In this paper we describe the design of B-spline surface models by means of curves and tangency c... more In this paper we describe the design of B-spline surface models by means of curves and tangency conditions. The intended application is the conceptual constraint-driven design of surfaces from hand-sketched curves. The solving of generalized curve surface constraints means to find the control points of the surface from one or several curves, incident on the surface, and possibly additional tangency and smoothness conditions. This is accomplished by solving large, and generally under-constrained, and badly conditioned linear systems of equations. For this class of linear systems, no unique solution exists and straight forward methods such as Gaussian elimination, QR-decomposition, or even blindly applied Singular Value Decomposition (SVD) will fail. We propose to use regularization approaches, based on the so-called L-curve. The L-curve, which can be seen as a numerical high frequency filter, helps to determine the regularization parameter such that a numerically stable solution is obtained. Additional smoothness conditions are defined for the surface to filter out aliasing artifacts, which are due to the discrete structure of the piece-wise polynomial structure of the B-spline surface. This leads to a constrained optimization problem, which is solved by Modified Truncated SVD: a L-curve based regularization algorithm which takes into account a user defined smoothing constraint.
We discuss the problem of adding features to a free form surface by applying one or several user ... more We discuss the problem of adding features to a free form surface by applying one or several user defined surface curves. The curves are seen as editable parameters, while the surface is to be changed automatically, keeping a predefined set of constraints satisfied, specifically the curve--surface incidence and derivatives. We review and update our approach presented earlier [18] and show, how the curve--surface composition can be expressed as a linear transformation. We describe the so--called "aliasing" effect caused by an incompatibility of a general curve on a surface with the rectangular mesh of degrees of freedom of a tensor product surface. A solution is proposed by locally changing the parametrization of the original surface which takes a domain curve to an iso-- line in either parameter direction. 1 Introduction The relational modeling paradigm is a very powerful method of creating models without an exact a--priori knowledge of all parameters. The user "roughl...
We investigate methods of using constraint--based modeling in a free--form curve and surface envi... more We investigate methods of using constraint--based modeling in a free--form curve and surface enviroment. In this work we concentrate on a problem of maintaining the curve-surface incidence relation while the curve is edited. We formulate the relation between the degrees of freedom (DOF) and parameters (control points of the surface and curve resp.) as an explicit functional prescription. We show how the polynomial composition algorithm based on blossoming and a data structures for efficient storage of intermediate results, significantly speeds up the computation. We also discuss how to insert new DOFs, if the incidence condition cannot be satisfied, and sketch how a common solution space for several constraints of this type can be found. Keywords: Incidence, Constraints, Free--form surfaces and curves, Algorithm, Blossoming, B--spline algebra. 1 Introduction The relational modeling paradigm has achieved much progress in recent years especially in 2--D. Editing ge- Tel. ++49 3677 69...
We investigate methods of using constraint{based modeling in a free{form curve and surface enviro... more We investigate methods of using constraint{based modeling in a free{form curve and surface enviroment. In this work we concentrate on a problem of maintaining the curve-surface incidence relation while the curve is edited. We formulate the relation between the degrees of freedom (DOF) and parameters (control points of the surface and curve resp.) as an explicit functional prescription. We show how the polynomial composition algorithm based on blossoming and a data structures for eecient storage of intermediate results, signii-cantly speeds up the computation. We also discuss how to insert new DOFs, if the incidence condition cannot be satissed, and sketch how a common solution space for several constraints of this type can be found.
We investigate methods of using constraint--basedmodeling in a free--form curve and surface envir... more We investigate methods of using constraint--basedmodeling in a free--form curve and surface enviroment.In this work we concentrate on a problem of maintainingthe curve-surface incidence relation while the curveis edited.We formulate the relation between the degrees offreedom (DOF) and parameters (control points of thesurface and curve resp.) as an explicit functional prescription.We show how the polynomial compositionalgorithm based on blossoming and
We discuss the problem of adding features to a free form surface by applyingone or several user d... more We discuss the problem of adding features to a free form surface by applyingone or several user defined surface curves. The curves are seen as editableparameters, while the surface is to be changed automatically, keeping a predefinedset of constraints satisfied, specifically the curve--surface incidence andderivatives. We review and update our approach presented earlier [18] andshow, how the curve--surface composition can be expressed as a linear transformation.We describe the...
