Recent posts

The intersection area of two circles


Posted by Diego Assencio on 2017.07.12 under Mathematics (Geometry)

Let $C_1$ and $C_2$ be two circles of radii $r_1$ and $r_2$ respectively whose centers are at a distance $d$ from each other. Assume, without loss of generality, that $r_1 \geq r_2$. What is the intersection area of these two circles?

If $d \geq r_1 + r_2$, the circles intersect at most up to a point (when $d = r_1 + r_2$) and therefore the intersection area is zero. On the other extreme, if $d + r_2 \leq r_1$, circle $C_2$ is entirely contained within $C_1$ and the intersection area is the area of $C_2$ itself: $\pi r_2^2$. The challenging case happens when both $d \lt r_1 + r_2$ and $d + r_2 \gt r_1$ are satisfied, i.e., when the the circles intersect only partially but the intersection area is more than simply a point. Rearranging the second inequality, we obtain $r_1 - r_2 \lt d \lt r_1 + r_2$, so we will assume this to be the case from now on.

To solve this problem, we will make use of a Cartesian coordinate system with origin at the center of circle $C_1$ such that the center of $C_2$ is at $(d,0)$ as shown on figure 1.

Fig. 1: Two intersecting circles $C_1$ (blue) and $C_2$ (red) of radii $r_1$ and $r_2$ respectively. The distance between the centers of the circles is $d = d_1 + d_2$, where $d_1$ is the $x$ coordinate of the intersection points and $d_2 = d - d_1$. Notice that $d_1 \geq 0$ since these points are always located to the right of the center of $C_1$, but $d_2$ may be negative when $r_2 \lt r_1$ since, in this case, the intersection points will eventually fall to the right of the center of $C_2$ as we move $C_2$ to the left, making $d \lt d_1$ and therefore $d_2 \lt 0$.

The circles $C_1$ and $C_2$ are described by the following equations respectively: $$ \begin{eqnarray} x^2 + y^2 &=& r_1^2 \label{post_8d6ca3d82151bad815f78addf9b5c1c6_c1}\\[5pt] (x - d)^2 + y^2 &=& r_2^2 \\[5pt] \end{eqnarray} $$ At the intersection points, we have $x = d_1$. To determine $d_1$, we can replace $x$ with $d_1$ and isolate $y^2$ on both equations above to get: $$ r_1^2 - d_1^2 = r_2^2 - (d_1 - d)^2 $$ Solving for $d_1$ is a simple task: $$ r_1^2 - d_1^2 = r_2^2 - d_1^2 + 2d_1d - d^2 \Longrightarrow d_1 = \displaystyle\frac{r_1^2 - r_2^2 + d^2}{2d} \label{post_8d6ca3d82151bad815f78addf9b5c1c6_eq_d1} $$ From equation \eqref{post_8d6ca3d82151bad815f78addf9b5c1c6_eq_d1}, we can see that $d_1 \geq 0$ since $r_1 \geq r_2$. The intersection area is the sum of the blue and red areas shown on figure 1, which we refer to as $A_1$ and $A_2$ respectively. We then have that: $$ \begin{eqnarray} A_1 &=& 2\int_{d_1}^{r_1} \sqrt{r_1^2 - x^2}dx \label{%INDEX_eq_A1_def} \\[5pt] A_2 &=& 2\int_{d - r_2}^{d_1} \sqrt{r_2^2 - (x - d)^2}dx \end{eqnarray} $$ where the factors of $2$ come from the fact that each integral above accounts for only half of the area of the associated region (only points on and above the $x$ axis are taken into account); the results must then be multiplied by two so that the areas below the $x$ axis are taken into account as well.

