Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given precision and preserve the geometric features of the given curve such as the to...

In this paper, we present the exact Hausdorff distance between the offset curve of quadratic Bézier curve and its quadratic GC1 approximation. To illustrate the formula for the Hausdorff distance, we give an example of the quadratic GC1 approximation of the offset curve of a quadratic Bézier curve. 1. Preliminaries Quadratic Bézier curves and their offset curves are widely used in CAD/CAM or Co...

We reprove the result of Stone and DeRose, which gives the geometric classification of the affine type of an untrimmed Bézier curve, using classical algebraic geometry. We show how to derive the characterization of Stone and DeRose from three classical results: Bézout theorem, polynomial parametrization criterion and classification of the singularity type of an algebraic curve given in Weierstr...

. Parametric curves are represented using these modified Bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve. We get Bézier curve defined on [0, 1] when we set the parameter α, β to the value 0. We also present a de Casteljau algorithm to compute Bernstein Bézier curves and surfaces with shifted knots. The new curves have some properti...

Analogous to the cubic Bézier curve, a quartic trigonometric Bézier curve with a shape parameter is presented in this paper. Each curve segment is generated by four consecutive control points. The shape of the curve can be adjusted by altering the values of shape parameters while the control polygon is kept unchanged. These curves are closer to the control polygon than the cubic Bézier curves, ...

In this paper we present an efficient technique for piecewise cubic Bézier approximation of digitized curve. An adaptive breakpoint detection method divides a digital curve into a number of segments and each segment is approximated by a cubic Bézier curve so that the approximation error is minimized. Initial approximated Bézier control points for each of the segments are obtained by interpolati...

This paper presents an algorithm for optimal multi-degree reduction of rational disk Bézier curve in the L norm. We start by introducing a novel disk rational Bézier based on parallel projection, whose properties are also discussed. Then we transform multi-degree reduction of the error radius curve and the weight curve of the disk rational Bézier curve into solving a constrained quadratic progr...

A geometric interpretation of the diagonal of a tensor-product trivariate Bézier volume using degree elevation of Bézier triangles is given. A brief discussion of the diagonal curve of a tensor-product bivariate Bézier surface will help to motivate our geometric interpretation for the diagonal curve of a Bézier volume. Suppose we have a tensor-product Bézier surface, x(u, v), given by x(u, v) =...

1. An image’s outline cannot be fitted by a single cubic Bézier curve piece if it contains corners. Corners are points In this paper, a new curve-fitting algorithm is presented. This algorithm can automatically fit a set of data points with at which the outline’s directions take a sharp turn, or the piecewise geometrically continuous (G) cubic Bézier curves. outlines have discontinuous tangent ...

This paper presents an accurate and efficient method to solve both point projection and point inversion for NURBS curves and surfaces. We first subdivide the NURBS curve or surface into a set of Bézier subcurves or patches. Based on the relationship between the test point and the control polygon of Bézier curve or the control point net of the Bézier patch, we extract candidate Bézier subcurves ...