Representation of Geometry in Model Space Óbuda University John von Neumann Faculty of Informatics Institute of Applied Mathematics Master in Engineering Informatics Course Modeling and design Lecture and laboratory No. 4 Representation of Geometry in Model Space Dr. László Horváth http://users.nik.uni-obuda.hu/lhorvath/

László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/ This presentation is intellectual property. It is available only for students in my courses. The screen shots in tis presentation was made in the CATIA V5 és V6 PLM systems the Laboratory of Intelligent Engineering systems, in real modeling process. The CATIA V5 és V6 PLM systems operate in the above laboratory by the help of Dassult Systémes Inc. and CAD-Terv Ltd. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/ Contents Lecture Interpolation and approximation curve Short history Parametric curve and its global and local parameters Parametric equation of surface B-spline curve Curve representation using spline base functions Parameterization of B-spline curve Control of B-spline curve Rational B-spline curve Laboratory tasks 4.1 Building contextual chains 4.2 Definition and analysis of contextual surfaces László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

Interpolation and approximation curve Linear Hermite (applied by: Ferguson and Coons) Circular Cubic t2 t2’ t1 Approximation P 1 2 3 Convex hull P 1 2 3 Convex hull Control polygon P 1 2 3 , Control point László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/ Short history French mathematician Paul Bezier (Renault factory): car body design using the method of approximation Paul de Casteljau applied the same method at the Citroen factory. However, the method became known under the name of Paul Bezier. Paul Bezier introduced the control polygon. Characteristics of the Bezier curve is provided by the Bernstein polynomial basic (blending) functions The curve passes over the first and last control points. The first and last segments of control polygon are tangents of curve. Global control. Degree of curve = number of control points - 1. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

n t b Parametric curve and its local parameters u max P (x,y,z) ( u ) General form of parametric equation of curve P(u)=[x(u) y(u) z(u)], where umin <= u <= umax Pu is the position vector to point P. Coordinates of point P in the function of parameter u : x=x(u), y=y(u) és z=z(u) t n b u max P (x,y,z) ( u ) min Z X Y P Local parameters of curve Accompanying trihedron (Vector units): t – tangent, n- normal b –binormal Tangent plane: t and b Normal plane: b and n Curvature Cartesian space László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

Parametric curve and its global parameters Control polygon consists of a single line. Control Local Global Degree (D) Class (N=D+1) Free or engaged end Shape of line is not allowed to change László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

Parametric curve and its global parameters Control polygon consists of 3 lines. Degree was changed to 2. Shape of line is allowed to change accordingly. Degree was changed to 3. Shape of line is allowed to change accordingly László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

Parametric equation of surface v=1 u=1 v=0 v=0,8 Isoperimetric curves P v u (x,y,z) Y X Z ( u, v ) Model coordinate system , General form of parametric equation of surface: P(u,v)=[x(u,v) y(u,v) z(u,v)] where umin <= u <= umax and vmin <= v <= vmax x=x(u,v), y=y(u,v) és z=z(u,v) László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

Parameter line and space Model space Parameter line u i u i u min u max Curve Parameter space v max v j u i v j u i v min u min u max Surface László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

Paraméter egyenes és tér B-spline curve Flexible steel ribbon in ship building. It was modeled as B spline. B-spline curve Consists of segments. Continuity at segment borders. Local control. Spline base functions. Degree of the curve is same as degree of the base (blending) function. Different degree of segments is allowed. Curve goes through of the first and last control points only in case of special parameterization. t=0 t=1 Sequence of intervals u u u u u 1 2 3 4 Knot vector László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

{ } å ( ) Representation of curve using spline base functions P : i = Analytical definition of B-spline curve: ( ) P = i 1 n u N k å , where: The normalized B-spline base function: N i, k ( u) The control points: { } P : i = 0,1, . , n The B-spline curve includes polynomial represented segments Order of segment k, its degree k-1. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

