Interactive design of constrained variational curves

A constrained variational curve is a curve that minimizes some energy functional under certain interpolation constraints. Modeling curves using constrained variational principles is attractive, because the designer is not bothered with the precise representation of the curve (e.g. control points). Until now, the modeling of variational curves is mainly done by means of constraints. If such a curve of least energy is deformed locally (e.g. by moving its control points) the concept of energy minimization is lost. In this paper we introduce deform operators with built-in energy terms. We have tested our ideas in a prototype system for modeling uniform B-spline curves

Constructing G2 Continuous Curve on Freeform Surface with Normal Projection

AbstractThis article presents a new method for G2 continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with prescribed tangent direction and curvature vector, respectively, at every point. First, a G2 continuous curve is constructed in three-dimensional space. Then the curve is projected normally onto the given surface. The desired interpolation curve is just the projection curve, which can be obtained by numerically solving the initial- value problems for a system of first-order ordinary differential equations in the parametric domain for parametric case or in three-dimensional space for implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification so that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface. Numerical experiments demonstrate that it is effective and potentially useful in numerical control (NC) machining, path planning for robotic fibre placement, patterns design on surface and other industrial and research fields

Constructing G2 Continuous Curve on Freeform Surface with Normal Projection

AbstractThis article presents a new method for G2 continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with prescribed tangent direction and curvature vector, respectively, at every point. First, a G2 continuous curve is constructed in three-dimensional space. Then the curve is projected normally onto the given surface. The desired interpolation curve is just the projection curve, which can be obtained by numerically solving the initial- value problems for a system of first-order ordinary differential equations in the parametric domain for parametric case or in three-dimensional space for implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification so that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface. Numerical experiments demonstrate that it is effective and potentially useful in numerical control (NC) machining, path planning for robotic fibre placement, patterns design on surface and other industrial and research fields

Simboličko spajanje B-splajn krivulja

We present an algorithm for stitching B-spline curves, which is different from the generally used least square method. Our aim is to find a symbolic solution for unifying the control polygons of arcs separately described as 4th degree B-spline curves. We show the effect of interpolation conditions and fairing functions as well.Predstavljamo algoritam za spajanje B-splajn krivulja, koji se razlikuje od općenito upotrebljavane metode najmanjih kvadrata. Naš cilj je naći simboličko rješenje za ujedinjavanje kontrolnih poligona lukova koji se svaki zasebno opisuju kao B-splajn krivulje 4. stupnja. Također pokazujemo utjecaj uvjeta interpolacije i postizanja glatkih funkcija

Geometric modeling with rational B-spline surfaces

Issued as Final report, Project no. E-25-67