点积叉乘的运用与原理(整理)

大二的时候特意花了几天时间看点积叉乘,时间一久又忘了!!!这一次把之前看的代码和找的资料收集起来,下次再忘了就直接看了！！
叉乘的公式记忆技巧
x1 y1 z1
x2 y2 z2
x:【y1,开始小叉】y:【左上与右下大叉】z:【x1开始,小叉】
叉乘的意义:可以判断方向性,可以算面积

//我们是以A点作为dist的判断点
inline bool LineIntersection2D(Vector2D   A,
                               Vector2D   B,
                               Vector2D   C, 
                               Vector2D   D,
                               double&     dist,
                               Vector2D&  point)
{

  	double rTop = (A.y-C.y)*(D.x-C.x)-(A.x-C.x)*(D.y-C.y);
	double rBot = (B.x-A.x)*(D.y-C.y)-(B.y-A.y)*(D.x-C.x);

	double sTop = (A.y-C.y)*(B.x-A.x)-(A.x-C.x)*(B.y-A.y);
	double sBot = (B.x-A.x)*(D.y-C.y)-(B.y-A.y)*(D.x-C.x);
	//上面4个判断相交性,其实只要最上面两个就可以算出来坐标点了
	//下面两个是用来判断相交的

	if ( (rBot == 0) || (sBot == 0))
	{
		//lines are parallel
		return false;
	}
	//cal scal
	double r = rTop/rBot;
	double s = sTop/sBot;

	if( (r > 0) && (r < 1) && (s > 0) && (s < 1) )
  {
  	dist = Vec2DDistance(A,B) * r;

    point = A + r * (B - A);

    return true;
  }
	else
  {
		dist = 0;
    return false;
  }
}

这是两张推导的图片具体的出处找不到了,写的非常好(如果有时间建议自己推导一下,仿佛像回到中学的时代)(感谢作者)
第一张是算比例的:

算交点的推导

强烈建议跟着推导一下

反射向量

这个博主写的简单明了,因为怕作者的链接失效截个图保存一下.

代码实现

inline void Vector2D::Reflect(const Vector2D& norm)
{
  *this += 2.0 * this->Dot(norm) * norm.GetReverse();
}

反射矩阵的推导

世界空间转局部空间

这个花了很多时间去理解逆矩阵的空间变换,ABA.当时觉得这辈子都忘不了结果还是忘了…
以下代码是2d的,所以没有用专门的逆矩阵,如果以后需要用到相关的东西,可以由下面的函数得到启发,3d是4x4的不能这么写!!!

//--------------------- PointToLocalSpace --------------------------------
//
//------------------------------------------------------------------------
inline Vector2D PointToLocalSpace(const Vector2D &point,
                             Vector2D &AgentHeading,
                             Vector2D &AgentSide,
                             Vector2D &AgentPosition)
{

	//make a copy of the point
  Vector2D TransPoint = point;
  
  //create a transformation matrix
	C2DMatrix matTransform;

  double Tx = -AgentPosition.Dot(AgentHeading);
  double Ty = -AgentPosition.Dot(AgentSide);

  //create the transformation matrix
  matTransform._11(AgentHeading.x); matTransform._12(AgentSide.x);
  matTransform._21(AgentHeading.y); matTransform._22(AgentSide.y);
  matTransform._31(Tx);           matTransform._32(Ty);
	
  //now transform the vertices
  matTransform.TransformVector2Ds(TransPoint);

  return TransPoint;
}

//--------------------- VectorToLocalSpace --------------------------------
//
//------------------------------------------------------------------------
inline Vector2D VectorToLocalSpace(const Vector2D &vec,
                             const Vector2D &AgentHeading,
                             const Vector2D &AgentSide)
{ 

	//make a copy of the point
  Vector2D TransPoint = vec;
  
  //create a transformation matrix
	C2DMatrix matTransform;

  //create the transformation matrix
  matTransform._11(AgentHeading.x); matTransform._12(AgentSide.x);
  matTransform._21(AgentHeading.y); matTransform._22(AgentSide.y);
	
  //now transform the vertices
  matTransform.TransformVector2Ds(TransPoint);

  return TransPoint;
}