Parameterization of B-spline curve The parameter range of curve is divided into intervals for segments: The knot vector: { } u : i = 0,1, . , n k + u0 u1 u3 u2 where: u i £ + 1 The curve passes over given number of segment border points. These points are called as knots. Knots carry the parameter values for segment border points. In case of control points n + 1 , order k, degree k-1, number of knots m: (m+1) = (n+1) + k Number of knots: m = n + k Periodic curve: The parameter intervals are repeated. The uniform B-spline is periodic. Non-periodic curve: Inside knots are uniform. However, intervals are repeated at beginning and end of the vector. Maximum number of repetition is the order of curve. The allocation of parameter range can be uniform or non-uniform. Non uniform allocation serves more sophisticated representation of geometry. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

Examples for knot vector "a" curve: k=2, degree=1 b "b" curve: k=3, degree=2 "c" curve: k=4, degree=3 c Knot vectors: "a" curve: 001233 "b" curve: 0001222 "c" curve: 00001111 László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

Control of B-spline curve A base (blending) function acts only on a segment of the B-spline curve (local control). However, second order continuity (same tangent and curvature) should be ensured at segment borders automatically. This continuity must be saved during any modification of curve. Consequently, the real control is: V 1 2 3 4 5 Uniform, periodic Control points which act on the highlighted segment Segments, e.g. 1 V - V 2 2 V - V 1 3 6 V - V 5 1 Effect of B-spline base functions László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

( ) ( ) Q = wx , wy , wz , w w ³ x , y , z P w Rational B-spline curve Representation of rational B-splines uses the concept of homogenous coordinates. This was originally applied at representation of transformation matrices. Pont defined in the three dimensional Euclidean space: ( ) x , y , z P Representation of this point in the four dimensional homogenous space w ( ) Q = wx , wy , wz , w where w is the homogenous coordinate which is often called as weight: w ³ Values of w are included for the control points in the weight vector. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

( ) å Q u w = , ( ) Q u N V = x y z Rational B-spline curve V V w = B-spline curve with homogenous coordinates: ( ) Q u N V w i k n = å , where: Qw(u) is the point of curve as expressed using four dimensional homogenous coordinates: ( ) Q u w x y z = , N i,k ( u ) is the spline base function, and V is control point in the four dimensional homogenous space: w V w i = , consequently László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

Rational B-spline curve Three dimensional projection of the four dimensional control points by dividing the first three coordinates by the homogenous coordinate: Similarly, the Q(u) point of curve is represented by as projection of four dimension space to three dimensional space: x w y z = ( ) Q u N w V i k n = , å The rational B-spline curves are characterized by knot and weight vectors. At representation of analytical curves value of w determines the shape of curve: straight line, elliptic, parabolic, or hyperbolic the segment. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Building contextual chains László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Definition of contexts for point definitions László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Definition of point in the context of curve László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Definition of point in the context of reference plane László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Definition of point in the context of curve László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Definition of point in the context of surface in boundary László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Definition of curve in the context of points. At one of the points context of curve tangent is defined. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Activation of close parameter of curve. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Definition of surface by tabulation of the closed curve. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Definition of reference plane in the context of three points. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Definition of point in the context of the reference surface Plane2. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise One point of the earlier defined curve was replaced by the Point4. One of the ends of the modified tabulated surface was closed by the first order contextual surface Fill1. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise The other end of modified tabulated surface was similarly closed by the surface Fill2. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise The three highlighted surfaces are joined in the feature Join1 in order to make them eligible as common context. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Solid feature was defined between the Join1 feature and its offset. For further development of solid body new reference plane will be required. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise The tabulated solid form feature was regenerated into shell shaped by activation of the relevant parameter. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Additional line definition is needed. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Rib form feature is defined. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Pipe stub is required in order to provide connection to the interior. The solution to the problem begins by defining a hole (topologically breakthrough!) László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise The hole feature extended the boundary by a cylindrical surface which is extrapolated. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Solid form feature is defined between the extrapolate surface and its offset. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Fillet features are defined. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Model is ready. Survey the contextual chains! László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Analysis of curvature in an important context. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Visualization of control polygon mesh of one of the surfaces. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.1 laboratory exercise Curve parallel with the closed Spline1 is defined for later use. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise Definition and analysis of contextual surfaces No comments included. Please understand model using screen shots. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

MD 4.2 laboratory exercise László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/