Curves and Surfaces

Pfaffian differentials 22-3. 608-11 in Second law of thermodynamics 63-7 three or more variables 67 two variables 64-6 solution curves and surfaces 610-11 in three dimensions 609 in two dimensions 611 phase equilibria criteria for 231-7... 

Some investigations in the geometry of curve and surface elements, Proc. London Math. Soc. 4(3) (1954), 24-49. 

Although the geometry can be expressed entirely mathematically in terms of formulas and relations, to do so may be awkward for those proficient in mathematics and quite incomprehensible for those who are not. It is often the case that curves and surfaces must be specified functionally, parametrically, or as piecewise sections, all of which add burden and potential error in specification. 

See Appendix C for a brief survey of mathematical relations for curves and surfaces. 5See standard references on dislocation mechanics [2, 4, 5],... 

The second approach to treating nondynamical correlation has an air of the ostrich about it ignore the spin symmetry of the wave function and use unrestricted Haxtree-Fock (UHF) theory as the single configuration description [7]. Since the UHF wave function comprises one spin-orbital for each electron, a molecular UHF wave function should dissociate to atomic UHF wave functions, for example. This is certainly not the case for spin-restricted Hartree-Fock (RHF) molecules and atoms in general. And there is an attractive simplicity about UHF — no active orbitals to identify, and so forth. However, where nondynamical correlation would be important in an RHF-based treatment, the UHF method will suffer from severe spin-contamination, while where nondynamical correlation is not important the RHF solution may be lower in energy than any broken-symmetry UHF solution, so potential curves and surfaces may have steps or kinks where the spin symmetry is broken in the UHF treatment. 

Lancaster P, Salkauskas K (1986), Curve and Surface Fitting, Academic Press, London. 

A second generalization of the many-body perturbation theory, which would certainly be useful in the calculation of potential energy curves and surfaces, would be to use a reference function constructed from non-orthogonal orbitals. 

Pade Approximants to Energy Derivatives. The evaluation of derivatives of the energy, with respect to the nuclear geometry, forms the basis of an efficient approach to the calculation of potential energy curves and surfaces. It is easily shown that... 

M.P. de Carmo, Differential Geometry of Curves and Surfaces, 1976, Prentice-Hall, New Jersey. 

The recent availability of working curves and surfaces for a range of conunon mechanisms at a number of electrode geometries (Alden and Compton, 1997a) allows a broad quantitative comparison of the kinetic discrimination of conunon electrode geometries for both first- and second-order homogeneous processes. 

FIGURE 10.6 Polarization curve and surface topography after anodic dissolution in... 

Electron density contour diagrams have been derived " (Figure 2) and potential energy curves and surfaces calculated In addition to this impressive quantity of accurate work on the parent compound a number of simplified schemes were proposed by Forster, Coulson and Moffit " and by Walsh these schemes have been widely employed for the interpretation of the photoelectron spectra and other properties of the derivatives of cyclopropane. 

M. do Carmo, "Differential geometry of curves and surfaces . (1976), Eaglewood Cliffe, N.J. Prentice-Hall Inc. 

A. Gray, "Modem Differential Geometry of Curves and Surfaces". Studies in Advanced Mathematics, ed. S. Krantz. (1993), Boca Raton, FLA. CRC Press. 

M. Berger and B. Gostiaux, Manifolds, Curves and Surfaces". Graduate Texts in Mathematics, Vol. 115. (1985), Berlin Springer Verlag. 

B.T. Sutcliffe, Potential energy curves and surfaces, in Handbook of Molecular Physics and Quantum Chemistry, vol. 1 Fundamentals, ed. S. Wilson, P.F. Bemath and R. McWeeny, chapter 34, Wiley, Chichester, 2003. 

This turns out to be an extremely important result. Being able to construct limit points without actually doing an infinite number of refinements is a key to practical use of subdivision curves and surfaces. 

C.A.Micchelli Subdivision algorithms for curves and surfaces. SIG-GRAPH 1986... 

N. Dyn and D.Levin Interpolating subdivision schemes for the generation of curves and surfaces. pp91-106 in Multivariate Interpolation and Approximation (eds Hausman and Jetter), Birkhauser Verlag, Basel. 1990... 

N.Dyn, D.Levin and I.Yad-Shalom Regularity conditions for a class of geometrically continuous curves and surfaces. ppl69-176 in Curves and Surfaces (ed Laurent, Le-Mehaute and Schumaker), Academic Press 1991 ISBN 0-12-438660-1... 

J.Warren Binary Subdivision Schemes for Functions over Irregular Knot Sequences. pp543-562 in Mathematical Methods for Curves and Surfaces (eds Daehlen, Lyche and Schumaker), Vanderbilt University Press 1995 ISBN 0-8265-1268-2... 

D. P.Kacso and H-J.Wenz On an Almost-Convex-hull property, pp 217-222 in Curves and Surfaces with applications in CAGD (eds Le Mehaute, Rabut and Schumaker), Vanderbilt University Press 1997 ISBN 0-8265-1293-3... 

M. F.Hassan and N.A.Dodgson Ternary and Three-point univariate subdivision schemes, pp 199-208 in Curve and Surface Fitting, (eds Cohen, Merrien and Schumaker), 2003... 

N.Dyn, M.S.Floater, K.Hormann A C2 four-point subdivision scheme with fourth order accuracy and its extensions. ppl45-156 in Mathematical Methods for Curves and Surfaces, (eds Daehlen, Morken and schumaker), Nashboro Press 2005 ISBN 0-9728482-4-X... 

C.Beccari, G.Gasciola and L.Romani A Vector Outline Descriptor using interpolatory subdivision curves. pp31-40 in Curve and Surface Design, (eds Chenin, Lyche and Schumaker), Nashboro Press 2007, ISBN 978-0-9728482-7-5... 

Then a chapter introducing the concepts for the reader who has not encountered subdivision curves and surfaces before and how the rules are described in ways that we can apply mathematical arguments to. 

Eisenhart, L.P. A Treatise on the Differential Geometry of Curves and Surfaces Ginn and Co., 1909 Dover Publications, 1960. 

Nonlinear extensions of PCA have been proposed by using autoasso-ciative neural networks discussed in Section 3.6.1 (an illustrative example is provided in Section 7.7.1) or by using principal curves and surfaces [106, 161]. 

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