B Spline（B样条曲线）

文章目录

说明
B样条曲线
代码

说明

在阅读此博客前，请访问2018级山东大学计算机学院图形学实验汇总。

原笔记通过latex编写，csdn只支持latex部分功能，所以下面主要是将pdf截屏上传。部分内容参考课本。

B样条曲线

前面讲解了Bézier曲线，这个曲线有很多用处，但是也有不少缺点：

一旦确定了特征多边形的顶点数(n+1个)，也就决定了曲线的阶次(n次)
Bézier曲线或曲面的拼接比较复杂
Bezier曲线或曲面不能作局部修改，移动一个控制顶点，整个曲线都会变化（因为每个Bernstein多项式在整个[0,1]区间上都有支撑（函数值不为0），并且曲线是这些函数的混合，所以每个控制项对0到1之间t值处的曲线都有影响。）

B样条曲线保留了Bézier曲线的所有优点，同时克服了缺点。所谓样条，就是分段连续多项式。

整条曲线用一个完整的表达形式，但是内在是一段一段的，比如三次曲线，两条之间满足二次连续。这样不仅克服了波动，并且曲线是低次的。有统一的表达和统一的算法。

比如5个点，通过Bézier是四阶多项式。但是B样条可以构造四段曲线，每一段都是三次的，段与段之间满足二阶几何连续。

代码

#include <glad/glad.h>
#include <GLFW/glfw3.h>
#include "shader.h"
#include <iostream>
#include <cstdio>
#include <cstdlib>
#include <cmath>
#include <vector>
using namespace std;struct Point{//像素点double x,y;Point(double _x=0,double _y=0):x(_x),y(_y){};Point&operator=(const Point &p){x=p.x; y=p.y;return *this;};bool operator ==(const Point &p){return (x==p.x&&y==p.y);}
}templine[2],LockPoint(-1,-1);//临时线和锁定点
const unsigned int SCR_WIDTH = 800;
const unsigned int SCR_HEIGHT = 600;
const unsigned int MAXN=600+1;
const char* vertexShaderPath="/Users/longzhengtian/Desktop/代码/C++代码/ComputerGraphics/shader.vs";
const char* fragmentShaderPath="/Users/longzhengtian/Desktop/代码/C++代码/ComputerGraphics/shader.fs";
vector<Point> ControlPloy;//控制多边形
vector<Point> ControlLine;//控制线
vector<Point> BSplinePoint;//BSpline曲线上的点bool DrawControlPoly=false,DrawControlPolyEnd=false;
bool change=false;//曲线要发生变化
bool moving=false;unsigned int VBO[4], VAO[4];
unsigned int k=0;//阶数
int LockPos=-1;void framebuffer_size_callback(GLFWwindow* window, int width, int height);
void processInput(GLFWwindow *window);
void key_callback(GLFWwindow* window, int key, int scancode, int action, int mods);
void mouse_button_callback(GLFWwindow* window, int button, int action, int mods);
void CalcuBspline(double &t,double pre);
float transX(double x){return (float)((float)(2*x)-SCR_WIDTH)/(SCR_WIDTH*1.0f);}
float transY(double y){return (float)(SCR_HEIGHT-2*(float)y)/(SCR_HEIGHT*1.0f);}int main(){//实例化glfw函数glfwInit();glfwWindowHint(GLFW_CONTEXT_VERSION_MAJOR, 3);glfwWindowHint(GLFW_CONTEXT_VERSION_MINOR, 3);glfwWindowHint(GLFW_OPENGL_PROFILE, GLFW_OPENGL_CORE_PROFILE);
#ifdef __APPLE__glfwWindowHint(GLFW_OPENGL_FORWARD_COMPAT, GL_TRUE);
#endif//创建glfw窗口GLFWwindow* window = glfwCreateWindow(SCR_WIDTH, SCR_HEIGHT, "B-Spline", nullptr, nullptr);if (window == nullptr){cout << "Failed to create GLFW window" << endl;glfwTerminate();return -1;}glfwMakeContextCurrent(window);//将window设置为接下来操作的主窗口glfwSetFramebufferSizeCallback(window, framebuffer_size_callback);glfwSetKeyCallback(window, key_callback);glfwSetMouseButtonCallback(window, mouse_button_callback);if (!gladLoadGLLoader((GLADloadproc)glfwGetProcAddress)){//使用glad加载glfw的所有函数指针cout << "Failed to initialize GLAD" << endl;return -1;}Shader