In this paper we describe the design of B-spline surface models by means of curves and tangency c... more In this paper we describe the design of B-spline surface models by means of curves and tangency conditions. The intended application is the conceptual constraint-driven design of surfaces from hand-sketched curves. The solving of generalized curve surface constraints means to find the control points of the surface from one or several curves, incident on the surface, and possibly additional tangency and smoothness conditions. This is accomplished by solving large, and generally under-constrained, and badly conditioned linear systems of equations. For this class of linear systems, no unique solution exists and straight forward methods such as Gaussian elimination, QR-decomposition, or even blindly applied Singular Value Decomposition (SVD) will fail. We propose to use regularization approaches, based on the so-called L-curve. The L-curve, which can be seen as a numerical high frequency filter, helps to determine the regularization parameter such that a numerically stable solution is obtained. Additional smoothness conditions are defined for the surface to filter out aliasing artifacts, which are due to the discrete structure of the piece-wise polynomial structure of the B-spline surface. This leads to a constrained optimization problem, which is solved by Modified Truncated SVD: a L-curve based regularization algorithm which takes into account a user defined smoothing constraint.
We discuss the problem of adding features to a free form surface by applying one or several user ... more We discuss the problem of adding features to a free form surface by applying one or several user defined surface curves. The curves are seen as editable parameters, while the surface is to be changed automatically, keeping a predefined set of constraints satisfied, specifically the curve--surface incidence and derivatives. We review and update our approach presented earlier [18] and show, how the curve--surface composition can be expressed as a linear transformation. We describe the so--called "aliasing" effect caused by an incompatibility of a general curve on a surface with the rectangular mesh of degrees of freedom of a tensor product surface. A solution is proposed by locally changing the parametrization of the original surface which takes a domain curve to an iso-- line in either parameter direction. 1 Introduction The relational modeling paradigm is a very powerful method of creating models without an exact a--priori knowledge of all parameters. The user "roughl...
We investigate methods of using constraint--based modeling in a free--form curve and surface envi... more We investigate methods of using constraint--based modeling in a free--form curve and surface enviroment. In this work we concentrate on a problem of maintaining the curve-surface incidence relation while the curve is edited. We formulate the relation between the degrees of freedom (DOF) and parameters (control points of the surface and curve resp.) as an explicit functional prescription. We show how the polynomial composition algorithm based on blossoming and a data structures for efficient storage of intermediate results, significantly speeds up the computation. We also discuss how to insert new DOFs, if the incidence condition cannot be satisfied, and sketch how a common solution space for several constraints of this type can be found. Keywords: Incidence, Constraints, Free--form surfaces and curves, Algorithm, Blossoming, B--spline algebra. 1 Introduction The relational modeling paradigm has achieved much progress in recent years especially in 2--D. Editing ge- Tel. ++49 3677 69...
We investigate methods of using constraint{based modeling in a free{form curve and surface enviro... more We investigate methods of using constraint{based modeling in a free{form curve and surface enviroment. In this work we concentrate on a problem of maintaining the curve-surface incidence relation while the curve is edited. We formulate the relation between the degrees of freedom (DOF) and parameters (control points of the surface and curve resp.) as an explicit functional prescription. We show how the polynomial composition algorithm based on blossoming and a data structures for eecient storage of intermediate results, signii-cantly speeds up the computation. We also discuss how to insert new DOFs, if the incidence condition cannot be satissed, and sketch how a common solution space for several constraints of this type can be found.
We investigate methods of using constraint--basedmodeling in a free--form curve and surface envir... more We investigate methods of using constraint--basedmodeling in a free--form curve and surface enviroment.In this work we concentrate on a problem of maintainingthe curve-surface incidence relation while the curveis edited.We formulate the relation between the degrees offreedom (DOF) and parameters (control points of thesurface and curve resp.) as an explicit functional prescription.We show how the polynomial compositionalgorithm based on blossoming and
We discuss the problem of adding features to a free form surface by applyingone or several user d... more We discuss the problem of adding features to a free form surface by applyingone or several user defined surface curves. The curves are seen as editableparameters, while the surface is to be changed automatically, keeping a predefinedset of constraints satisfied, specifically the curve--surface incidence andderivatives. We review and update our approach presented earlier [18] andshow, how the curve--surface composition can be expressed as a linear transformation.We describe the...
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Papers by Paul Michalik