The computation of these integrals is straightforward. Before we proceed, notice first that: $$ \begin{eqnarray} A_2 &=& 2\int_{d - r_2}^{d_1} \sqrt{r_2^2 - (x - d)^2}dx \nonumber \\[5pt] &=& 2\int_{- r_2}^{d_1 - d} \sqrt{r_2^2 - x^2}dx \nonumber \\[5pt] &=& 2\int_{d - d_1}^{r_2} \sqrt{r_2^2 - x^2}dx \nonumber \\[5pt] &=& 2\int_{d_2}^{r_2} \sqrt{r_2^2 - x^2}dx \label{%INDEX_eq_A2} \end{eqnarray} $$ where above we used the fact that $d_2 = d - d_1$. This is the same as equation \eqref{%INDEX_eq_A1_def} if we apply the substitutions $d_1 \rightarrow d_2$ and $r_1 \rightarrow r_2$. Therefore, by computing $A_1$, we will immediately obtain $A_2$ as well. Let's then compute $A_1$ first: $$ \begin{eqnarray} A_1 &=& 2\int_{d_1}^{r_1} \sqrt{r_1^2 - x^2}dx \nonumber\\[5pt] &=& 2r_1 \int_{d_1}^{r_1} \sqrt{1 - \left(\frac{x}{r_1}\right)^2}dx \nonumber\\[5pt] &=& 2r_1^2 \int_{d_1/r_1}^{1} \sqrt{1 - x^2}dx \label{%INDEX_eq_A1} \end{eqnarray} $$ All we need to do now is to integrate $\sqrt{1 - x^2}$. The process is straightforward if we use integration by parts: $$ \begin{eqnarray} \int \sqrt{1 - x^2}dx &=& x \sqrt{1 - x^2} - \int x \left(\frac{-x}{\sqrt{1 - x^2}}\right) dx \nonumber\\[5pt] &=& x \sqrt{1 - x^2} + \int \frac{x^2 - 1}{\sqrt{1 - x^2}} dx + \int \frac{1}{\sqrt{1 - x^2}} dx \nonumber\\[5pt] &=& x \sqrt{1 - x^2} - \int \sqrt{1 - x^2} dx + \sin^{-1}(x) \end{eqnarray} $$ Therefore: $$ \int \sqrt{1 - x^2}dx = \frac{1}{2}\left( x \sqrt{1 - x^2} + \sin^{-1}(x) \right) \label{post_8d6ca3d82151bad815f78addf9b5c1c6_int_for_A1_A2} $$ Using equation \eqref{post_8d6ca3d82151bad815f78addf9b5c1c6_int_for_A1_A2} on equation \eqref{%INDEX_eq_A1} yields: $$ \begin{eqnarray} A_1 &=& r_1^2 \left( \frac{\pi}{2} - \frac{d_1}{r_1}\sqrt{1 - \left(\frac{d_1}{r_1}\right)^2} - \sin^{-1}\left(\frac{d_1}{r_1}\right) \right) \nonumber\\[5pt] &=& r_1^2 \left( \cos^{-1}\left(\frac{d_1}{r_1}\right) - \frac{d_1}{r_1}\sqrt{1 - \left(\frac{d_1}{r_1}\right)^2} \right) \nonumber\\[5pt] &=& r_1^2 \cos^{-1}\left(\frac{d_1}{r_1}\right) - d_1 \sqrt{r_1^2 - d_1^2} \label{post_8d6ca3d82151bad815f78addf9b5c1c6_eq_A1_final} \end{eqnarray} $$ where above we used the fact that $\pi/2 - \sin^{-1}(\alpha) = \cos^{-1}(\alpha)$ for any $\alpha$ in $[-1,1]$. This fact is easy to prove: $$ \cos\left(\frac{\pi}{2} - \sin^{-1}(\alpha)\right) = \cos\left(\frac{\pi}{2}\right)\cos(\sin^{-1}(\alpha)) + \sin\left(\frac{\pi}{2}\right)\sin(\sin^{-1}(\alpha)) = \alpha $$ and therefore $\pi/2 - \sin^{-1}(\alpha) = \cos^{-1}(\alpha)$. As discussed above, we can now obtain $A_2$ directly by doing the substitutions $d_1 \rightarrow d_2$ and $r_1 \rightarrow r_2$ on the expression for $A_1$ on equation \eqref{post_8d6ca3d82151bad815f78addf9b5c1c6_eq_A1_final}: $$ A_2 = r_2^2 \cos^{-1}\left(\frac{d_2}{r_2}\right) - d_2 \sqrt{r_2^2 - d_2^2} $$ The sum of $A_1$ and $A_2$ is the intersection area of the circles: $$ \boxed{ \begin{eqnarray} A_{\textrm{intersection}} &=& r_1^2 \cos^{-1}\left(\frac{d_1}{r_1}\right) - d_1\sqrt{r_1^2 - d_1^2} \nonumber \\[5pt] &+& r_2^2\cos^{-1}\left(\frac{d_2}{r_2}\right) - d_2\sqrt{r_2^2 - d_2^2} \nonumber \end{eqnarray} } \label{post_8d6ca3d82151bad815f78addf9b5c1c6_A_intersection} $$ where: $$ \boxed{ d_1 = \displaystyle\frac{r_1^2 - r_2^2 + d^2}{2d} } \quad \textrm{ and } \quad \boxed{ d_2 = d - d_1 = \displaystyle\frac{r_2^2 - r_1^2 + d^2}{2d} } \label{post_8d6ca3d82151bad815f78addf9b5c1c6_eq_d1_final} $$

