背景知识：

在数学中，向量（也称为欧几里得向量、几何向量、矢量），指具有大小（magnitude）和方向的量.

本例实现一个二维向量。二维向量通常表示的方法有 直角坐标（x,y）和极坐标(a,m) // (a为长度，m为角度)。

创建一个类首先考虑类包含的数据。

一个向量(Vector)可以使用两个double表示直角坐标的x,y；用两个double表示极坐标的a,m。

还需要一个东西用于存储向量的形式（直角or极坐标）。使用枚举enum 。

则有：

public:
    enum Mode{RECT,POL};

private:
    double x;
    double y;
    double mag;
    double ang;
    Mode mode;

接着考虑对向量的操作，一个类最基本的 应该有构造函数和析构函数。

在创建构造函数时，我们遇到了一个问题，由于同一个向量有两个形式，那么创建时应该怎么做呢？

最容易想到的办法就是选择其一，如直角坐标。那么，构造函数应该像是这种形式，

Vector(double n1, double n2);

然后让x=n1,y=n2;这样就可以设置向量的直角坐标，但是，极坐标就被忽略了。

考虑到向量的直角坐标和极坐标是一一对应的，于是想让程序自动设置对应的极坐标。

在Vector类中添加函数(在private中，因为不需要给外部提供接口)

void set_mag();
    void set_ang();

具体实现为：

void Vector::set_mag()
  {
    mag = sqrt(x*x + y * y);
  }
  void Vector::set_ang()
  {
    if (x == 0.0 && y == 0.0)
      ang = 0.0;
    else
      ang = atan2(y, x);
  }

然而，如果用户想要用极坐标形式输入向量，当前的构造函数无法满足需求。

于是，改造构造函数，添加一个参数 form 用来选择形式。

Vector(double n1, double n2, Mode form= RECT);

相应的，构造函数的实现改为：

Vector::Vector(double n1, double n2, Mode form )
  {
    mode = form;
    if (form == RECT)
    {
      x = n1;
      y = n2;
      set_mag();
      set_ang();
    }
    else if (form == POL)
    {
      mag = n1;
      ang = n2/Rad_to_deg;
      set_x();
      set_y();
    }
    else
    {
      cout << "Incorrect 3rd argument to Vector() --";
      cout << "Vector set to 0";
      x = y = mag = ang = 0;
      mode = RECT;
    }
  }

相应的，在Vector类添加set_x();set_t()方法。

void set_x();
  void set_y();

void Vector::set_x()
  {
    x = mag * cos(ang);
  }
  void Vector::set_y()
  {
    y = mag * sin(ang);
  }

如此，完成了构造函数。

析构函数使用默认的即可。或者自己加上一个构造函数 ~Vector(); 然后在对应的实现函数函数体中留下空格即可。

                                                                                            Vector::~Vector()   { }

完成了构造函数和析构函数，考虑向量需要的其他功能/接口。

两个向量相加减，向量的数乘，改变向量的坐标。输出向量。

获取向量的坐标值(提供接口才可访问，否则在private中不能访问)

设置向量的形式

......

具体如下：

void reset(double n1, double n2, Mode form = RECT);
    double xval()const { return x; }
    double yval()const { return y; }
    double magval()const { return mag; }
    double angval()const { return ang; }
    void polar_mode();
    void rect_mode();
    //operator
    Vector operator+(const Vector&b)const;
    Vector operator-(const Vector&b)const;
    Vector operator-()const;
    Vector operator*(double n)const;
    //friends
    friend Vector operator*(double n, const Vector &a);
    friend std::ostream & operator<<(std::ostream &os, const Vector &v);

实现：

void Vector::reset(double n1, double n2, Mode form )
  {
    mode = form;
    if (form == RECT)
    {
      x = n1;
      y = n2;
      set_mag();
      set_ang();
    }
    else if (form == POL)
    {
      mag = n1;
      ang = n2 / Rad_to_deg;
      set_x();
      set_y();
    }
    else
    {
      cout << "Incorrect 3rd argument to Vector() --";
      cout << "Vector set to 0";
      x = y = mag = ang = 0;
      mode = RECT;
    }
  }

  void Vector::polar_mode()
  {
    mode = POL;
  }
  void Vector::rect_mode()
  {
    mode = RECT;
  }
  //operator
  Vector Vector::operator+(const Vector&b)const
  {
    return Vector(x + b.x, y + b.y );
  }
  Vector Vector::operator-(const Vector&b)const
  {
    return Vector(x - b.x, y - b.y);
  }
  Vector Vector::operator-()const
  {
    return Vector(-x, -y);
  }
  Vector Vector::operator*(double n)const
  {
    return Vector(x*n, y*n);
  }
  //friends
  Vector operator*(double n, const Vector &a)
  {
    return a * n;
  }
  std::ostream & operator<<(std::ostream &os, const Vector &v)
  {
    if (v.mode == Vector::RECT)
      os << "(x,y) = (" << v.x << ", " << v.y << ")";
    else if (v.mode == Vector::POL)
    {
      os << "(m,a) = (" << v.mag << ", " << v.ang*Rad_to_deg << ")";
    }
    else
      os << "Vector Object mode is invalid";
    return os;
  }

