|
|
左手坐標系的直觀表示：  向量的表示（軸對齊包圍盒（AABB（axially aligned bounding box）））：  2D向量的長度： ||v|| = sqrt(vx*vx + vy*vy)3D向量的長度： ||v|| = sqrt(vx*vx + vy*vy + vz*vz)
標準化向量＝此向量/此向量的長度＝vx / ||v|| , vy / ||v|| , vz / ||v||
標準化后的向量的頭接觸到圓心在原點的單位圓（單位圓的半徑為1）
向量加法的幾何意義（三角形法則）：
計算一個點到另一個點的位移可以使用三角形法則和向量減法來解決這個問題:
兩點間距離的公式：設兩點分別為3D向量a和b
則距離（a，b）=||b-a||=sqrt((bx - ax)2 +(by - ay)2 +(bz - az)2 )
向量點乘：點乘等于向量大小與向量夾角的cos值的積，其中c為兩點a和b的夾角：
a點乘b=||a||*||b||*cos(c)
計算向量的夾角：c =acos( (a*b)/(||a|| * ||b||))
如果a和b是單位向量，則c=acos(a*b)
當點乘結果大于0，則夾角小于90度
當點乘結果等于0，則夾角等于90度
當點乘結果小于0，則夾角大于90度
向量差乘：
向量的封裝類：
  /**////////////////////////////////////////////////////////////////////////////// //
// 3D Math Primer for Games and Graphics Development
//
// Vector3.h - Declarations for 3D vector class
//
// Visit gamemath.com for the latest version of this file.
//
// For additional comments, see Chapter 6.
//
/**//////////////////////////////////////////////////////////////////////////////
#ifndef __VECTOR3_H_INCLUDED__
#define __VECTOR3_H_INCLUDED__
#include <math.h>
/**////////////////////////////////////////////////////////////////////////////////
// class Vector3 - a simple 3D vector class
//
/**//////////////////////////////////////////////////////////////////////////////
class Vector3  {
 public:
// Public representation: Not many options here.
float x,y,z;
// Constructors
// Default constructor leaves vector in
// an indeterminate state
  Vector3() {}
// Copy constructor
Vector3(const Vector3 &a) : x(a.x), y(a.y), z(a.z) {}
// Construct given three values
Vector3(float nx, float ny, float nz) : x(nx), y(ny), z(nz) {}
// Standard object maintenance
// Assignment. We adhere to C convention and
// return reference to the lvalue
Vector3 &operator =(const Vector3 &a) {
x = a.x; y = a.y; z = a.z;
return *this;
 }
// Check for equality
bool operator ==(const Vector3 &a) const {
return x==a.x && y==a.y && z==a.z;
}
bool operator !=(const Vector3 &a) const {
return x!=a.x || y!=a.y || z!=a.z;
}
// Vector operations
// Set the vector to zero
void zero() { x = y = z = 0.0f; }
// Unary minus returns the negative of the vector
Vector3 operator -() const { return Vector3(-x,-y,-z); }
// Binary + and - add and subtract vectors
Vector3 operator +(const Vector3 &a) const {
return Vector3(x + a.x, y + a.y, z + a.z);
}
Vector3 operator -(const Vector3 &a) const {
return Vector3(x - a.x, y - a.y, z - a.z);
}
// Multiplication and division by scalar
Vector3 operator *(float a) const {
return Vector3(x*a, y*a, z*a);
}
Vector3 operator /(float a) const {
float oneOverA = 1.0f / a; // NOTE: no check for divide by zero here
return Vector3(x*oneOverA, y*oneOverA, z*oneOverA);
}
// Combined assignment operators to conform to
// C notation convention
Vector3 &operator +=(const Vector3 &a) {
x += a.x; y += a.y; z += a.z;
return *this;
}
Vector3 &operator -=(const Vector3 &a) {
x -= a.x; y -= a.y; z -= a.z;
return *this;
}
Vector3 &operator *=(float a) {
x *= a; y *= a; z *= a;
return *this;
}
Vector3 &operator /=(float a) {
float oneOverA = 1.0f / a;
x *= oneOverA; y *= oneOverA; z *= oneOverA;
return *this;
}
// Normalize the vector
void normalize() {
float magSq = x*x + y*y + z*z;
if (magSq > 0.0f) { // check for divide-by-zero
float oneOverMag = 1.0f / sqrt(magSq);
x *= oneOverMag;
y *= oneOverMag;
z *= oneOverMag;
}
}
// Vector dot product. We overload the standard
// multiplication symbol to do this
float operator *(const Vector3 &a) const {
return x*a.x + y*a.y + z*a.z;
}
 };
/**////////////////////////////////////////////////////////////////////////////////
// Nonmember functions
//
/**//////////////////////////////////////////////////////////////////////////////
// Compute the magnitude of a vector
inline float vectorMag(const Vector3 &a) {
return sqrt(a.x*a.x + a.y*a.y + a.z*a.z);
}
// Compute the cross product of two vectors
inline Vector3 crossProduct(const Vector3 &a, const Vector3 &b) {
return Vector3(
a.y*b.z - a.z*b.y,
a.z*b.x - a.x*b.z,
a.x*b.y - a.y*b.x
);
}
// Scalar on the left multiplication, for symmetry
inline Vector3 operator *(float k, const Vector3 &v) {
return Vector3(k*v.x, k*v.y, k*v.z);
}
// Compute the distance between two points
inline float distance(const Vector3 &a, const Vector3 &b) {
float dx = a.x - b.x;
float dy = a.y - b.y;
float dz = a.z - b.z;
return sqrt(dx*dx + dy*dy + dz*dz);
}
// Compute the distance between two points, squared. Often useful
// when comparing distances, since the square root is slow
inline float distanceSquared(const Vector3 &a, const Vector3 &b) {
float dx = a.x - b.x;
float dy = a.y - b.y;
float dz = a.z - b.z;
return dx*dx + dy*dy + dz*dz;
}
/**////////////////////////////////////////////////////////////////////////////////
// Global variables
//
/**//////////////////////////////////////////////////////////////////////////////
// We provide a global zero vector constant
extern const Vector3 kZeroVector;
/**//////////////////////////////////////////////////////////////////////////////#endif // #ifndef __VECTOR3_H_INCLUDED__
|