标签: vector

Vector是C语言里面没有的概念。 Vector是Container（容器）的一种，有关容器更多的留在后面再说。 要使用Vector，首先要包含头文件，并且声明命名空间： #include using std::vector; Vector是一个类的模板（template）。 C++有类的模板，也有函数的模板。以后会学习怎样创建模板，现在只需要了解如何使用它。 模板自身并不是类或者函数，它们是用来创建类或者函数的指令。这个过程称为实例化。 为了将模板实例化，我们需要提供额外的信息。 为了将Vector实例化，我们需要提供的额外信息是Vector持有的对象类型： vector ivec; vector sales_vec; vector file;  //旧版的C++需要写成 我们可以创建任何类型的Vector，整型、类或者Vector自身。 引用并不是对象，因此不能为引用创建Vector。 有很多种方法可以初始化Vector： vector v1 vector v2(v1) vector v2 = v1 vector v3(n, val) vector v4(n) vector v5{a,b,c...} vector v5 = {a,b,c...} vector v1；//它定义了一个空的向量 空向量看起来没什么用，但是我们可以动态的往向量里面添加内容。 实际上，大部分使用都是这么做的。 Vector初始化的时候，使用括号（parentheses）和使用大括号（curly braces）， 有非常有意思的区别。编译器会聪明的配置它们。 ---------------- 如何为Vector增加元素？ 使用push_back增加新的最后一个。 先来一个整型数的例子： vector v2; for (int i = 0; i != 100; i++)     v2.push_back(i); 再来一个string的例子： string word; vector text; while (cin word)     text.push_back(word); 由于Vector的长度可变，因此在使用range for的时候，不要在循环里面增加它的长度。 ---------------- v.empty() v.size() v.push_back() v v1 = v2 v1 = {a,b,c...} v1 == v2 v1 != v2 ,=,,= 这些操作和string类似。 size_type的用法也和string类似，除了vector必须指定元素类型： vector::size_type 同string一样，vector也可以使用下标来访问元素； 同string一样，vector的下标越界也是不会被检查的。

  A 3×3 matrix class? It's this character flaw that keeps me from being able to show you a completed 3×3 matrix class. Until recently, I've been stuck in one of those while-loops, and this time the ex-wife isn't here to assert the interrupt. I wasn't able to exit the loop until I began writing this column. I'd like to tell you what I have, and see what you think.   Clearly, a 3×3 matrix is not just any old matrix, even any old matrix with one dimension of three. It's a square matrix, and square matrices have special properties. For one thing, they have well-defined diagonals. A square matrix might, in fact, be diagonal. That is, its off-diagonal elements may all be zero. That's important, because a diagonal matrix has a trivial inverse: A square matrix might also be symmetric. If so, it can easily be transformed to a diagonal matrix. When I'm doing arithmetic with 3×3 matrices, they're almost always associated with coordinate rotations. For example, to convert a vector from a rotating coordinate system to a fixed one, I might write:   The matrix T is even more specialized than most. It's a rotation matrix , also called an orthonormal matrix . It has two very important attributes: Its determinant is 1, and its inverse is the same as its transpose:   This is very important, because a transpose of a 3×3 matrix is trivial to generate, whereas an inverse takes a lot longer. Also, because the determinant is known, we don't have to worry about the matrix being singular.   Finally, there exist more than one way to represent a rotation. The matrix is one choice, but not the only one, or even the best one. Other choices include Euler angles and quaternions. Euler angles are useful because people tend to be able to visualize the rotations better. But they're terrible choices for computations. Quaternions are the best and most compact, but good luck trying to visualize them.   Side Comment: You know you've arrived as a guru when you realize you can visualize a quaternion. Since it's a four-dimensional vector, that takes a little practice.   Decisions, decisions Now that I've given you the background, I can tell you why I was stuck in the while-loop. I know that I want to use 3×3 matrices mostly to effect coordinate rotations. But is that always going to be the case? Maybe not. And if it is, may I include constructors to convert a set of 3-vectors to a matrix? Or a set of Euler angles, or quaternions? If I demand that the matrix must be orthonormal, I'd better not try to create one from vectors.   For that matter, if I'm dealing exclusively with rotation matrices, maybe I need a function to make sure they stay orthonormal.   Here's why. When we're writing dynamic simulations involving rotations, we have to numerically integrate the elements that make up the state vector. If one of those elements is a rotation matrix (or equivalent structure), numerical roundoff errors can cause the matrix to drift away from its nominally orthonormal state. Those of us who write such simulations often include functions to re-normalize the matrix. It's not an easy thing to do, and doing it in an optimal way is even harder.   So a re-normalizing function is going to be almost a necessity for rotation matrices, but it's going to look really, really strange in a general-purpose matrix package—even one specialized to 3×3 matrices.   Of course, you know the conceptual solution to problems such as this: inheritance. Write a C++ base class for 3×3 matrices. Then define a derived class that specializes the class even further, to rotation matrices. Most of the operations will be the same, but a few functions, such as re-normalization, can be added, and the inverse function can be changed to invoke the simpler transpose function.   There's only one problem with this approach. A rotation in 3-space is a very unique thing, and the function that it performs is vastly different from the more generic matrix product. Heck, the member data for a "rotation matrix" need not even include a matrix. Rotations can also be described by a quaternion or a set of Euler angles.   And because there are three possible ways to describe the rotation, we're also going to need conversion functions to transform one set of descriptors to another.   FYI, a couple of years ago I wrote a library of Matlab functions capable of converting between angle, matrix, and quaternion representations. In doing so, I even surprised myself, coming up with killer algorithms that resulted in code that was tight, accurate, and bulletproof. I'll be sharing the algorithms with you soon.   Most people who work every day with rotation-related problems choose the quaternion, because it's more efficient to do so (no angles involved, and fewer elements to store). One has to ask: what's the point having a rotation matrix be a derived class of the 3×3 matrix, if it doesn't even hold a matrix anywhere in its structure?   Off and on, I've wrestled with questions like this many times. I can't say that I've ever come up with the definitive solution. More often, I find myself in need of that priority interrupt.   But this time, I've finally got the structure I need. The trick is, there should be no class called RotationMatrix . Instead, there's a class called Rotation . When applied to a vector or another matrix (or Rotation ), the operation may look like a matrix product, but the appearance is only superficial. Internally, a Rotation should be a separate type of object, worthy of its own class. Heck, it may not—and probably won't—even have a matrix as part of its member data. And in the best tradition of object-oriented programming, the way the rotation class is implemented should be transparent to the user. If I decide to change the internal representation from matrix to quaternion, that decision shouldn't matter one iota to users of the class.   When you think about it, the fact that I can define a rotation operation that looks like a matrix product is no different than defining products of integers or real numbers. I can declare:   int i, j, k; double x, y, z; Vector X, Y; Matrix A; Rotation R;   and write:   k = i*j; z = x*y; Y = A*X; Y = R*X;   Yes, the statements look the same and mean much the same. That's the elegance of operator overloading, and it makes life incredibly easy for the programmer. But we have no particular reason for a rotation object to inherit the '*' operator from a matrix, any more than we might have it do so from class int . Separating the class Rotation into a full-blown separate class is beginning to make a lot of sense.   Wrapping up I'm glad we had this little talk. So often in life, I've found that as I try to explain a knotty problem to someone else, I realize that I suddenly know the solution. That seems to be what has happened here. Thanks for listening.