Summary

Given two circles $C_1$ and $C_2$ of radii $r_1$ and $r_2$ respectively (with $r_1 \geq r_2$) whose center points are at a distance $d$ from each other, the intersection area of the circles is:

1.zero, if $d \geq r_1 + r_2$, since in this case the circles intersect at most up to a point.
2.$\pi r_2^2$, if $d \leq r_1 - r_2$, since in this case $C_2$ is entirely contained within $C_1$.
3.given by equation \eqref{post_8d6ca3d82151bad815f78addf9b5c1c6_A_intersection} in all other cases.
Comments (0) Direct link

The nature of the "this" pointer in C++


Posted by Diego Assencio on 2017.04.01 under Programming (C/C++)

Whenever you call a non-static member function of a class, you call it through an existing object of that class type. Inside the definition of such a member function, you can refer to this object through the this pointer. Unless there is a need to disambiguate the use of a certain variable name (for instance, if a class data member has the same name as a local variable of the member function), the this pointer is often not used by developers to explicitly refer to class data members. This is almost always not a problem, but as I will discuss in this post, there are situations which require special care in order to avoid certain pitfalls.

To start, consider the following piece of code:

class AddNumber
{
public:

	...

	int add(const int other) const;

private:
	int number;
};

int AddNumber::add(const int other) const
{
	return number + other;
}

When the compiler parses the code above, it will understand that on the definition of AddNumber::add, number refers to the class data member with that name, i.e., that the code above is equivalent to this:

class AddNumber
{
public:

	...

	int add(const int other) const;

private:
	int number;
};

int AddNumber::add(const int other) const
{
	return this->number + other;
}

However, if we change the name of the parameter other of AddNumber::add to number, the compiler will interpret any occurrence of number inside its definition as the function parameter number instead of the data member this->number:

class AddNumber
{
public:

	...

	int add(const int number) const;

private:
	int number;
};

int AddNumber::add(const int number) const
{
	return number + number; /* here number is not this->number! */
}

To fix this ambiguity, we can use the this pointer to indicate to the compiler that the first occurrence of number actually refers to the class data member instead of the function parameter:

class AddNumber
{
public:

	...

	int add(const int number) const;

private:
	int number;
};

int AddNumber::add(const int number) const
{
	return this->number + number; /* this is what we originally had */
}

I hope there was nothing new for you on everything discussed so far, so let's move on to more interesting things.

One could argue that classes as we see them don't really exist: they are purely syntactic sugar for avoiding having to explicitly pass object pointers around as we do in C programs. To clarify this idea, take a look at the code below: it is conceptually equivalent to the one above except for the absence of the private access specifier. To prevent any desperation in advance, the code below is not valid C++; its purpose is merely to illustrate the concepts we are about to discuss:

struct AddNumber
{
	...

	int number;
};

int AddNumber::add(const AddNumber* this, const int number)
{
	return this->number + number;
}

Why is the code above not valid? Well, for two reasons: AddNumber::add is not a valid function name in this context (it is not a member of AddNumber), and this, being a reserved keyword, cannot be used as a parameter name. While in the original version, AddNumber:add is called through an existing object of type AddNumber:

AddNumber my_adder;

...

my_adder.add(3);

in our (invalid) non-class version, AddNumber:add is called with an object as argument:

AddNumber my_adder;

...

AddNumber::add(&my_adder, 3);

Were it not invalid, the non-class version would do exactly the same as the original one. But in any case, it better represents how the compiler actually interprets things. Indeed, it makes it obvious that if we remove the this-> prefix from the first occurrence of number, we will end up with the problem discussed earlier: number will be interpreted exclusively as the function parameter. But don't take my word for it, see it for yourself:

struct AddNumber
{
	...

	int number;
};

int AddNumber::add(const AddNumber* this, const int number)
{
	return number + number; /* this pointer not used, return 2*number */
}

This brings us to the first lesson of this post: whenever you see a non-static member function, try to always read it as a stand-alone (i.e., non-member) function containing a parameter called this which is a pointer to the object the function is doing its work for.

