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In this paper, we propose a new approach to the problem of degree reduction of Bézier curves based on the given endpoint constraints. A differential term is added for the purpose of controlling the smoothness to a certain extent. Considering the adjustment of second derivative in curve design, a modified objective function including two parts is constructed here. One part is a kind of measure of the distance between original high order Bézier curve and degree-reduced curve. The other part represents the second derivative of degree-reduced curve. We tackle two kinds of conditions which are position vector constraint and tangent vector constraint respectively. The explicit representations of unknown points are presented. Some examples are illustrated to show the influence of the differential terms to approximation and smoothness effect.

Degree reduction of Bézier curves is an important and classical problem in CAGD (Computer Aided Geometric Design). It is to approximate the given curve with a Bézier curve of a lower degree while the approximation error is minimized. Degree reduction of curves is needed for the convenience of data exchange and transmission. It is frequently used in data compression as well. Besides, it is also useful for computing roots of polynomials [

Many researches dealing with this problem have been done in recent years. These researches can be classified by norm which the distance between polynomials is measured in, such as _{1}-norm [

The modification of conventional optimal function has also become a research hotspot recently. Lu [_{2}-norm, and the modification optimal approximation is obtained by minimizing the objective function based on the L_{2}-error between the two curves. For solving the similar degree reduction problem, Przemysław [

In this paper, we add a differential item which is the second derivative of the degree-reduced curve based on the conventional optimal function. It is well known that second derivative of a curve plays a leading role in curvature decision. In other words, second derivative can reflect curvature to a great extent. What is more, curvature and smoothness are closely linked and a sudden change of curvature may influence the smoothness of the curve. So the smoothness can be controlled to a certain extent by using the additional term. The distance part and the differential part are combined with a weight here. L_{2}-norm is taken to measure the distance between the degree-reduced curve and the given one. It turns to be the conventional optimal function when ω = 0 and ω = 1 represents square of the norm of second derivative of the degree reduced curve.

In this paper,

where

The integral of a Bernstein polynomial is

and the derivative is

In this paper, we define a

The matrix

The forward differential operator

where

The problem we are considering is to find a Bézier curve of degree m

which is the multi-degree reduction curve of

In previous researches, least square error which represents the distance between the given Bézier curve and degree-reduced curve is always taken as an optimal function [

Besides, there are some continuity constraint conditions attend points. [

There are however some other factors needed to consider in curve design, such as smoothness or energy. Note that in this paper, as second derivative has decisive effect on curvature, we add a differential term which is the second derivative of degree-reduced curve based on the conventional optimal function mentioned above. So the smoothness or energy can be controlled to a certain extent by using the differential term. What is more, for example, the process of adjusting this term becomes useful when global or local acceleration in movement needs to be lowered so that the speed variation can be more uniform in certain part.

For the convenience of adjusting the smoothness degree of degree-reduced curve, we denote an objective function

with a weight

To solve the unknown control points of

It is obvious that Equation (8) turns to the conventional optimal problem Equation (6) when

In this section, the explicit matrix expressions of unknown points are given. Two kinds of parametric continuity are taken into consideration. For

Firstly, we simplify the derivative part of the modified objective function in the following by using Equation (2) and Equation (3),

where

and

For the second part of the modified optimal function, notice that

where

where the four matrixs

So for the modified optimal function

For constraint of position at endpoints to curves

which means the points

So there are

where

It is obvious that

where

In this section, for constraint of tangent vector at endpoints to curves

and we get

Then the vector

where

Same as 3.1, the positions of remaining

We first consider a given planar Bézier curve of degree eleven (

(8). In

6 | 0 | 61.04 | 18.97 | 118.58 | 21.40 | 84.89 | 17.57 |

6 | 0.001 | 61.04 | 18.97 | 34.03 | 13.40 | 85.04 | 17.57 |

5 | 0.001 | 61.04 | 18.97 | 44.15 | 13.32 | 66.71 | 19.51 |

6 | 0.02 | 61.04 | 18.97 | 57.89 | 15.05 | 88.00 | 17.68 |

5 | 0.02 | 61.04 | 18.97 | 8.16 | 2.82 | 67.68 | 19.76 |

The maximum of second derivative of a curve can represents its smooth degree to some extent while the mean can reflect its energy. We can see that curves in

The following example is a Bézier curve of degree nineteen (

We consider a closed curve with degree of thirteen (

In this paper, we introduce a new approach to generate Bézier curve with an additional smoothing term based on the conventional objective function. Sometimes when the smoothness of a curve needs to be adjusted globally or locally within a smaller range, adding this term becomes required. Therefore, our objective function includes two parts: a conventional approximation error part and a smoothing part. They are organized with a weight . The explicit representations of unknown points with conditions of two kinds of endpoint constraints are presented respectively. It is obvious from the examples that the shape of degree-reduced curve depends on three elements, namely endpoint condition, smoothing term and approximation error. Under the premise of guaranteeing endpoint condition, as the names imply, smoothing term determines the smoothness and flatness of curve while the approximation error is the main factor of proximity between degree-reduced curve and the given one. In addition, the smoothing effects are different under different endpoint constraint conditions.

This work is supported by the National Natural Science Foundation of China (No. 11271376).

Xuli Han,Jing Yang, (2016) Multi-Degree Reduction of Bézier Curves with Distance and Energy Optimization. Journal of Applied Mathematics and Physics,04,8-15. doi: 10.4236/jamp.2016.41002