ourShader(vertexShaderPath,fragmentShaderPath);//创建着色器程序GLfloat vertices[MAXN*MAXN];//B样条曲线上的点GLfloat lintVertices[7];//临时线GLfloat linVertices[MAXN];//现存线GLfloat ControlVertices[MAXN*MAXN];//控制线glGenVertexArrays(4, VAO);glGenBuffers(4, VBO);glBindVertexArray(VAO[0]);glBindBuffer(GL_ARRAY_BUFFER, VBO[0]);glBufferData(GL_ARRAY_BUFFER, sizeof(vertices), vertices, GL_DYNAMIC_DRAW);glVertexAttribPointer(0, 3, GL_FLOAT, GL_FALSE, 3 * sizeof(float), (void *) 0);glEnableVertexAttribArray(0);glBindVertexArray(VAO[1]);//临时边glBindBuffer(GL_ARRAY_BUFFER, VBO[1]);glBufferData(GL_ARRAY_BUFFER, sizeof(lintVertices), lintVertices, GL_DYNAMIC_DRAW);glVertexAttribPointer(0, 3, GL_FLOAT, GL_FALSE, 3 * sizeof(float), (void *) 0);glEnableVertexAttribArray(0);glBindVertexArray(VAO[2]);//现存边glBindBuffer(GL_ARRAY_BUFFER, VBO[2]);glBufferData(GL_ARRAY_BUFFER, sizeof(linVertices), linVertices, GL_DYNAMIC_DRAW);glVertexAttribPointer(0, 3, GL_FLOAT, GL_FALSE, 3 * sizeof(float), (void *) 0);glEnableVertexAttribArray(0);glBindVertexArray(VAO[3]);//控制边glBindBuffer(GL_ARRAY_BUFFER, VBO[3]);glBufferData(GL_ARRAY_BUFFER, sizeof(ControlVertices), ControlVertices, GL_DYNAMIC_DRAW);glVertexAttribPointer(0, 3, GL_FLOAT, GL_FALSE, 3 * sizeof(float), (void *) 0);glEnableVertexAttribArray(0);double precision=0.005;double t=-precision;//时间控制while (!glfwWindowShouldClose(window)){//输入processInput(window);//渲染命令glClearColor(0.2f, 0.3f, 0.3f, 1.0f);glClear(GL_COLOR_BUFFER_BIT);//临时边ourShader.setVec4("ourColor", 1.0f, 1.0f, 1.0f, 1.0f);ourShader.use();glBindVertexArray(VAO[1]);glDrawArrays(GL_LINES, 0, 2);//现存边ourShader.setVec4("ourColor", 1.0f, 1.0f, 1.0f, 1.0f);ourShader.use();glBindVertexArray(VAO[2]);glDrawArrays(GL_LINE_STRIP, 0, ControlPloy.size());//BSpline Curveif (change && k!=0) {if(t==-precision){//初次绘画ControlLine.clear();BSplinePoint.clear();}CalcuBspline(t, precision);vector<float> tempvecV;GLfloat *tempvec;tempvecV.clear();for (int i = 0; i < BSplinePoint.size(); i++) {tempvecV.push_back(transX(BSplinePoint[i].x));tempvecV.push_back(transY(BSplinePoint[i].y));tempvecV.push_back(1.0f);}tempvec = tempvecV.data();glBindBuffer(GL_ARRAY_BUFFER, VBO[0]);glBufferSubData(GL_ARRAY_BUFFER, 0, tempvecV.size() * sizeof(float), tempvec);glBindBuffer(GL_ARRAY_BUFFER, 0);if(t<ControlPloy.size()){tempvecV.clear();for (int i = 0; i < ControlLine.size(); i++) {tempvecV.push_back(transX(ControlLine[i].x));tempvecV.push_back(transY(ControlLine[i].y));tempvecV.push_back(1.0f);}tempvec = tempvecV.data();glBindBuffer(GL_ARRAY_BUFFER, VBO[3]);glBufferSubData(GL_ARRAY_BUFFER, 0, tempvecV.size() * sizeof(float), tempvec);glBindBuffer(GL_ARRAY_BUFFER, 0);}else {t=-precision;change=false;//不写会无限循环}}if(t!=-precision){ourShader.setVec4("ourColor", 1.0f, 0.0f, 0.0f, 1.0f);ourShader.use();glBindVertexArray(VAO[3]);glDrawArrays(GL_LINES, 0, ControlLine.size());}ourShader.setVec4("ourColor", 0.0f, 1.0f, 0.0f, 1.0f);ourShader.use();glBindVertexArray(VAO[0]);glDrawArrays(GL_LINE_STRIP, 0, BSplinePoint.size());//检查并调用事件，交换缓冲glfwSwapBuffers(window);glfwPollEvents();}glDeleteVertexArrays(4, VAO);glDeleteBuffers(4, VBO);glfwTerminate();return 0;
