具有形状参数的新型广义混合三角贝齐尔样曲线（GHT_Bezier_curves） matlab实现

参考文献：Geometric Modeling of Novel Generalized Hybrid Trigonometric Bézier-Like Curve with Shape Parameters and Its Applications
       
本博客是对上述参考文献的部分程序实现

一、n维 GHT-Bernstein 基函数

1、定义

       论文中给出的$q_{2,2}(\theta )$定义不对，不满足端点性质。需要修改一点。
       下面是我修改的定义，以及写出的三阶公式：

2、性质：

1. 单位分解性。
2. 正性。
3. 对称性。
4. 端点插值性。
   
   

二、基函数的matlab程序实现

1、程序：

function GHT_Bernstein
%  三阶GHT_Bernstein基函数图像
clear;clc;
% Gma = 1;        %Gma = γ
% v = 0.2;
% beta = 0;         %β
Gma = 0.5;                                   %  Gma = γ
beta = 0;                                    %  β
v = -0.5;
x = (0:0.01:1)';
y = F0_3(x,v,Gma);
y1 = F2_3(x,v,beta,Gma);
y2 = F1_3(x,v,beta,Gma);
y3 = F3_3(x,v,Gma);
plot(x,y,'y')
title('三阶GHT-Bernstein基函数')
hold on
plot(x,y1,'b')
hold on
plot(x,y2,'g')
hold on
plot(x,y3,'r')% 二阶GHT_Bernstein基函数图像  （可以直接在命令行使用）
Gma = 0.5;                                   %  Gma = γ
beta = 0;                                    %  β
v = -0.5;x = (0:0.01:1)';
y = F1_2(x,v,beta,Gma);
y1 = F2_2(x,beta,Gma);
y2 = F0_2(x,v,Gma);
figure(2)
plot(x,y,'y')
title('二阶GHT-Bernstein基函数')
hold on
plot(x,y1,'b')
hold on
plot(x,y2,'r')
endfunction g = F0_2(x,v,Gma)
g = (1-sin(0.5*pi*x)).*(1-v*sin(0.5*pi*x)).*exp(Gma*x);
endfunction g = F1_2(x,v,beta,Gma)
g = 1 - F0_2(x,v,Gma) - F2_2(x,beta,Gma);
endfunction g = F2_2(x,beta,Gma)
g = (1-cos(0.5*pi*x)).*(1-beta*cos(0.5*pi*x)).*exp((1-x)*Gma);
endfunction g = F0_3(x,v,Gma)
g = (1-x).*F0_2(x,v,Gma);
endfunction g = F1_3(x,v,beta,Gma)
g = (1-x).*F1_2(x,v,beta,Gma) + x.*F0_2(x,v,Gma);
endfunction g = F2_3(x,v,beta,Gma)
g = (1-x).*F2_2(x,v,Gma)+x.*F1_2(x,v,beta,Gma);
endfunction g = F3_3(x,v,Gma)
g =  x.*F2_2(x,v,Gma);
end

2.2 基函数图像

三、n维GHT-Bézier曲线（GHT-Bézier Curves of Degree n）

相应程序：

function [Px,Py,px,py] = GHT_Bezier_curves(v,beta,Gma)
px=[0,2,3,5];
py=[0,1,0,1];                              %  控制顶点(0,0),(2,1),(3,0),(5,1)x = (0:0.005:1)';
y0 = F0_3(x,v,Gma);
y1 = F1_3(x,v,beta,Gma);
y2 = F2_3(x,v,beta,Gma);
y3 = F3_3(x,v,Gma);Px = y0*px(1) + y1.*px(2) + y2*px(3) + y3.*px(4);
Py = y0*py(1) + y1.*py(2) + y2*py(3) + y3.*py(4);endfunction g = F0_2(x,v,Gma)
g = (1-sin(0.5*pi*x)).*(1-v*sin(0.5*pi*x)).*exp(Gma*x);
endfunction g = F1_2(x,v,beta,Gma)
g = 1 - F0_2(x,v,Gma) - F2_2(x,beta,Gma);
endfunction g = F2_2(x,beta,Gma)
g = (1-cos(0.5*pi*x)).*(1-beta*cos(0.5*pi*x)).*exp((1-x)*Gma);
endfunction g = F0_3(x,v,Gma)
g = (1-x).*F0_2(x,v,Gma);
endfunction g = F1_3(x,v,beta,Gma)
g = (1-x).*F1_2(x,v,beta,Gma) + x.*F0_2(x,v,Gma);
endfunction g = F2_3(x,v,beta,Gma)
g = (1-x).*F2_2(x,v,Gma)+x.*F1_2(x,v,beta,Gma);
endfunction g = F3_3(x,v,Gma)
g =  x.*F2_2(x,v,Gma);
end

