Objective

2

The Bézier curve is a widely used tool in the representation of parametric curves. At present

most computer aided design (CAD) systems take the Bézier curve as a basic module. When using the Bézier curve for geometric modeling

it is often necessary to link multiple curves to meet the needs of designing complex curves. In order to satisfy the continuity between the linked Bézier curves

the control points that satisfy the corresponding continuity conditions need to be selected in advance. For linked Bézier curves with low order continuity

the control points of the curves can be adjusted to improve the continuity so as to smooth the linked curves. In theory

the continuity between linked Bézier curves can be improved by arbitrarily adjusting the control points of the curves to make the corresponding continuity conditions hold. However

this kind of adjustment of control points without specific objectives is often unable to meet the needs of practical applications. Given that the continuity can be improved by adjusting the control points of the linked Bézier curves

the control points that need to be adjusted can be optimized according to some specific targets

so that the smooth linked curves meet the corresponding requirements. In practical applications

if the control points of the curves are taken from the real objects

it is often hoped that the distance between the new control points and the original control points is as small as possible. Hence

the minimum distance between the new control points and the original control points can be used as the target to optimize the control points that need to be adjusted. In addition

energy minimization has become a common method for constructing curves and surfaces in CAD and related fields. Thus

the minimum energy can be used as a target to optimize the control points that need to be adjusted. In this study

three methods for smoothing the linked Bézier curves from

C

0

to

C

1

and from

C

1

to

C

2

by distance and internal energy minimization are given.

Method

2

First

the problems to be discussed are described

and the control points to be adjusted are pointed out when smoothing the linked Bézier curves from

C

0

to

C

1

and from

C

1

to

C

2

. Two control points need to be adjusted when smoothing the linked Bézier curves from

C

0

to

C

1

or from

C

1

to

C

2

. However

only one of them needs to be optimized due to the relationship between the two control points that need to be adjusted. Then

the distance minimizations for smoothing the linked Bézier curves from

C

0

to

C

1

and from

C

1

to

C

2

are presented. Next

the internal energy minimizations for smoothing the linked Bézier curves from

C

0

to

C

1

and from

C

1

to

C

2

are given. Then

the simultaneous minimization of the distance and the internal energy for smoothing the linked Bézier curves from

C

0

to

C

1

and from

C

1

to

C

2

are provided. The optimal solutions of the control points that need to be adjusted can be easily obtained by solving the corresponding unconstrained optimization problems. Finally

the comparison of the three minimization methods is given

and the applicable occasions of different methods are pointed out. The distance between the new control points and the original control points by the distance minimization is relatively small

which is suitable for the application when the control points are taken from the real object; the internal energy of the linked curves obtained by the internal energy minimization is relatively small

which is suitable for the application where the energy of the curve is required to be as small as possible; the distance and internal energy minimization simultaneously takes into account the distance between the new control points and the original control points and the internal energy of the linked curves

which is suitable for applications where both targets are required. Some numerical examples are presented to illustrate the effectiveness of the proposed methods.

Result

2

By using the distance minimization

the total distance between the new control points and the original control points is obviously smaller than that of the other two methods. When the control points of Bézier curves are all taken from the real object

the distance between the new control points and the original control points should not be too large

so the distance minimization method is more suitable. By using the internal energy minimization

the total internal energy of the curves is obviously smaller than that of the other two methods. When the energy of the curve is required to be as small as possible

the internal energy minimization is more suitable. In addition

given that the stretch energy

strain energy

and curvature variation energy correspond to the arc length

curvature

and curvature variation of the curve

the corresponding minimization method can be selected according to the specific requirements. By using the simultaneous minimization of distance and internal energy

the total distance between the new control points and the original control points and the total internal energy of the curves are between the other two methods. When the distance between the new control points and the original control points should not be too large and the energy of the curve should be as small as possible

the method of simultaneous minimization of distance and internal energy is a better choice.

Conclusion

2

The proposed methods provide three effective means for smoothing the linked Bézier curves and are easy to implement. The linked Bézier curves can be effectively smoothed from

C

0

to

C

1

and from

C

1

to

C

2

by using the proposed methods according to the demand of minimum distance or internal energy. When smoothing other types of link curves

the distance minimization

internal energy minimization

and simultaneous minimization of distance and internal energy proposed in this study maybe useful.