至此已经完成了Vector类的定义。（借助了cmath库进行了坐标两种形式转换）

完整代码：

//vector.h
#pragma once
#include<iostream>
namespace VECTOR
{
  class Vector
  {
  public:
    enum Mode{RECT,POL};
  private:
    double x;
    double y;
    double mag;
    double ang;
    Mode mode;
    void set_mag();
    void set_ang();
    void set_x();
    void set_y();
  public:
    Vector();
    Vector(double n1, double n2, Mode form= RECT);
    ~Vector();
    void reset(double n1, double n2, Mode form = RECT);
    double xval()const { return x; }
    double yval()const { return y; }
    double magval()const { return mag; }
    double angval()const { return ang; }
    void polar_mode();
    void rect_mode();
    //operator
    Vector operator+(const Vector&b)const;
    Vector operator-(const Vector&b)const;
    Vector operator-()const;
    Vector operator*(double n)const;
    //friends
    friend Vector operator*(double n, const Vector &a);
    friend std::ostream & operator<<(std::ostream &os, const Vector &v);
  };
}

#include<cmath>
#include"vector.h"
using std::sqrt;
using std::sin;
using std::cos;
using std::atan;
using std::atan2;
using std::cout;

namespace VECTOR
{
  const double Rad_to_deg = 45.0 / atan(1.0);

  //private methods
  void Vector::set_mag()
  {
    mag = sqrt(x*x + y * y);
  }
  void Vector::set_ang()
  {
    if (x == 0.0 && y == 0.0)
      ang = 0.0;
    else
      ang = atan2(y, x);
  }
  void Vector::set_x()
  {
    x = mag * cos(ang);
  }
  void Vector::set_y()
  {
    y = mag * sin(ang);
  }
  //public methods
  Vector::Vector()
  {
    x = y = mag = ang = 0.0;
    mode = RECT;
  }
  Vector::Vector(double n1, double n2, Mode form )
  {
    mode = form;
    if (form == RECT)
    {
      x = n1;
      y = n2;
      set_mag();
      set_ang();
    }
    else if (form == POL)
    {
      mag = n1;
      ang = n2/Rad_to_deg;
      set_x();
      set_y();
    }
    else
    {
      cout << "Incorrect 3rd argument to Vector() --";
      cout << "Vector set to 0";
      x = y = mag = ang = 0;
      mode = RECT;
    }
  }
  Vector::~Vector()
  {
  }
  void Vector::reset(double n1, double n2, Mode form )
  {
    mode = form;
    if (form == RECT)
    {
      x = n1;
      y = n2;
      set_mag();
      set_ang();
    }
    else if (form == POL)
    {
      mag = n1;
      ang = n2 / Rad_to_deg;
      set_x();
      set_y();
    }
    else
    {
      cout << "Incorrect 3rd argument to Vector() --";
      cout << "Vector set to 0";
      x = y = mag = ang = 0;
      mode = RECT;
    }
  }

  void Vector::polar_mode()
  {
    mode = POL;
  }
  void Vector::rect_mode()
  {
    mode = RECT;
  }
  //operator
  Vector Vector::operator+(const Vector&b)const
  {
    return Vector(x + b.x, y + b.y );
  }
  Vector Vector::operator-(const Vector&b)const
  {
    return Vector(x - b.x, y - b.y);
  }
  Vector Vector::operator-()const
  {
    return Vector(-x, -y);
  }
  Vector Vector::operator*(double n)const
  {
    return Vector(x*n, y*n);
  }
  //friends
  Vector operator*(double n, const Vector &a)
  {
    return a * n;
  }
  std::ostream & operator<<(std::ostream &os, const Vector &v)
  {
    if (v.mode == Vector::RECT)
      os << "(x,y) = (" << v.x << ", " << v.y << ")";
    else if (v.mode == Vector::POL)
    {
      os << "(m,a) = (" << v.mag << ", " << v.ang*Rad_to_deg << ")";
    }
    else
      os << "Vector Object mode is invalid";
    return os;
  }
}