Up \( \bf{v} \) の \( (S_a, P_a) \) 表現 作成: 2022-06-26
更新: 2022-10-11

    \( {\bf v} \) は，\( \vec{OP},\ \vec{OT} \) とつぎの関係にある：
      \[ {\bf v} = v\ \frac{ \vec{OT} \times \vec{OP} }{ | \vec{OT} \times \vec{OP} |} \\ \]


    そして \[ T_x = 0 \\ T_y = - R\ sin( S_a ) \\ T_z = R\ cos( S_a ) \ \\ P_x = R\ cos( P_a ) \\ P_y = R\ sin( P_a )\ cos( S_a ) \\ P_z = R\ sin( P_a )\ sin( S_a ) \] なので， \[ \begin{align} & \vec{OT} \times \vec{OP} \\ \ \\ = & ( T_x,\ T_y,\ T_z ) \times (P_x,\ P_y,\ P_z )\\ \ \\ = & (T_y\ P_z - T_z\ P_y,\ T_z\ P_x - T_x\ P_z,\ T_x\ P_y - T_y\ P_x ) \\ \ \\ = & (- R\ sin( S_a )\ R\ sin( P_a )\ sin( S_a ) - R\ cos( S_a )\ R\ sin( P_a )\ cos( S_a ), \\ & R\ cos( S_a )\ R\ cos( P_a ) - 0\ R\ sin( P_a )\ sin( S_a ), \\ & 0\ R\ sin( P_a )\ cos( S_a ) + R\ sin( S_a )\ R\ cos( P_a ) ) \\ \ \\ = & (- R^2\ sin( P_a )\ ( ( sin( S_a ) )^2 + ( cos( S_a ) )^2 ), \\ & R\ cos( S_a )\ R\ cos( P_a ), \\ & R\ sin( S_a )\ R\ cos( P_a ) ) \\ \ \\ = & R^2\ ( - \ sin( P_a ), \\ & cos( S_a )\ cos( P_a ), \\ & sin( S_a )\ cos( P_a ) ) \\ ) \\ \end{align} \] また， \[ | \vec{OT} \times \vec{OP} | = | \vec{OT} |\ | \vec{OP} |\ sin( \pi / 2 ) = R^2 \\ \]
    よって， \[ v_x = - v\ sin( P_a ) \\ v_y = v\ cos( S_a )\ cos( P_a ) \\ v_z = v\ sin( S_a )\ cos( P_a ) \\ \]