Survey of Continuities of Curves and Surfaces
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paper_title: Geometric continuity of parametric curves: constructions of geometrically continuous splines

paper_content:

Some observations are made concerning the source and nature of shape parameters. It is then described how Bezier curve segments can be stitched together with G/sup 1/ or G/sup 2/ continuity, using geometric constructions. These constructions lead to the development of geometric constructions for quadratic G/sup 1/ and cubic G/sup 2/ Beta-splines. A geometrically continuous subclass of Catmull-Rom splines based on geometric continuity and possessing shape parameters is discussed. >
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paper_title: Geometric continuity of parametric curves: three equivalent characterizations

paper_content:

Some of the important basic results on geometric continuity of curves are presented in a self-contained manner. The paper covers parametric representation and smoothness, parametric continuity, reparameterization and equivalent parameterization, beta-constraints, and arc-length parameterization. >
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paper_title: Scattered Data Interpolation in Three or More Variables

paper_content:

This is a survey of techniques for the interpolation of scattered data in three or more independent variables. It covers schemes that can be used for any number of variables as well as schemes specifically designed for three variables. Emphasis is on breadth rather than depth, but there are explicit illustrations of different techniques used in the solution of multivariate interpolation problems.
---
paper_title: Scattered Data Interpolation in Three or More Variables

paper_content:

This is a survey of techniques for the interpolation of scattered data in three or more independent variables. It covers schemes that can be used for any number of variables as well as schemes specifically designed for three variables. Emphasis is on breadth rather than depth, but there are explicit illustrations of different techniques used in the solution of multivariate interpolation problems.
---
paper_title: Computer Aided Geometric Design.

paper_content:

Abstract : The purpose of this contract was to develop and apply subdivision techniques to Computer Aided Geometric Design, using the conceptual key based on the theory of discrete splines in the Oslo Algorithm computational engine. The basic validity of the theory and algorithms were established by the development of new modeling, graphical, and interactive schemes based on them and also the use of many of these ideas by other researchers and the use of the published results by industry. Related new theory and algorithms were developed; the mathematical model was extended. New styles of modeling were proposed and theory and algorithms for graphics and modeling of nonrectangular surface generalizations were developed. The role of graphics in design and modeling as a tool has been considered.
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paper_title: Algebraic aspects of geometric continuity

paper_content:

Let C ( M , τ) denote the set of all scalar valued functions with connection matrix M at τ. We show that C ( M , τ) is closed under multiplication and division if and only if M is a reparametrization matrix. We conclude that reparametrization is the most general form of geometric continuity for which the shape parameters remain invariant under lifting and projection. We go on to show that Frenet frame continuity is also invariant under projection, even though the shape parameters are not preserved. We also investigate curves which are not smooth, but which become smooth under projection.
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paper_title: Curvature continuous curves and surfaces

paper_content:

Abstract The well-known construction of the Bezier points of a cubic spline curve or surface is generalized to curvature continuous curves and surfaces. Special examples of this kind of new splines are Nu-splines and Beta-splines.
---
paper_title: Scattered Data Interpolation in Three or More Variables

paper_content:

This is a survey of techniques for the interpolation of scattered data in three or more independent variables. It covers schemes that can be used for any number of variables as well as schemes specifically designed for three variables. Emphasis is on breadth rather than depth, but there are explicit illustrations of different techniques used in the solution of multivariate interpolation problems.
---
paper_title: Explicit continuity conditions for adjacent Bézier surface patches

paper_content:

Abstract Liu and Hoschek recently gave necessary and sufficient conditions for G 1 -continuity of two adjacent Bezier surface patches. In this paper, not only the common edge but also the position of the tangent planes at its points are considered to be given. Then one obtains by algebraic methods explicit representations of the first order cross-boundary tangent vectors of the two patches. These methods can be extended to the G 2 case. In combination with new results of J. Hahn, a complete and explicit solution is derived for G 2 -continuity too.
---
paper_title: Scattered Data Interpolation in Three or More Variables

paper_content:

This is a survey of techniques for the interpolation of scattered data in three or more independent variables. It covers schemes that can be used for any number of variables as well as schemes specifically designed for three variables. Emphasis is on breadth rather than depth, but there are explicit illustrations of different techniques used in the solution of multivariate interpolation problems.
---
paper_title: Computer Aided Geometric Design.

paper_content:

Abstract : The purpose of this contract was to develop and apply subdivision techniques to Computer Aided Geometric Design, using the conceptual key based on the theory of discrete splines in the Oslo Algorithm computational engine. The basic validity of the theory and algorithms were established by the development of new modeling, graphical, and interactive schemes based on them and also the use of many of these ideas by other researchers and the use of the published results by industry. Related new theory and algorithms were developed; the mathematical model was extended. New styles of modeling were proposed and theory and algorithms for graphics and modeling of nonrectangular surface generalizations were developed. The role of graphics in design and modeling as a tool has been considered.
---
paper_title: Computer Aided Geometric Design.

paper_content:

Abstract : The purpose of this contract was to develop and apply subdivision techniques to Computer Aided Geometric Design, using the conceptual key based on the theory of discrete splines in the Oslo Algorithm computational engine. The basic validity of the theory and algorithms were established by the development of new modeling, graphical, and interactive schemes based on them and also the use of many of these ideas by other researchers and the use of the published results by industry. Related new theory and algorithms were developed; the mathematical model was extended. New styles of modeling were proposed and theory and algorithms for graphics and modeling of nonrectangular surface generalizations were developed. The role of graphics in design and modeling as a tool has been considered.
---
paper_title: Geometric Modeling Algorithms And New Trends

paper_content:

Thank you very much for downloading geometric modeling algorithms and new trends. Maybe you have knowledge that, people have look numerous times for their favorite readings like this geometric modeling algorithms and new trends, but end up in malicious downloads. Rather than reading a good book with a cup of coffee in the afternoon, instead they juggled with some infectious virus inside their computer.
---
paper_title: Scattered Data Interpolation in Three or More Variables

paper_content:

This is a survey of techniques for the interpolation of scattered data in three or more independent variables. It covers schemes that can be used for any number of variables as well as schemes specifically designed for three variables. Emphasis is on breadth rather than depth, but there are explicit illustrations of different techniques used in the solution of multivariate interpolation problems.
---

Title: Survey of Continuities of Curves and Surfaces

Section 1: Introduction
Description 1: Provide an overview of the importance of continuity in curves and surfaces, and state the purpose and structure of the paper.

Section 2: Continuity of Curves
Description 2: Discuss the fundamental concepts and different types of continuity (parametric, geometric, Frenet frame, tangent surface) in the context of curves.

Section 3: Continuity of Surfaces
Description 3: Explore the various types of continuity as applied to surfaces, including parametric and geometric continuity.

Section 4: Modeling
Description 4: Explain how continuity concepts apply to curve and surface modeling, including specific types of splines and patches.

Section 5: Visual Aspects of Continuity
Description 5: Relate the notions of continuity to visual effects in rendering, illumination models, and shading algorithms.

Section 6: Conclusions
Description 6: Summarize the key points of the survey, highlighting the different notions of continuity, their implications for modeling, and their relationship to rendering and perception.