Quaternion(四元數)和旋轉
作者: 劉鵬
日期: 2011-06-10
本文介紹了四元數以及如何在OpenGL中使用四元數表示旋轉。
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Quaternion 的定義四元數一般定義如下: q=w+xi+yj+zk 其中 w,x,y,z是實數。同時,有: i*i=-1 j*j=-1 k*k=-1 四元數也可以表示爲: q=[w,v] 其中v=(x,y,z)是矢量,w是標量,雖然v是矢量,但不能簡單的理解爲3D空間的矢量,它是4維空間中的的矢量,也是非常不容易想像的。 通俗的講,一個四元數(Quaternion)描述了一個旋轉軸和一個旋轉角度。這個旋轉軸和這個角度可以通過 Quaternion::ToAngleAxis轉換得到。當然也可以隨意指定一個角度一個旋轉軸來構造一個Quaternion。這個角度是相對於單位四元數而言的,也可以說是相對於物體的初始方向而言的。 當用一個四元數乘以一個向量時,實際上就是讓該向量圍繞着這個四元數所描述的旋轉軸,轉動這個四元數所描述的角度而得到的向量。 四元組的優點1有多種方式可表示旋轉,如 axis/angle、歐拉角(Euler angles)、矩陣(matrix)、四元組等。 相對於其它方法,四元組有其本身的優點:
Quaternion 的基本運算1Normalizing a quaternion// normalising a quaternion works similar to a vector. This method will not do anything // if the quaternion is close enough to being unit-length. define TOLERANCE as something // small like 0.00001f to get accurate results void Quaternion::normalise() { // Don't normalize if we don't have to float mag2 = w * w + x * x + y * y + z * z; if ( mag2!=0.f && (fabs(mag2 - 1.0f) > TOLERANCE)) { float mag = sqrt(mag2); w /= mag; x /= mag; y /= mag; z /= mag; } } The complex conjugate of a quaternion// We need to get the inverse of a quaternion to properly apply a quaternion-rotation to a vector // The conjugate of a quaternion is the same as the inverse, as long as the quaternion is unit-length Quaternion Quaternion::getConjugate() { return Quaternion(-x, -y, -z, w); } Multiplying quaternions// Multiplying q1 with q2 applies the rotation q2 to q1 Quaternion Quaternion::operator* (const Quaternion &rq) const { // the constructor takes its arguments as (x, y, z, w) return Quaternion(w * rq.x + x * rq.w + y * rq.z - z * rq.y, w * rq.y + y * rq.w + z * rq.x - x * rq.z, w * rq.z + z * rq.w + x * rq.y - y * rq.x, w * rq.w - x * rq.x - y * rq.y - z * rq.z); } Rotating vectors// Multiplying a quaternion q with a vector v applies the q-rotation to v Vector3 Quaternion::operator* (const Vector3 &vec) const { Vector3 vn(vec); vn.normalise(); Quaternion vecQuat, resQuat; vecQuat.x = vn.x; vecQuat.y = vn.y; vecQuat.z = vn.z; vecQuat.w = 0.0f; resQuat = vecQuat * getConjugate(); resQuat = *this * resQuat; return (Vector3(resQuat.x, resQuat.y, resQuat.z)); } How to convert to/from quaternions1Quaternion from axis-angle// Convert from Axis Angle void Quaternion::FromAxis(const Vector3 &v, float angle) { float sinAngle; angle *= 0.5f; Vector3 vn(v); vn.normalise(); sinAngle = sin(angle); x = (vn.x * sinAngle); y = (vn.y * sinAngle); z = (vn.z * sinAngle); w = cos(angle); } Quaternion from Euler angles// Convert from Euler Angles void Quaternion::FromEuler(float pitch, float yaw, float roll) { // Basically we create 3 Quaternions, one for pitch, one for yaw, one for roll // and multiply those together. // the calculation below does the same, just shorter float p = pitch * PIOVER180 / 2.0; float y = yaw * PIOVER180 / 2.0; float r = roll * PIOVER180 / 2.0; float sinp = sin(p); float siny = sin(y); float sinr = sin(r); float cosp = cos(p); float cosy = cos(y); float cosr = cos(r); this->x = sinr * cosp * cosy - cosr * sinp * siny; this->y = cosr * sinp * cosy + sinr * cosp * siny; this->z = cosr * cosp * siny - sinr * sinp * cosy; this->w = cosr * cosp * cosy + sinr * sinp * siny; normalise(); } Quaternion to Matrix// Convert to Matrix Matrix4 Quaternion::getMatrix() const { float x2 = x * x; float y2 = y * y; float z2 = z * z; float xy = x * y; float xz = x * z; float yz = y * z; float wx = w * x; float