埃尔米特插值 原理

#pragma once
#include <vector>
#include <functional>
/*

埃尔米特插值

*/
struct InterpolationPoint {
    double x; // 插值点的横坐标
    double y; // 插值点的纵坐标
    double derivative; // 插值点的导数值
     // 默认构造函数
    InterpolationPoint() : x(0.0), y(0.0), derivative(0.0) {}

    // 带参数的构造函数
    InterpolationPoint(double x_val, double y_val, double derivative_val) : x(x_val), y(y_val), derivative(derivative_val) {}

    // 拷贝构造函数
    InterpolationPoint(const InterpolationPoint& other) : x(other.x), y(other.y), derivative(other.derivative) {}

    // 移动构造函数
    InterpolationPoint(InterpolationPoint&& other) noexcept : x(other.x), y(other.y), derivative(other.derivative) {
        other.x = 0.0;
        other.y = 0.0;
        other.derivative = 0.0;
    }
    // Copy assignment operator
    InterpolationPoint& operator=(const InterpolationPoint& other) {
        if (this != &other) {
            x = other.x;
            y = other.y;
            derivative = other.derivative;
        }
        return *this;
    }

    // 设置插值点的值
    void set(double x_val, double y_val, double derivative_val) {
        x = x_val;
        y = y_val;
        derivative = derivative_val;
    }

    // 获取插值点的横坐标
    double get_x() const {
        return x;
    }

    // 获取插值点的纵坐标
    double get_y() const {
        return y;
    }

    // 获取插值点的导数值
    double get_derivative() const {
        return derivative;
    }
};

class HermiteInterpolator {
public:
    HermiteInterpolator(const std::vector<InterpolationPoint>& points);
    HermiteInterpolator(int width, std::vector<int> &adjPoints);
    void setPoints(const std::vector<InterpolationPoint>& points);
    double interpolate(double x) ;

private:
    // 返回连接两点的线段函数
    std::function<double(double)> getLineFunction( InterpolationPoint& p1,  InterpolationPoint& p2);

private:
    std::vector<InterpolationPoint> points_;
};
#include "pch.h"
#include "HermiteInterpolator.h"
#include <fstream>
HermiteInterpolator::HermiteInterpolator(const std::vector<InterpolationPoint>& points) 
	: points_(points)
{
}
HermiteInterpolator::HermiteInterpolator(int width, std::vector<int>& adjPoints)
{
	float step = width / adjPoints.size();
	for (int i = 0; i < adjPoints.size(); i++)
	{
		InterpolationPoint point(step*i, adjPoints[i] , 0);
		points_.push_back(point);
	}
}
void HermiteInterpolator::setPoints(const std::vector<InterpolationPoint>& points)
{
	points_ = points;
}

// 返回连接两点的线段函数
std::function<double(double)> HermiteInterpolator::getLineFunction( InterpolationPoint& p1,  InterpolationPoint& p2) {
    // 计算线段的斜率和截距
    double slope = (p2.y - p1.y) / (p2.x - p1.x);
    double intercept = p1.y - slope * p1.x;

    // 返回线段的lambda表达式
    return [slope, intercept](double x) {
        return slope * x + intercept;
    };
}
// 计算三次分段Hermite插值函数的值
double HermiteInterpolator::interpolate(double x)  {
    int y = 0;
    
    int n = points_.size();
    if (n < 3)
    {
        // 获取线段函数
        std::function<double(double)> lineFunction = getLineFunction(points_[0], points_[1]);
        y= lineFunction(x);
    }
    else
    {
        for (int i = 0; i < n - 1; i++) {
            if (x >= points_[i].x && x <= points_[i + 1].x) {
                double h = points_[i + 1].x - points_[i].x;
                double t = (x - points_[i].x) / h;// (x-x_k)/(x_{k+1} - x_k)
                double tk = (x - points_[i + 1].x) / (-h); // (x - x_{ k + 1 }) / (x_k - x_{ k + 1 }) 
                double y0 = (1 + 2 * t) * tk * tk;
                double y1 = (1 + 2 * tk) * t * t;
                double y2 = (x - points_[i].x) * tk * tk;
                double y3 = (x - points_[i + 1].x) * t * t;

                y= points_[i].y * y0 + points_[i + 1].y * y1 + points_[i].derivative * y2 + points_[i + 1].derivative * y3;
            }
        }
    }
    
    //ofstream  f;
    //f.open("D:\\work\\documentation\\HermiteInterpolator.txt", ios::app);
    //f <<x<<"," << y << endl;
    //f.close();
    return y; // 如果找不到对应的插值段，返回默认值
}

为了可视化效果可以把结果写到HermiteInterpolator.txt
画图python代码：

import matplotlib.pyplot as plt

# 打开文本文件进行读取
with open('D:\\work\\documentation\\HermiteInterpolator.txt') as f:
    data = f.readlines()

# 定义两个列表分别存储横坐标和纵坐标的数据    
x = []
y = [] 

# 遍历每一行
for i, line in enumerate(data):
    # 去除换行符
    if line:
        user_pwd_list = line.strip().split(',')
    
        # 横坐标是行号
        x.append(float(user_pwd_list[0]))
        
        # 纵坐标是数值数据
        y.append(float(user_pwd_list[1]))
    
# 创建散点图    
plt.scatter(x, y)

# 添加标题和轴标签
plt.title('Scatter Plot')  
plt.xlabel('Line')
plt.ylabel('Value')

# 显示并保存图像
#plt.savefig('plot.png')
plt.show()