}
void CalcuBspline(double &t,double precision){t+=precision;//时间分为1/precison份if(t<k-1) t=k-1;//直接跳到合适的区间，[t_{k-1},t_{n+1}]int j=t;//直接向下取整获得区间vector<vector<Point>> tempConLine;//tempConLine[i][j]是指第i层，第j个点vector<Point> tempv;for (int i=j-k+1; i<=j; i++){//第0次迭代，将P_{j-k+1},...P_j都加入控制线中tempv.push_back(ControlPloy[i]);}tempConLine.push_back(tempv);for (int r=1; r<=k-1; r++){//迭代k-1次tempv.clear();for (int i=j-k+r+1,ii=1; i<=j,ii<tempConLine[r-1].size(); i++,ii++){double alpha=t-i;double dev=(k-r);//t_{i+k-r}-t_{i};alpha=(dev!=0)?alpha/dev:0;//获得alpha的值tempv.emplace_back((1.0-alpha)*tempConLine[r-1][ii-1].x+alpha*tempConLine[r-1][ii].x,(1.0-alpha)*tempConLine[r-1][ii-1].y+alpha*tempConLine[r-1][ii].y);}tempConLine.push_back(tempv);}ControlLine.clear();for (int i=1; i<=k-1; i++){ControlLine.push_back(tempConLine[i][0]);for (int l=1; l<tempConLine[i].size(); l++){ControlLine.push_back(tempConLine[i][l]);ControlLine.push_back(tempConLine[i][l]);}ControlLine.push_back(tempConLine[i][tempConLine[i].size()-1]);}BSplinePoint.push_back(tempConLine[k-1][0]);
}
void processInput(GLFWwindow *window){if(DrawControlPoly){//此时处于画直线状态double xpos,ypos;glfwGetCursorPos(window,&xpos,&ypos);templine[0]=ControlPloy[ControlPloy.size()-1];templine[1]=Point(xpos,ypos);float tempvec[]={transX(templine[0].x),transY(templine[0].y),1.0f,transX(templine[1].x),transY(templine[1].y),1.0f};glBindBuffer(GL_ARRAY_BUFFER, VBO[1]);glBufferSubData(GL_ARRAY_BUFFER, 0, sizeof(tempvec), &tempvec);glBindBuffer(GL_ARRAY_BUFFER, 0);}if(moving){double xpos,ypos;glfwGetCursorPos(window,&xpos,&ypos);float tempvec[]={transX(xpos),transY(ypos),1.0f};glBindBuffer(GL_ARRAY_BUFFER, VBO[2]);glBufferSubData(GL_ARRAY_BUFFER, LockPos*3*sizeof(float), sizeof(tempvec), &tempvec);glBindBuffer(GL_ARRAY_BUFFER, 0);}
}
void framebuffer_size_callback(GLFWwindow* window, int width, int height){glViewport(0, 0, width, height);
}
void key_callback(GLFWwindow* window, int key, int scancode, int action, int mods){if (key == GLFW_KEY_ESCAPE && action == GLFW_PRESS)glfwSetWindowShouldClose(window, GL_TRUE);else if (key == GLFW_KEY_P && action == GLFW_PRESS){//如果按下"P"，开始画控制多边形，也可以用于增加点if(!DrawControlPoly){DrawControlPoly=true;double xpos,ypos;glfwGetCursorPos(window,&xpos,&ypos);float tempvec[]={transX(xpos),transY(ypos),1.0f};glBindBuffer(GL_ARRAY_BUFFER, VBO[2]);glBufferSubData(GL_ARRAY_BUFFER, ControlPloy.size()*3*sizeof(float), sizeof(tempvec), &tempvec);glBindBuffer(GL_ARRAY_BUFFER, 0);ControlPloy.emplace_back(xpos,ypos);}else {//这次是为了结束DrawControlPoly=false;templine[0]=Point(0,0);templine[1]=Point(0,0);float tempvec[]={0.0f,0.0f,1.0f,0.0f,0.0f,1.0f};glBindBuffer(GL_ARRAY_BUFFER, VBO[1]);glBufferSubData(GL_ARRAY_BUFFER, 0, sizeof(tempvec), &tempvec);glBindBuffer(GL_ARRAY_BUFFER, 0);change=true;//发生改变，需要重新计算控制线}}else if (key == GLFW_KEY_M && action == GLFW_PRESS) {//如果按下"M"，开始移动if(!moving && LockPos!