheme: is generalization of spatial enumeration scheme with variable cell size.
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