An Introduction to Differential Geometry. Luther Pfahler Eisenhart

The purpose of the study is to show how the features of geocartographic way of thinking are manifested in the meta-theory of knowledge based on mathematical formalisms. General cartographic concepts and regularities are considered in the view of metatheoretic analysis using cognitive procedures of fiber bundle from differential geometry. On levels of metainformation generalization, the geocartographic metatheoretic approach to the study of reality is higher than the system-theoretical one. It regulates the type of equations, models, and methods of each intertheory expressed in its own system terms. There is a balance between the state of any system and its geographical environment; therefore the observed phenomena are only explained theoretically in a metatheoretic projection on the corresponding system-thematic layer of the knowledge map. Metatheoretic research enables passing from the systematization of already known patterns to the formation of new knowledge through the scientific stratification of reality. General methods of metatheoretic analysis are mathematically distinguished

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Computation of Self-Intersections of Offsets of Be´zier Surface Patches

Journal of Mechanical Design ◽

10.1115/1.2826247 ◽

1997 ◽

Vol 119 (2) ◽

pp. 275-283 ◽

Cited By ~ 23

Author(s):

Takashi Maekawa ◽

Wonjoon Cho ◽

Nicholas M. Patrikalakis

Keyword(s):

Differential Geometry ◽

Distance Function ◽

Polynomial Equations ◽

Intersection Curve ◽

Parameter Domain ◽

Surface Patches ◽

Intersection Curves

Self-intersection of offsets of regular Be´zier surface patches due to local differential geometry and global distance function properties is investigated. The problem of computing starting points for tracing self-intersection curves of offsets is formulated in terms of a system of nonlinear polynomial equations and solved robustly by the interval projected polyhedron algorithm. Trivial solutions are excluded by evaluating the normal bounding pyramids of the surface subpatches mapped from the parameter boxes computed by the polynomial solver with a coarse tolerance. A technique to detect and trace self-intersection curve loops in the parameter domain is also discussed. The method has been successfully tested in tracing complex self-intersection curves of offsets of Be´zier surface patches. Examples illustrate the principal features and robustness characteristics of the method.

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