Sergei Shaykin (original) (raw)

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�� XVIII ��. �� 4-� �� , �� . �� q �� (x, y, z, w):

w + xi + yj + zk
, �� i2 = j2 = k2 = �1.

�� :

[w,v]
, �� w �� , � v = (x, y, z) � ��.

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�� : q[w,v] � q'[w',v']. �� :

q+q' = [w+w',v+v']

qq' = [ww' � v�v', vxv' + wv' + w'v]
, �� x - �� , � � � ��.

norm(q) = sqrt(w2+x2+y2+z2) (��)

q*=[w, �v] (��)

\(q^{-1}=\frac {q*} {norm(q)^2}\)

norm(q)=1 => q�1=q*

�� . �� (norm(q) = 1). �� , �� , �� norm(q) � �� .

�� 4-� �� . �� 4-� �� 1. �� , �� , (�� - ��, ��, �� ).

�� q, �� v �� v':

V'=q V q�1
, �� V = [o,v], V'=[0,v']

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�� Q �� , �� v. �� :
q=[cos(Q/2),sin(Q/2)v]

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struct Quaternion{ float x,y,z; float w;
};

void SphericalToQuaternion(Quaternion * q, float latitude, float longitude, float angle) { float sin_a = sin( angle / 2 ); float cos_a = cos( angle / 2 );

float sin_lat = sin( latitude ); float cos_lat = cos( latitude );

float sin_long = sin( longitude ); float cos_long = cos( longitude );

q->x = sin_a * cos_lat * sin_long; q->y = sin_a * sin_lat; q->z = sin_a * sin_lat * cos_long; q->w = cos_a; }

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�� yaw, pitch � roll. �� :

qroll = [cos (y/2), (sin(y/2), 0, 0)]
qpitch = [cos (q/2), (0, sin(q/2), 0)]
qyaw = [cos (f/2), (0, 0, sin(f/2))]
q = qyaw qpitch qroll

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�� 3x3. ��, �� 3D API �� 4x4, �� . �� , �� , �� , �� , �� 1. �� . ��, �� , �� .

void MatrixToQuaternion(Quaternion * quat, float m[4][4]) { float tr, s, q[4]; int i, j, k;

int nxt[3] = {1, 2, 0};

tr = m[0][0] + m[1][1] + m[2][2];

if (tr > 0.0) { s = sqrt (tr + 1.0); quat->w = s / 2.0; s = 0.5 / s; quat->x = (m[1][2] - m[2][1]) * s; quat->y = (m[2][0] - m[0][2]) * s; quat->z = (m[0][1] - m[1][0]) * s; } else { i = 0; if (m[1][1] > m[0][0]) i = 1; if (m[2][2] > m[i][i]) i = 2; j = nxt[i]; k = nxt[j];

s = sqrt ((m[i][i] - (m[j][j] + m[k][k])) + 1.0);

q[i] = s * 0.5;

if (s != 0.0) s = 0.5 / s;

q[3] = (m[j][k] - m[k][j]) * s;
q[j] = (m[i][j] + m[j][i]) * s;
q[k] = (m[i][k] + m[k][i]) * s;

quat->x = q[0];
quat->y = q[1];
quat->z = q[2];
quat->w = q[3];

} }

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TRot = [ 1-2y2-2z2 2xy-2wz 2xz+2wy
2xy+2wz 1-2x2-2z2 2yz-2wx
2xz-2wy 2yz+2wx 1-2x2-2y2 ]

void QuaternionToMatrix(float m[4][4], const Quaternion * quat) { float wx, wy, wz, xx, yy, yz, xy, xz, zz, x2, y2, z2; x2 = quat->x + quat->x; y2 = quat->y + quat->y; z2 = quat->z + quat->z; xx = quat->x * x2; xy = quat->x * y2; xz = quat->x * z2; yy = quat->y * y2; yz = quat->y * z2; zz = quat->z * z2; wx = quat->w * x2; wy = quat->w * y2; wz = quat->w * z2;

m[0][0]=1.0f-(yy+zz); m[0][1]=xy-wz; m[0][2]=xz+wy; m[1][0]=xy+wz; m[1][1]=1.0f-(xx+zz); m[1][2]=yz-wx; m[2][0]=xz-wy; m[2][1]=yz+wx; m[2][2]=1.0f-(xx+yy);

m[0][3] = m[1][3] = m[2][3] = 0; m[3][0] = m[3][1] = m[3][2] = 0; m[3][3] = 1; }

�� . � �� , �� . �� , �� R1 � R2, �� : R = R1 x R2. ��, �� q1 � q2, �� q = q1 x q2. ��, �� , �� . �� :

void MulQuaternions(Quaternion *res, const Quaternion *q1, const Quaternion *q2) { float A, B, C, D, E, F, G, H;

A = (q1->w + q1->x) * (q2->w + q2->x); B = (q1->z - q1->y) * (q2->y - q2->z); C = (q1->x - q1->w) * (q2->y + q2->z); D = (q1->y + q1->z) * (q2->x - q2->w); E = (q1->x + q1->z) * (q2->x + q2->y); F = (q1->x - q1->z) * (q2->x - q2->y); G = (q1->w + q1->y) * (q2->w - q2->z); H = (q1->w - q1->y) * (q2->w + q2->z);

res->w = B + (-E - F + G + H) * 0.5; res->x = A - ( E + F + G + H) * 0.5; res->y =-C + ( E - F + G - H) * 0.5; res->z =-D + ( E - F - G + H) * 0.5; }

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5 �� 2001 (��: 29 �� 2013)

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