Objective

2

The parameter of a conventional C-Bézier curve is often limited in a closed interval. In this study

we focus on an integral one made up of the traditional inner segment in a finite closed interval (such as[0

$$α$$

]) and a part out of the interval. However

in computer-aided design

the modeling of an integral curve is often expressed as different stages and results in redundant data. In fact

when modeling an entire curve

the control points of different segments may have relations with one another. Therefore

if the control points of one segment and some shape parameters are stored

then the entire curve may be obtained

and the curve data may be decreased. We need to determine the relations among different segments and judge whether they are on the same integral curve to decrease the final redundancy. We raise two questions:1) Given an inner curve

can any segment of its integral curve be presented as an inner form? and 2) Are two neighboring inner C-Bézier curves on the same integral curve? We select a C-Bézier curve for our research. The focus of this study is to consider the changes in control vertices for the C-Bézier curve when the original parametric region[0

$$α$$

] is scaled on (-∞

+∞).

Method

2

Any C-Bézier curve is divided into two arcs from geometric point of view:a center Bézier curve and a trigonometric part. On the basis of their movements

any segment of an integral C-Bézier curve can be represented as an inner form. We can analyze relations of control vertices from algebra perspective and give three forms (direct

subdivision

and linear interpolation forms) between newly produced control points in the movement and old ones. First

we represent certain segments of the integral C-Bézier curve as an inner form

consider basis functions recursively

and compare them to obtain the direct form of original control points. Second

one endpoint of the moving segment is considered

which relates to a subdivision scheme. The scheme subdivides the inner curve into two neighboring C-Bézier segments. Similar to the direct form

expressions can be easily worked out by using recursive evaluation. Third

we consider a corner-cutting form under special subdivision situation to identify linear interpolation from easy to difficult. The corner cutting is an alternative of the direct form

and the corner-cutting form can be obtained by the knot-inserting process. However

the NUAT-B-spline generated after interpolation cannot adapt to the integral case because it is piecewise and is zero out of the interval. We use the area between a corner-cutting scheme and t-axis to extend the scheme. Subdivision with a corner-cutting form is obtained on the basis of recursion and the relations between a basis and the subdivision scheme. The linear interpolation form is considered to move along an integral C-Bézier curve for general case. The length of the parameter interval of an inner C-Bézier curve needs to be less than π; thus

the corner-cutting form under special subdivision situation cannot be directly extended. Although the parameter interval of the C-Bézier curve changes

the position of each point on the integral curve never changes. We utilize an evaluation scheme to solve the extension problem. Consistent with Bézier curve

results of the movement along an integral C-Bézier curve with a linear interpolation form are obtained by using recursive evaluation twice. Finally

we establish an algorithm to judge if two given inner C-Bézier curves are on the same integral curve by considering that integral curve can be used to reduce redundant data. The error of the C-Bézier curve can be limited by the error of its control points; hence

we use an error item to control the judgment accuracy after calculating control points by direct form.

Result

2

This study focuses on C-Bézier curves and regards the traditional inner part and the extended part out of an interval as integrals. An inner C-Bézier curve can be moved along the integral curve while its parameter integral length is less than

$${\rm{ \mathsf{ π} }}$$

and motion curves can be represented as an inner C-Bézier form when its parameter interval length is in (0

$${\rm{ \mathsf{ π} }}$$

). New control points can be obtained by a direct linear combination or stepwise linear interpolation (including traditional interpolation and extrapolation) form of the old ones. A subdivision scheme

including direct and corner-cutting forms

of the inner C-Bézier curve is included as a subcase. The integral curve and the movement along it may be considered to reduce redundant storage data.

Conclusion

2

The applications are as follows:First

the movement along an integral C-Bézier can be used to scale the parameter interval of a given C-Bézier curve. Second

integral curve can be considered to reduce redundancy by focusing on the part and extending the parameter interval. Third

two neighboring C-Bézier curves are judged on whether they are on the same integral curve under permissible error. If they are on the same curve

then data of one curve may be reduced while storing

whereas data of the other one can be saved. However

the movement process is slow because of the recursive integral definition of the C-Bézier basis. In the future

we may consider the acceleration of the movement method or other types of Bézier-like curves.