One question which must be asked at this point is: what about static member functions? Do they also implicitly contain a this pointer? The answer is no, they don't. If they did, they would inevitably be associated with some existing object of the class, but static member functions, like static data members, belong to the class itself and can be invoked directly, i.e., without the need for an an existing class object. In this regard, a static member function is in no way special: the compiler will neither implicitly add a this parameter to its declaration nor introduce this-> prefixes anywhere on its definition.

Static member functions have, however, access to the internals of a class like any other member or friend function, provided it is given a pointer to a class object. This means the following code is valid:

class AddNumber
{
public:

	...

	static int add(const AddNumber* adder, const int number);

private:
	int number;
};

int AddNumber::add(const AddNumber* adder, const int number)
{
	return adder->number + number;
}

There is one type of situation in which the implicit presence of the this pointer on non-static member functions can cause a lot of headache to the innocent developer. Here it is, in its full "glory":

/* a global array of callable warning objects */
std::vector<std::function<void()>> warnings;

class WarningManager
{
public:

	...

	void add_warning(const std::string& message) const;

private:
	std::string name;
};

void WarningManager::add_warning(const std::string& message) const
{
	warnings.emplace_back([=]() {
		std::cout << name << ": " << message << "\n";
	});
}

The purpose of the code above is simple: WarningManager::add_warning populates the global array warnings with lambda functions which print some warning message when invoked. Regardless of how silly the purpose of this code may seem, scenarios like these do happen in practice. And being so, do you see what is the problem here?

If the problem is unclear to you, consider the advice given earlier: read the member function WarningManager::add_warning as a non-member function which takes a pointer called this to a WarningManager object:

/* a global array of callable warning objects */
std::vector<std::function<void()>> warnings;

struct WarningManager
{
	...

	std::string name;
};

void WarningManager::add_warning(const WarningManager* this,
                                 const std::string& message)
{
	warnings.emplace_back([=]() {
		std::cout << this->name << ": " << message << "\n";
	});
}

You may be puzzled with the fact that name on the original version of the code was replaced by this->name on the (remember, invalid) second version. Perhaps you are asking yourself: "isn't name itself actually copied by the capture list on the lambda function"? The answer is no. A "capture all by value" capture list (i.e., [=]) captures all non-static local variables which are visible in the scope where the lambda is created and nothing else. Function parameters fall into this category, but class data members don't. Therefore, the code above is conceptually identical to the following one:

/* a global array of callable warning objects */
std::vector<std::function<void()>> warnings;

struct WarningManager
{
	...

	std::string name;
};

void WarningManager::add_warning(const WarningManager* this,
                                 const std::string& message)
{
	warnings.emplace_back([this, message]() {
		std::cout << this->name << ": " << message << "\n";
	});
}

The problem is now easier to spot: in the original example, the name data member is not being captured directly by value, but is instead accessed through a copy of the this pointer to the WarningManager object for which WarningManager::add_warning is called. Since the lambda may be invoked at a point at which that object may no longer exist, the code above is a recipe for disaster. The lifetime of the lambda is independent from the lifetime of the WarningManager object which creates it, and the implicit replacement of name by this->name on the definition of the lambda means we can find ourselves debugging an obscure program crash.

A simple way to fix the problem just discussed is by being explicit about what we want: we want to capture name by value, so let's go ahead and make that very clear to everyone:

/* a global array of callable warning objects */
std::vector<std::function<void()>> warnings;

class WarningManager
{
public:

	...

	void add_warning(const std::string& message) const;

private:
	std::string name;
};

void WarningManager::add_warning(const std::string& message) const
{
	const std::string& manager_name = this->name;

	warnings.emplace_back([manager_name, message]() {
		std::cout << manager_name << ": " << message << "\n";
	});
}

Inside the capture list, the string this->name will be copied through its reference manager_name, and the lambda will therefore own a copy of this->name under the name manager_name. In C++14, this code can be simplified using the init capture capability which was added to lambda functions:

/* a global array of callable warning objects */
std::vector<std::function<void()>> warnings;

class WarningManager
{
public:

	...

	void add_warning(const std::string& message) const;

private:
	std::string name;
};

void WarningManager::add_warning(const std::string& message) const
{
	warnings.emplace_back([manager_name = this->name, message]() {
		std::cout << manager_name << ": " << message << "\n";
	});
}

In this case, we are explicitly coping this->name into a string called manager_name which is then accessible inside the lambda function. As discussed in a previous post, lambda functions are equivalent to functor classes, and in this case, manager_name is a data member of such a class which is initialized as a copy of this->name.

