When a Bézier curve is used to describe complex shapes

the problem of joining curve segments smoothly has to be solved. To maintain the continuity of the whole curve

adjacent curve segments must meet strict continuity conditions. A higher the requirement for continuity usually causes conditions to become more complex and involves a larger number of control points. This study improves the modifiable Bézier curve in the literature to achieve smooth connection between curves automatically and to construct a piecewise composite curve with numerous merits. We first present a sufficient condition of continuity for two curves with continuous position.On the basis of the sufficient condition

we prove that the modifiable Bézier curve can achieve a smooth connection under conditions that usually guarantee continuity only for the usual Bézier curve and most Bézier-like curves in the literature. We then use a transition matrix to convert the modifiable Bézier basis to a new set of basis functions. We employ this set of basis functions to define a new kind of curve according to the definition mode of the standard Bézier curve. We then analyze the smooth connection conditions of the new curve.Considering theses smooth connection conditions and by using a special definition mode

we construct a kind of piecewise composite curve. The connection of the control points between adjacent curve segmentsis apparently similar to that of the classical B-spline curve. However

the connections are actually different. The B-spline curve only has one edge between the control polygons of two adjacent curve segments

that is

only one control point is different. Nevertheless

the composite curve defined in this study only has one edge. Thus

only two control points are the same. The new curve defined by the new basis function has relatively simple and special smooth connection conditions. Two neighboring curves can be smoothly joined automatically as long as the last control edge of the former coincides with the first control edge of the latter. Furthermore

the degree of smoothness at the meeting point can be freely adjusted by simply changing the value of the parameter. The piecewise composite curve of the new curve possesses numerous desirable properties

such as geometric invariance

symmetry

automatic smoothness property

and local control capability similar to that of the classical B-spline curve. However

the main difference between the newly defined piecewise curve and the standard B-spline curve is that in the new composite curve

each segment can be defined by different numbers of control points. By contrast

the number of segments must be equal forthe usual B-spline curve.This difference is the main reason that the new piecewise curve has a stronger local control capability than the B-spline curve. Furthermore

a suitable parameter can enable the new composite curve to achieve the expected smoothness at each connection point. In addition

changing one control point can alter the shape of two curve segments. When the parameter of one curve segment is changed

only the shape of the current curve and the degree of smoothness at two connection points will change. This study presents the general Method to construct curves. This Method can easily achieve a smooth connection. We further present the general Method to construct composite curves. This Method can achieve a smooth connection automatically.