Miklós Hoffmann

Constrained shape modification of cubic B-spline curves by means of knots

Abstract The effect of the modification of knot values on the shape of B-spline curves is examined in this paper. The modification of a knot of a B-spline curve of order k generates a one-parameter family of curves.This family has an envelope which is also a B-spline curve with the same control polygon and of order k−1. Applying this theoretical result, three shape control methods are provided for cubic B-spline curves, that are based on the modification of three consecutive knots. The proposed methods enable local shape modifications subject to position and/or tangent constraints that can be specified within well defined limits.

Shape control of cubic B-spline and NURBS curves by knot modifications

Presents shape control methods for cubic B-spline and NURBS curves by the modification of their knot values and by the simultaneous modification of weights and knots. Theoretical aspects of knot modification are also discussed, concerning the paths of points on a curve and the existence of an envelope for the family of curves resulting from a knot modification for curves of degree k.

Modifying a knot of B-spline curves

The modification of a knot of a B-spline curve of order k generates a family of B-spline curves. We show that an envelope of this family is a B-spline curve defined by the same control polygon, and its order is k - m, where m is the multiplicity of the modified knot. Moreover, their arbitrary order derivatives differ only in a multiplier.

On interpolation by spline curves with shape parameters

Interpolation of a sequence of points by spline curves generally requires the solution of a large system of equations. In this paper we provide a method which requires only local computation instead of a global system of equations and works for a large class of curves. This is a generalization of a method which previously developed for Bspline, NURBS and trigonometric CB-spline curves. Moreover, instead of numerical shape parameters we provide intuitive, user-friendly, control point based modification of the interpolating curve and the possibility of optimization as well.

A cyclic basis for closed curve and surface modeling

We define a cyclic basis for the vectorspace of truncated Fourier series. The basis has several nice properties, such as positivity, summing to 1, that are often required in computer aided design, and that are used by designers in order to control curves by manipulating control points. Our curves have cyclic symmetry, i.e. the control points can be cyclically arranged and the curve does not change when the control points are cyclically permuted. We provide an explicit formula for the elevation of the degree from n to n+r, r>=1 and prove that the control polygon of the degree elevated curve converges to the curve itself if r tends to infinity. Variation diminishing property of the curve is also verified. The proposed basis functions are suitable for the description of closed curves and surfaces with C^~ continuity at all of their points.

Paths of C-Bézier and C-B-spline curves

C-Bezier and C-B-spline curves, as the trigonometric extensions of cubic uniform spline curves are well-known in geometric modeling. These curves depend on a shape parameter α ∈ (0, π] about what the only fact we know is that α → 0 yields the cubic polynomial curves. The geometric effect of the alteration of this parameter is discussed in this paper by the help of relative parametrization and linear approximation.

Skinning of circles and spheres

Skinning of an ordered set of discrete circles is discussed in this paper. By skinning we mean the geometric construction of two G^1 continuous curves touching each of the circles at a point, separately. After precisely defining the admissible configuration of initial circles and the desired geometric properties of the skin, we construct the touching points and tangents of the skin by applying classical geometric methods, like cyclography and the ancient problem of Apollonius, finding touching circles of three given circles. Comparing the proposed method to a recent technique (Slabaugh et al., 2008, 2010), larger class of admissible data set and fast computation are the main advantages. Spatial extension of the problem for skinning of spheres by a surface is also discussed in detail.

Isoptics of Bézier curves

Given a planar curve s(t), the locus of those points from which the curve can be seen under a fixed angle is called isoptic curve of s(t). Isoptics are well-known and widely studied, especially for some classical curves such as e.g. conics (Loria, 1911). They can theoretically be computed for a large class of parametric curves by the help of their support functions or by direct computation based on the definition, but unfortunately these computations are extremely complicated even for simple curves. Our purpose is to describe the isoptics of those curves which are still frequently used in geometric modeling - the Bezier curves. It turns out that for low degree Bezier curves the direct computation is possible, but already for degree 4 or 5 the formulas are getting too complicated even for computer algebra systems. Thus we provide a new way to solve the problem, proving some geometric relations of the curve and their isoptics, and computing the isoptics as the envelope of envelopes of families of isoptic circles over the chords of the curve.

KSpheres - an efficient algorithm for joining skinning surfaces

Besides classical point based surface design, sphere based creation of characters and other surfaces has been introduced by some of the recently developed modeling tools in computer graphics. ZSpheres? by Pixologic, or Spore? by Electronic Arts are just two prominent examples of these softwares. In this paper we introduce a new sphere based modeling tool, which allows us to create smooth, tubular-like surfaces by skinning a user-defined set of spheres. The main advantage of the new method is to provide a parametric surface with more natural and smoother shape, especially at the connection of branches than the surfaces provided by the existing softwares and methods.

Constrained Shape Control of Bicubic B-Spline Surfaces by Knots

Constrained modification of B-spline surfaces is essential in geometric design. Modification tools by purely knot alteration are presented in this paper: moving an isoparametric line to a specified location and dragging a point of the surface to a predefined point. Both methods yield smoother change than the well-known control point repositioning, while in the first case symmetric bodies can be deformed in a constrained way by automatically preserving their symmetry. Since the modified surface remains in the original convex hull, these techniques can be useful tools in the fine tuning phase, when users do not want to change the overall shape of the body