=-1){//本来不移动，结果点击开始移动了moving=true;}else{//移动停止cout<<"You moved the "<<LockPos+1<<"th vertex in the control polygon."<<endl;moving=false;double xpos,ypos;glfwGetCursorPos(window,&xpos,&ypos);ControlPloy[LockPos]=Point(xpos,ypos);cout<<"New ControlPoly:"<<endl;for (int i=0; i<ControlPloy.size(); i++)cout<<"("<<ControlPloy[i].x<<','<<ControlPloy[i].y<<") ";cout<<endl;change=true;//发生改变，需要重新计算控制线}}else if (key == GLFW_KEY_D && action == GLFW_PRESS) {//如果按下"D"，删除点if(LockPos!=-1){//此时刚刚锁定了一个点，可以删除了int tempsize=(ControlPloy.size()-LockPos)*3,tempcnt=0;float tempvec[tempsize];memset(tempvec,0,sizeof(tempvec));for (int i=LockPos+1; i<ControlPloy.size(); i++){tempvec[tempcnt++]=transX(ControlPloy[i].x);tempvec[tempcnt++]=transY(ControlPloy[i].y);tempvec[tempcnt++]=1.0f;}//控制多边形数组中删除auto it=find(ControlPloy.begin(),ControlPloy.end(),LockPoint);ControlPloy.erase(it);//缓冲区中删除glBindBuffer(GL_ARRAY_BUFFER, VBO[2]);glBufferSubData(GL_ARRAY_BUFFER, LockPos*3*sizeof(float), sizeof(tempvec), &tempvec);glBindBuffer(GL_ARRAY_BUFFER, 0);cout<<"You deleted the "<<LockPos+1<<"th vertex in the control polygon."<<endl;cout<<"New ControlPoly:"<<endl;for (int i=0; i<ControlPloy.size(); i++)cout<<"("<<ControlPloy[i].x<<','<<ControlPloy[i].y<<") ";cout<<endl;//解除锁定LockPoint=Point(-1,-1); LockPos=-1;change=true;//发生改变，需要重新计算控制线DrawControlPolyEnd=true;}}else if (key == GLFW_KEY_K && action == GLFW_PRESS) {//如果按下"K"，改变阶数cout<<"Please enter the order of the B-spline："<<endl;cin>>k;//输入阶数。我们规定，节点表从0开始，逐次+1。t向下取整可以直接确定区间。change=true;}
}
void mouse_button_callback(GLFWwindow* window, int button, int action, int mods){if(button == GLFW_MOUSE_BUTTON_LEFT && action == GLFW_PRESS){//鼠标点击了左键double xpos,ypos;glfwGetCursorPos(window,&xpos,&ypos);if(DrawControlPoly){//此时是画线状态，需要增加线float tempvec[]={transX(xpos),transY(ypos),1.0f};glBindBuffer(GL_ARRAY_BUFFER, VBO[2]);glBufferSubData(GL_ARRAY_BUFFER, ControlPloy.size()*3*sizeof(float), sizeof(tempvec), &tempvec);glBindBuffer(GL_ARRAY_BUFFER, 0);ControlPloy.emplace_back(xpos,ypos);}else if(!moving){//此时锁定点，看情况是否删除LockPoint=Point(-1,-1); LockPos=-1;//初始化for (int i=0; i<ControlPloy.size(); i++){if(fabs(xpos-ControlPloy[i].x)<=5 && fabs(ypos-ControlPloy[i].y)<=5){//给定一个误差范围LockPoint=ControlPloy[i]; LockPos=i;cout<<"You locked the "<<LockPos+1<<"th vertex in the control polygon."<<endl;break;}}}}
}

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B Spline（B样条曲线）

文章目录

说明

B样条曲线

代码

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