四元数乘法的几何意义是什么？

Question

我目前正在编写一个Arcball (用于模型视图相机)，我正在研究MS DXUT代码，它们有一个Arcball实现，如下所示

#include "ArcBall.h"

ArcBall::ArcBall(void)
{
        Reset() ;
        m_vDownPt = D3DXVECTOR3( 0, 0, 0 ) ;
        m_vCurrentPt = D3DXVECTOR3( 0, 0, 0 ) ;
        m_Offset.x = 0 ;
        m_Offset.y = 0 ;
}

ArcBall::~ArcBall(void)
{
}

void ArcBall::Reset()
{
        D3DXQuaternionIdentity( &m_qDown );
        D3DXQuaternionIdentity( &m_qNow );
        D3DXMatrixIdentity( &m_mRotation );
        m_bDrag = FALSE;
        m_fRadius = 1.0f;

        RECT rc ;
        GetClientRect(GetForegroundWindow(), &rc) ;
        SetWindow(rc.right, rc.bottom) ;
}

void ArcBall::OnBegin(int nX, int nY)
{
        // enter drag state only if user click the window's client area
        if( nX >= m_Offset.x && 
                nX <= m_Offset.x + m_nWidth &&
                nY >= m_Offset.y &&
                nY < m_Offset.y + m_nHeight )
        {
                m_bDrag = true ; // begin drag state
                m_qDown = m_qNow ;
                m_vDownPt = ScreenToVector((float)nX, (float)nY) ;
        }
}

void ArcBall::OnMove(int nX, int nY)
{
        if(m_bDrag)
        {
                m_vCurrentPt = ScreenToVector((float)nX, (float)nY) ;
                m_qNow = m_qDown * QuatFromBallPoints( m_vDownPt, m_vCurrentPt ) ;
        }
}

void ArcBall::OnEnd()
{
        m_bDrag = false ;
}

void ArcBall::SetOffset( INT nX, INT nY )
{
        m_Offset.x = nX ; 
        m_Offset.y = nY ;
}

void ArcBall::SetWindow( int nWidth, int nHeight, float fRadius)
{
         m_nWidth = nWidth; 
         m_nHeight = nHeight; 
         m_fRadius = fRadius; 
         m_vCenter = D3DXVECTOR2(m_nWidth / 2.0f, m_nHeight / 2.0f);
}

const D3DXMATRIX* ArcBall::GetRotationMatrix()
{
        return D3DXMatrixRotationQuaternion(&m_mRotation, &m_qNow) ;
}

D3DXQUATERNION ArcBall::QuatFromBallPoints(const D3DXVECTOR3 &vFrom, const D3DXVECTOR3 &vTo)
{
        D3DXVECTOR3 vPart;
        float fDot = D3DXVec3Dot( &vFrom, &vTo );       // rotation angle
        D3DXVec3Cross( &vPart, &vFrom, &vTo );          // rotation axis

        return D3DXQUATERNION( vPart.x, vPart.y, vPart.z, fDot );
}

D3DXVECTOR3 ArcBall::ScreenToVector(float fScreenPtX, float fScreenPtY)
{
        // Scale to screen
        // xֵΪ¸º£¬ÊÇÒòΪDirectXʹÓÃ×óÊÖϵ
        FLOAT x = -( fScreenPtX - m_Offset.x - m_nWidth / 2 ) / ( m_fRadius * m_nWidth / 2 );
        FLOAT y = ( fScreenPtY - m_Offset.y - m_nHeight / 2 ) / ( m_fRadius * m_nHeight / 2 );

        FLOAT z = 0.0f;
        FLOAT mag = x * x + y * y;

        if( mag > 1.0f )
        {
                FLOAT scale = 1.0f / sqrtf( mag );
                x *= scale;
                y *= scale;
        }
        else
                z = sqrtf( 1.0f - mag );

        // Return vector
        return D3DXVECTOR3( x, y, z );
}

在函数移动中，我们可以看到两个四元数的乘法

m_qNow = m_qDown * QuatFromBallPoints( m_vDownPt, m_vCurrentPt ) ; 

function QuatFromBallPoints根据两个点计算四元数的底数。我想知道这个产品是什么意思？

代码说明:当用户拖动鼠标时，当鼠标按键按下(假设是左键)时，OnBegin函数被调用。当鼠标按键打开时，调用OnEnd函数；当用户拖动鼠标时，调用OnMove函数。

Answer 1

乘法是指旋转的组合。QuatFromBallPoints返回将m_vDownPt映射到m_vCurrentPt的旋转。此新旋转必须与arcball的当前旋转(开始拖动时的旋转)组合在一起。