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1: Adjacent Double Tetrahedra, Functions and Control Points for two Non-Convex Adjacent Faces C 0 Continuity: If two tetrahedra share a common face, we equate the control points of the associated cubic polynomials on the common face(see Lemma 2.2):
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We present a sufficient criterion for the Bernstein-Bezier (BB) form of a trivariate polynomial within a tetrahedron, such that the real zero contour of the polynomial defines a smooth, connected and single sheeted algebraic surface patch. We call this an A-patch. We present algorithms to build a mesh of cubic A-patches to interpolate a given set o...
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... fact, since w 2 p 1 p 2 p 3 ], the coeecients on the same layer are C 1 related. For the 0-th layer (see Figure 5.2), the control points labeled are thus already determined. The control points are determined by a coplanar condition with surrounding . ...
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... the 1st layer (see Figure 5.2), the control points labeled and 2 are similarly determined as the 0-th layer. For the 2nd layer (see Figure 5.2), the control points are arbitrarily chosen and 2 is determined by the coplanar condition. ...
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... the 1st layer (see Figure 5.2), the control points labeled and 2 are similarly determined as the 0-th layer. For the 2nd layer (see Figure 5.2), the control points are arbitrarily chosen and 2 is determined by the coplanar condition. Finally, the 3rd layer coeecient is free. ...
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... assume (without loss of generality) that all the normals point to the same side of the surface triangulation T. That is the side on which p 4 and p 0 4 lie(see Figure 5.1). Under this assumption, it follows from Deenition 4.1 and equation (5.2) that, the control points on the edge, say a i 0210 ; a i 0120 on edge p 2 p 3 ](see Figure 5.1), are nonpositive if the edge is nonnegative convex, and nonnegative if the edge is nonpositive convex. ...
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... assume (without loss of generality) that all the normals point to the same side of the surface triangulation T. That is the side on which p 4 and p 0 4 lie(see Figure 5.1). Under this assumption, it follows from Deenition 4.1 and equation (5.2) that, the control points on the edge, say a i 0210 ; a i 0120 on edge p 2 p 3 ](see Figure 5.1), are nonpositive if the edge is nonnegative convex, and nonnegative if the edge is nonpositive convex. ...
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... the control points around the vertices of T are determined by the normals, the smooth vertex condition is obviously satissed. If the surface contains the edge p 2 p 3 ](see Figure 5.1), then since a i 1110 (or a i 0111 ) is freely chosen, the smooth edge condition is easily satissed(see the proof of Proposition 5.3). ...
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... the surface contains the edge p 2 p 3 ](see Figure 5.1), then since a i 1110 (or a i 0111 ) is freely chosen, the smooth edge condition is easily satissed(see the proof of Proposition 5.3). Referring to Figure 5.1, we prove in the following that the patches constructed over V 1 and W 1 are single sheeted. ...
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... any face of T = p 1 ; p 2 ; p 3 ], if it is non-convex and if the three inner products of the face normal and its three adjacent face normals have diierent signs, then subdivide the double face tetrahedra into 6 subtetrahedra by adding a vertex at the center w of the face (a Clough-Tocher split). The coeecients are speciied as before by regarding w as p 1 (see Figure 5.1). ...
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... Computer Aided Geometric Design focuses on the mathematical representation of complex surface geometries. There is a wide variety of side interpolating multisided free-form surfaces in the literature, including both parametric [4,11,7,17,18] and implicit [1,6,16] patches. They are popular in curvenet-based design, as a patchwork of smoothly connected complex N-sided patches can be automatically created from simple ribbon surfaces. ...
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... In this book, we are not going to go into much detail on the geometry modeling of curves. For more informative journals and standard textbooks of the topic, the reader can find more detail and other aspects in geometric modeling in Bajaj et al. (1995), Bernstein (1912), Faux and Pratt (1981), Mortenson (1985), Hoffmann (1989) and Beach (1991), Lei and Wang (2009), Piegl andTiller (1997, 2003), and Lorentz (1986). ...
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