Operator overloading

Bài này viết về toán tử chồng trong bài toán đa hình. Tìm mãi không có bài nào tiếng Việt đăng lên nên mình tạm xin trích dẫn toàn bộ bài viết:
You can make operators virtual just like other member functions. Implementing virtual operators often becomes confusing, however, because you may be operating on two objects, both with unknown types. This is usually the case with mathematical components (for which you often overload operators). For example, consider a system that deals with matrices, vectors and scalar values, all three of which are derived from class Math:

//: C15:OperatorPolymorphism.cpp
// Polymorphism with overloaded operators
#include 
using namespace std;

class Matrix;
class Scalar;
class Vector;

class Math {
public:
  virtual Math& operator*(Math& rv) = 0;
  virtual Math& multiply(Matrix*) = 0;
  virtual Math& multiply(Scalar*) = 0;
  virtual Math& multiply(Vector*) = 0;
  virtual ~Math() {}
};

class Matrix : public Math {
public:
  Math& operator*(Math& rv) {
    return rv.multiply(this); // 2nd dispatch
  }
  Math& multiply(Matrix*) {
    cout << "Matrix * Matrix" << endl;
    return *this;
  }
  Math& multiply(Scalar*) {
    cout << "Scalar * Matrix" << endl;
    return *this;
  }
  Math& multiply(Vector*) {
    cout << "Vector * Matrix" << endl;
    return *this;
  }
};

class Scalar : public Math  {
public:
  Math& operator*(Math& rv) {
    return rv.multiply(this); // 2nd dispatch
  }
  Math& multiply(Matrix*) {
    cout << "Matrix * Scalar" << endl;
    return *this;
  }
  Math& multiply(Scalar*) {
    cout << "Scalar * Scalar" << endl;
    return *this;
  }
  Math& multiply(Vector*) {
    cout << "Vector * Scalar" << endl;
    return *this;
  }
};

class Vector : public Math  {
public:
  Math& operator*(Math& rv) {
    return rv.multiply(this); // 2nd dispatch
  }
  Math& multiply(Matrix*) {
    cout << "Matrix * Vector" << endl;
    return *this;
  }
  Math& multiply(Scalar*) {
    cout << "Scalar * Vector" << endl;
    return *this;
  }
  Math& multiply(Vector*) {
    cout << "Vector * Vector" << endl;
    return *this;
  }
};

int main() {
  Matrix m; Vector v; Scalar s;
  Math* math[] = { &m, &v, &s };
  for(int i = 0; i < 3; i++)
    for(int j = 0; j < 3; j++) {
      Math& m1 = *math[i];
      Math& m2 = *math[j];
      m1 * m2;
    }
} ///:~

For simplicity, only the operator* has been overloaded. The goal is to be able to multiply any two Math objects and produce the desired result – and note that multiplying a matrix by a vector is a very different operation than multiplying a vector by a matrix.

The problem is that, in main( ), the expression m1 * m2 contains two upcast Math references, and thus two objects of unknown type. A virtual function is only capable of making a single dispatch – that is, determining the type of one unknown object. To determine both types a technique called multiple dispatching is used in this example, whereby what appears to be a single virtual function call results in a second virtual call. By the time this second call is made, you’ve determined both types of object, and can perform the proper activity. It’s not transparent at first, but if you stare at the example for awhile it should begin to make sense. This topic is explored in more depth in the Design Patterns chapter in Volume 2, which you can download at www.BruceEckel.com.