将上述代码保存为函数，用下面代码调用实现：

clear;clc;
%调用GHT_Bezier_curves(v,beta,Gma)函数
%  论文Figure2 中的(a)图像的实现
Gma = 1;                                   %  Gma = γ
beta = 0;                                    %  β
v = 0.95;
[Px,Py,px,py] = GHT_Bezier_curves(v,beta,Gma);
figure(1)
plot(Px,Py,'r',px,py,'b',px,py,'mo')
xlabel('x-axis'), ylabel('y-axes');
title('figure2 (a)')
hold on
v = 0.5;
[Px,Py,px,py] = GHT_Bezier_curves(v,beta,Gma);
plot(Px,Py,'y',px,py,'b',px,py,'m*')
hold on
v = -0.1;
[Px,Py,px,py] = GHT_Bezier_curves(v,beta,Gma);
plot(Px,Py,'m',px,py,'b',px,py,'m*')
hold on
v = -0.6;
[Px,Py,px,py] = GHT_Bezier_curves(v,beta,Gma);
plot(Px,Py,'g',px,py,'b',px,py,'m*')
hold on%  论文Figure2 中的(b)图像的实现
Gma = 0.5;                                   %  Gma = γ
beta = 0;                                    %  β
v = -0.5;
[Px,Py,px,py] = GHT_Bezier_curves(v,beta,Gma);
figure(2)
plot(Px,Py,'r',px,py,'b',px,py,'mo')
xlabel('x-axis'), ylabel('y-axes');
title('figure2 (b)')
hold on
v = 0;
[Px,Py,px,py] = GHT_Bezier_curves(v,beta,Gma);
plot(Px,Py,'y',px,py,'b',px,py,'m*')
hold on
v = 0.5;
[Px,Py,px,py] = GHT_Bezier_curves(v,beta,Gma);
plot(Px,Py,'m',px,py,'b',px,py,'m*')
hold on
v = 1;
[Px,Py,px,py] = GHT_Bezier_curves(v,beta,Gma);
plot(Px,Py,'g',px,py,'b',px,py,'m*')
hold on%  论文Figure2 中的(d)图像的实现
Gma = 0;                                   %  Gma = γ
beta = 1;                                    %  β
v = 1;
[Px,Py,px,py] = GHT_Bezier_curves(v,beta,Gma);
figure(4)
plot(Px,Py,'r',px,py,'b',px,py,'mo')
xlabel('x-axis'), ylabel('y-axes');
title('figure2 (d)')
hold on
Gma = 0.5;   
[Px,Py,px,py] = GHT_Bezier_curves(v,beta,Gma);
plot(Px,Py,'y',px,py,'b',px,py,'m*')
hold on
Gma = -0.5;   
[Px,Py,px,py] = GHT_Bezier_curves(v,beta,Gma);
plot(Px,Py,'m',px,py,'b',px,py,'m*')
hold on
Gma = -1;   
[Px,Py,px,py] = GHT_Bezier_curves(v,beta,Gma);
plot(Px,Py,'g',px,py,'b',px,py,'m*')
hold on

相应结果：

注：得到的曲线应该满足凸包性质，但是有几个参数得到的曲线不在控制多边形里面（如b图中红色曲线），程序检查了几遍没查出来原因，因为这篇文章就是借鉴一下想法，就不继续研究下去了。