wy = w * y; float wz = w * z; // This calculation would be a lot more complicated for non-unit length quaternions // Note: The constructor of Matrix4 expects the Matrix in column-major format like expected by // OpenGL return Matrix4( 1.0f - 2.0f * (y2 + z2), 2.0f * (xy - wz), 2.0f * (xz + wy), 0.0f, 2.0f * (xy + wz), 1.0f - 2.0f * (x2 + z2), 2.0f * (yz - wx), 0.0f, 2.0f * (xz - wy), 2.0f * (yz + wx), 1.0f - 2.0f * (x2 + y2), 0.0f, 0.0f, 0.0f, 0.0f, 1.0f) } Quaternion to axis-angle// Convert to Axis/Angles void Quaternion::getAxisAngle(Vector3 *axis, float *angle) { float scale = sqrt(x * x + y * y + z * z); axis->x = x / scale; axis->y = y / scale; axis->z = z / scale; *angle = acos(w) * 2.0f; } Quaternion 插值2線性插值最簡單的插值算法就是線性插值,公式如: q(t)=(1-t)q1 + t q2 但這個結果是需要規格化的,否則q(t)的單位長度會發生變化,所以 q(t)=(1-t)q1+t q2 / || (1-t)q1+t q2 || 球形線性插值儘管線性插值很有效,但不能以恆定的速率描述q1到q2之間的曲線,這也是其弊端,我們需要找到一種插值方法使得q1->q(t)之間的夾角θ是線性的,即θ(t)=(1-t)θ1+t*θ2,這樣我們得到了球形線性插值函數q(t),如下: q(t)=q1 * sinθ(1-t)/sinθ + q2 * sinθt/sineθ 如果使用D3D,可以直接使用 D3DXQuaternionSlerp 函數就可以完成這個插值過程。 用 Quaternion 實現 Camera 旋轉總體來講,Camera 的操作可分爲如下幾類:
下面是一個使用了 Quaternion 的 Camera 類: class Camera { private: Quaternion m_orientation; public: void rotate (const Quaternion& q); void rotate(const Vector3& axis, const Radian& angle); void roll (const GLfloat angle); void yaw (const GLfloat angle); void pitch (const GLfloat angle); }; void Camera::rotate(const Quaternion& q) { // Note the order of the mult, i.e. q comes after m_Orientation = q * m_Orientation; } void Camera::rotate(const Vector3& axis, const Radian& angle) { Quaternion q; q.FromAngleAxis(angle,axis); rotate(q); } void Camera::roll (const GLfloat angle) //in radian { Vector3 zAxis = m_Orientation * Vector3::UNIT_Z; rotate(zAxis, angleInRadian); } void Camera::yaw (const GLfloat angle) //in degree { Vector3 yAxis; { // Rotate around local Y axis yAxis = m_Orientation * Vector3::UNIT_Y; } rotate(yAxis, angleInRadian); } void Camera::pitch (const GLfloat angle) //in radian { Vector3 xAxis = m_Orientation * Vector3::UNIT_X; rotate(xAxis, angleInRadian); } void Camera::gluLookAt() { GLfloat m[4][4]; identf(&m[0][0]); m_Orientation.createMatrix (&m[0][0]); glMultMatrixf(&m[0][0]); glTranslatef(-m_eyex, -m_eyey, -m_eyez); } 用 Quaternion 實現 trackball用鼠標拖動物體在三維空間裏旋轉,一般設計一個 trackball,其內部實現也常用四元數。 class TrackBall { public: TrackBall(); void push(const QPointF& p); void move(const QPointF& p); void release(const QPointF& p); QQuaternion rotation() const; private: QQuaternion m_rotation; QVector3D m_axis; float m_angularVelocity; QPointF m_lastPos; }; void TrackBall::move(const QPointF& p) { if (!m_pressed) return; QVector3D lastPos3D = QVector3D(m_lastPos.x(), m_lastPos.y(), 0.0f); float sqrZ = 1 - QVector3D::dotProduct(lastPos3D, lastPos3D); if (sqrZ > 0) lastPos3D.setZ(sqrt(sqrZ)); else lastPos3D.normalize(); QVector3D currentPos3D = QVector3D(p.x(), p.y(), 0.0f); sqrZ = 1 - QVector3D::dotProduct(currentPos3D, currentPos3D); if (sqrZ > 0) currentPos3D.setZ(sqrt(sqrZ)); else currentPos3D.normalize(); m_axis = QVector3D::crossProduct(lastPos3D, currentPos3D); float angle = 180 / PI * asin(sqrt(QVector3D::dotProduct(m_axis, m_axis))); m_axis.normalize(); m_rotation = QQuaternion::fromAxisAndAngle(m_axis, angle) * m_rotation; m_lastPos = p; } Yaw, pitch, roll 的含義3Yaw – Vertical axis:
Pitch – Lateral axis
Roll – Longitudinal axis
The Position of All three axes
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