The calculation is described by “quatermion” which is frequently used in computer graphics. Quatermion provides the simplest means for coordinate translation and reduces the calculation cost. A quatemion $q$ is defined as follows. Display Formula
$q=a+ix+jy+kz=(a;x,y,z)=(a;v),i2=j2=k2=ijk=\u22121.$
Here, $\nu $ is a point or vector in 3-D coordinates. The conjugation of $q$ is described as $q\xaf$. Although the real number $a$ is arbitrary, $a=0$ is the simplest description. For the translation to be applied, two conditions must be satisfied. The first condition is that the oriented vector in the interaction coordinates must fit that in the simulation coordinates. The second is that the incident point and vector in the interaction coordinates must also fit those in the simulation coordinates. For the first condition, we use Eq. (8). $UpciMC(x,y,z)=(0;Ux,Uy,Uz)$ is translated to $UpciMC(r,s,t)=(0;Ur,Us,Ut)$ by the following process: Display Formula$UpciMC(r,s,t)=roll1\xb7UpciMC(x,y,z)\xb7roll1\xafroll1=(cos\u2009\theta 12;sin\u2009\theta 12\xb7y),cos\u2009\theta 1=oV\xb7xoVModel1(x,y,z)=(0,1,0)=yoVModel2(x,y,z)=(\u2212x,0,z0+z12\u2212z)|(\u2212x,0,z0+z12\u2212z)|oVModel3(x,y,z)=(z0+z12\u2212z,0,x)|(z0+z12\u2212z,0,x)|,$
where $oV$ is the vector in the direction of orientation. The vectors $x=(1,0,0)$ and $y=(0,1,0)$ are the unit vectors in the $x$- and $y$-directions in pciMC simulation coordinates. After the incident point $Pn$ and the vector $\chi \u2192m=(0;\chi x,\chi y,\chi z)$ in the photon-RBC interaction coordinates have been selected by Eq. (9), they are rotated around the $z$-axis so that $\chi x$ becomes $Ur$. The translated incident vector $tiV$ and the rotation $roll2$ are described by the following equations: Display Formula$tiV=(tiVx,tiVy,tiVz)=(0;Ur,\xb1(Us2+Ut2)\u2212\chi z2,\chi z)roll2=(cos\u2009\theta 22;sin\u2009\theta 22\xb7\chi XY\xd7tiVXY)cos\u2009\theta 2=\chi XY\xb7tiVXY\chi XY=(0;\chi x,\chi y,0)tiVXY=(0;tiVx,tiVy,0),$
where $tiVy$ is randomly selected to be either positive or negative so that the photon azimuthal angle is uniformly distributed. Next, for the second condition, the translated point and vector are additionally rotated so that $tiV$ becomes $UpciMC(r,s,t)$. The rotation is described as follows: Display Formula$roll3=(cos\u2009\theta 32;sin\u2009\theta 32\xb7UST\xd7tiVYZ)cos\u2009\theta 3=UST\xb7tiVYZUST=(0;0,Us,Ut)tiVYZ=(0;0,tiVy,tiVz).$
Finally, using these equations, the scattered point $sPpciMC(x,y,z)$ and the vector $sUpciMC(x,y,z)$ are calculated as follows. Display Formula$sPpciMC(x,y,z)=s(roll1\xafroll3roll2(Mk\u2212Pn)roll2\xafroll3\xafroll1)+iPpciMC(x,y,z)sUpciMC(x,y,z)=roll1\xafroll3roll2Nkroll2\xafroll3\xafroll1,$
where $iPpciMC(x,y,z)$ is the incident point prior to the scattering event, $s$ is the scale parameter to fit the scale of the interaction coordinates with those of the simulation coordinates.