To close this post, I strongly recommend you read the Zen of Python. Look at the second guiding principle: "Explicit is better than implicit". After reading this post, I hope you can better appreciate what a wise statement that is! :-)

Comments (0) Direct link

How are virtual function table pointers initialized?


Posted by Diego Assencio on 2017.03.07 under Programming (C/C++)

A class declaring or inheriting at least one virtual function contains a virtual function table (or vtable, for short). Such a class is said to be a polymorphic class. An object of a polymorphic class type contains a special data member (a "vtable pointer") which points to the vtable of this class. This pointer is an implementation detail and cannot be accessed directly by the programmer (at least not without resorting to some low-level trick). In this post, I will assume the reader is familiar with vtables on at least a basic level (for the uninitiated, here is a good place to learn about this topic).

I hope you learned that when you wish to make use of polymorphism, you need to access objects of derived types through pointers or references to a base type. For example, consider the code below:

#include <iostream>

struct Fruit
{
	virtual const char* name() const
	{
		return "Fruit";
	}
};

struct Apple: public Fruit
{
	virtual const char* name() const override
	{
		return "Apple";
	}
};

struct Banana: public Fruit
{
	virtual const char* name() const override
	{
		return "Banana";
	}
};

void analyze_fruit(const Fruit& f)
{
	std::cout << f.name() << "\n";
}

int main()
{
	Apple a;
	Banana b;

	analyze_fruit(a);   /* prints "Apple" */
	analyze_fruit(b);   /* prints "Banana" */

	return 0;
}

So far, no surprises here. But what will happen if instead of taking a reference to a Fruit object on analyze_fruit, we take a Fruit object by value?

Any experienced C++ developer will immediately see the word "slicing" written in front of their eyes. Indeed, taking a Fruit object by value means that inside analyze_fruit, the object f is truly a Fruit, and never an Apple, a Banana or any other derived type:

/* same code as before... */

void analyze_fruit(Fruit f)
{
	std::cout << f.name() << "\n";
}

int main()
{
	Apple a;
	Banana b;

	analyze_fruit(a);   /* prints "Fruit" */
	analyze_fruit(b);   /* prints "Fruit" */

	return 0;
}

This situation is worth analyzing in further detail, even if it seems trivial at first. On the calls to analyze_fruit, we pass objects of type Apple and Banana as arguments which are used to initialize its parameter f (of type Fruit). This is a copy initialization, i.e., the initialization of f in both of these cases is no different from the way f is initialized on the code fragment below:

Apple a;
Fruit f(a);

Even though Fruit does not define a copy constructor, one is provided by the compiler. This default copy constructor merely copies each data member of the source Fruit object into the corresponding data member of the Fruit object being created. In our case, Fruit has no data members, but it still has a vtable pointer. How is this pointer initialized? Is it copied directly from the input Fruit object? Before we answer these questions, let us look at what the compiler-generated copy constructor of Fruit looks like:

struct Fruit
{
	/* compiler-generated copy constructor */
	Fruit(const Fruit& sf): vptr(/* what goes in here? */)
	{
		/* nothing happens here */
	}

	virtual const char* name() const
	{
		return "Fruit";
	}
};

The signature of the Fruit copy constructor shows that is takes a reference to a source Fruit object, which means if we pass an Apple object to the copy constructor of Fruit, the vtable pointer of sf (for "source fruit"), will really point to the vtable of an Apple object. In other words, if this vtable pointer is directly copied into the vtable pointer of the Fruit object being constructed (represented under the name vptr on the code above), this object will behave like an Apple whenever any of its virtual functions are called!

But as we mentioned on the second code example above (the one in which analyze_fruit takes a Fruit object by value), the Fruit parameter f always behaves as a Fruit, and never as an Apple or as a Banana.

This brings us to the main lesson of this post: vtable pointers are not common data members which are directly copied or moved by copy and move constructors respectively. Instead, they are always initialized by any constructor used to build an object of a polymorphic class type T with the address of the vtable for the T class. Also, assignment operators will never touch the values stored by vtable pointers. In the context of our classes, the vtable pointer of a Fruit object will be initialized by any constructor of Fruit with the address of the vtable for the Fruit class and will retain this value throughout the entire lifetime of the object.

Comments (0) Direct link