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by: Aliyah Boyer


Aliyah Boyer
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Class Notes
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This 5 page Class Notes was uploaded by Aliyah Boyer on Friday September 18, 2015. The Class Notes belongs to CIS 6905 at University of Florida taught by Staff in Fall. Since its upload, it has received 27 views. For similar materials see /class/207034/cis-6905-university-of-florida in Comm Sciences and Disorders at University of Florida.

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Date Created: 09/18/15
Geometric continuity Geometric continuity GC is a constructive way of describing when two manifolds meet with C6 continuity along a common interface In essence two manifolds A and B join Gf continuity if there exists a parameterization of A and B so that partial derivatives agree along the interface While computeraided geometric design CAGD has relied heavily on mathematical descriptions of point sets based on parametric functions in recent years geometric continuity for parametric curves and surfaces actually needs a notion different from the direct matching of Taylor expansions which is used to define the continuity of piecewise functions 1 Background B zier representation Every polynomial curve segment can be represented by its socalled Bezier polygon A dimension count shows that the n1 linearly independent Bernstein polynomials Blquot form a basis for all polynomials of degree 3 n Hence every polynomial curve bu of degree 3 n has a unique nth degree B zier representation n bu 20131quot H 10 Any affine parameter transformation u al tbt 61 b leaves the degree of the curve b unchanged Consequently bul has also an nth degree Bezier representation n 11010 2 M31quot 0 10 The coefficients b are elements of Rd and are called B zier points They are the vertices of the B zier polygon of bu over the interval 1 b The parameter I is called the local and u the global parameter of b Derivatives From the definition the derivative of a Bernstein polynomial of degree n is 61 E370 nagquot 0 13 1 t for 0 n u a one obtains b a where 31 BZ 1 0 Thus given a curve bu ZblBlquot 1 10 for its derivatives b u n b n1 11111 ZAbIBIquot391twhere Ab b1 b1 dt a 10 See also Bernstein polynomial Bezier Surface 2 motivation Since the x y and 2 components of point sets curves and surfaces are functions we always tend to relate the continuity of functions Cr continuity to geometric continuity Ck continuity Two Cr function pieces join smoothly at a boundary to form a joint Cr function if at all common points their kth derivatives agree for k 0 l k But even if the derivatives of the component functions agree this criterion is neither sufficient nor necessary for characterizing the smoothness geometric continuity of curves or surfaces The following example illustrates the inadequacy of the standard notion of smoothness for functions when applied to curves Figure l Geometric continuity and Function continuity Matching derivatives of the component functions and geometric visual continuity are not the same the blue and pink shape is parameterized by two parabolic arcs with equal derivatives at the tip but the shape is not geometrically continuous the green and pink shape is parameterized by two parabolic arcs with unequal derivatives at their common point but the shape is geometrically continuous In Figure l the V shape of VC is parameterized by these two quadratic pieces u v 6 01 11 2021 02 91u1 u 0uu 0H 01 2 021 12 qzv 0v0 vv1v and 0 At the common point q1l Q q2 0 the derivatives of two pieces ofV shape agree 0 D611 1 0 D612 0 But obviously the V shape is not joining smoothly at this common point showing that matching derivatives do not always imply smoothness Conversely smoothness does not imply matching derivatives The green and pink shape of VC is parameterized by the two quadratic pieces u V 6 01 01 2 021 1 2 q3u 0 lt u 0 uu 1u 01 2 021 12 q4v 0v0 vv1v 0 The is visually and geometrically smooth at the common point q3 0 Q since the two pieces and have the same vertical tangent line but he derivatives do not agree 2 0 13900 2 0 13 0 In the case of surface the distinction between higherorder continuity of the component functions and geometric continuity of the surface is more subtle Dealing with two variables we contrast the geometric smoothness criteria with coplanarity condition of the edgeadj acent triangles of the control net The conclusion remains similar The coplanarity ofthe edge adjacent triangles of the control net does not imply geometric smoothness neither is it necessary for a smooth join 3 Geometric Continuity ofParametric Curves and Surfaces Geometric continuity is a relaxation of parameterization not a relaxation of smoothness In this section we ll define kth order geometric continuity Gk as agreement of derivatives after suitable reparametrization 31 Joining Parametric Curve Pieces De nition 21 ijoin Two Cr curve segments g and p join at p0 with geometric continuity Gk viathekaap p R HR if D g ploD pl0 160 Jc Dp logt0Dplo 0 The map p is called reparametrization If p is a rigid transformation the p and g are said to join parametrically C6 and if p id the identity map then p and gform a C6 map The constraint D p 0 gt 0 rules out cusps and other singularities With the abbreviation jkplO pit DPlo kal0T E RUM for p E Rquot the condition of k G continuity can be written as jkg 0 p lo Ajkg lpw jkp 1 0 Dp sz DMZ whereA 3 2 3 D p 3DpD p DP 0 W DpY The matrix A is called Gk connection matrix or i matrix and jkp is the k jet of p In one variable two regular maps p and q can both be reparametrized so that ppp and qpq have the preferred arclength parametrization ie unit increments in the parameter correspond to unit k k increments in the length of the curve then p 0 pp 0 J g 0 pg 0 Gk splines with different connection matrices do not form a common vector space For example if p1 and g1 join Gk via p1 at p10 and p2 and gzjoin Gk via p2 at p20 p10 then in general there does not exist a reparametrization p so that l o p1 02 joint Gk with l o g1 ogz at p20 p10 That is in terms of connection matrix there does not generally exit a connection matrix A such that A11 039jkg1 A2039jkgz A1 039jkg1 O39jkgz 11 amp lil g g22 Pi Pz2 g1 1 P2 13 1 l 3 Figure 2 The average bold lines of two curves whose pieces p and g Jom G can be tangent discontinuous ie its pieces do not evenJom G In particular the average of two curves that join Gk is not necessary Gk as illustrated in Figure 2 1 0 1 1 0 1 0 1 1 JP1 5A1 Jg1 suchthat fg1AJp1 0 1 0 3 0 13 0 1 11g2 le2 0 I 0 1 0 2 3 but While and there does not exist a G 2 0 1 2 0 1 3 0 connection matrix A such that jl M Ajl Dp 2 2 0 32 4


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