## 氏名: 管 輝 (d05756)

## 論文題目: The Design of Local Convex Curves by Cubic B-splines

## 論文概要

Interactive design of smooth curves with arbitrary shapes is useful in many
application areas, such as computer graphics, computer-aided geometric design, and
cartography. A natural approach is to decompose complex curve structures into
several convex parts and then approximate each part smoothly. Decomposing the
curve is generally the easier task. Thus the major challenge in this approach is how to
efficiently produce convex approximation curves on the assumption that some level
of continuity is maintained at all the joints of the curves.
Polynomial B-splines are usually used as approximation curves ininteractive curve
design, because of the simplicity of their representation and their ease of
manipulation. The prevalent method of designing such a curve is by direct
manipulations of a user over a set of control vertices defining a B-spline
approximation curve. This process, however, is often tedious and may not be able to
achieve the desired shape in a practical design situation, since it does not provide
direct control over geometric properties at certain points of the curve. Instead of
interactively adjusting the control vertices, an alternative is to interpolate some data
points on the intended curve. The problems with this approach are that it may
develop unwanted wiggles or undulations, and that the expensive cost of
computations required to solve a large system of equations does
not enable real-time
interactive curve design.

In this research, we present a simple yet efficient approach for B-spline-based
geometric design of smooth convex curves. Our approach adopts a subdivision
strategy: an intended curve is repeatedly split until it can finally be approximated by
a set of acceptable B-spline curve segments. To ensure the continuity of adjacent
curve segments, it uses first derivatives as end-conditions, since a curve with
continuous unit tangent vectors shows satisfactory visual smoothness for most
interactive graphics applications. Unlike existing interpolation
techniques, the new
method avoids solving a large system of equations to generate a B-spline
approximation curve. The basic idea is to construct
a convex curve, starting from a selected point and using information about
the behavior of end-points, based on the convexity-preserving property of
B-spline curves. Theoretical analysis indicates that all curves generated
by the method have the following features:
(1) geometric continuity, (2) convexity, and (3) interpolation
of the given points on the original curve. In addition, preliminary
experiments with the method show that it is both accurate enough and fast
enough to be used for interactive curve design. Currently, in order to
apply the present results in the industrial design, we are developing a
graphic editor.

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