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system-design/problem.md: fixed latex indentation issues
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‎system-design/problem.md‎

Lines changed: 73 additions & 80 deletions
Original file line numberDiff line numberDiff line change
@@ -82,34 +82,29 @@ The mathematical model is based on the vector rotational transformation model th
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* Let vector $\vec{V}$ be a vector in an $R^3$ vectorspace; such that:
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$$
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\begin{equation*}
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\vec{V} = \begin{pmatrix}
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\|\vec{V}\| \cdot \cos(\phi_{(\vec{V}, \vec{V}_x)}) \\
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\|\vec{V}_{yz}\| \cdot \cos\left(\frac{\pi}{2} - \phi_{(\vec{V}_{yz}, \vec{V}_z)}\right) \\
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\|\vec{V}_{yz}\| \cdot \cos(\phi_{(\vec{V}_{yz}, \vec{V}_z)})
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\end{pmatrix}
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\end{equation*}
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$$
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* Therefore, the rotated vector around the X-axis using angle $\phi_R$ with the positive direction of Z-axis (Z-convention) using the direction cosines methodology; could be rewritten as a function of the rotational angle by evaluating the addition trigonometric identity using the Euclidean Geometry:
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$$
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\begin{equation*}
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\vec{V}^{'} = \begin{pmatrix}
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\|\vec{V}\|\cos(\phi_{(\vec{V}, \vec{V}_x)}) \\
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\|\vec{V}_{yz}\|(\sin(\phi_{(\vec{V}_{yz}, \vec{V}_z)})\cos(\phi_{R}) \pm \cos(\phi_{(\vec{V}_{yz}, \vec{V}_z)})
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\sin(\phi_{R})) \\
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\|\vec{V}_{yz}\|(\cos(\phi_{(\vec{V}_{yz}, \vec{V}_z)})\cos(\phi_{R}) \mp \sin(\phi_{(\vec{V}_{yz}, \vec{V}_z)})
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\sin(\phi_{R})) \\
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\end{pmatrix}
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\end{equation*}
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$$
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* The rotated vector around the X-axis with Z-convention expressed in direction cosines; could be further rewritten using matrix algebra into a **matrix product of a rotation matrix and the vector column matrix**; where $R_x$ is the rotation matrix around X-Axis with Z-convetion:
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$$
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\begin{equation*}
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\vec{V}^{'} = R_x(\pm \hat{\Phi_R}) \times \vec{V} \\
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= \begin{pmatrix}
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1 & 0 & 0 \\
@@ -122,94 +117,92 @@ $$
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\|\vec{V}_{yz}\|\sin(\phi_{(\vec{V}_{yz}, \vec{V}_z)}) \\
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\|\vec{V}_{yz}\|\cos(\phi_{(\vec{V}_{yz}, \vec{V}_z)})
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\end{pmatrix}
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\\
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=
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\begin{pmatrix}
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1 & 0 & 0 \\
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0 & \cos(\phi_{R}) & \pm \sin(\phi_{R}) \\
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0 & \mp \sin(\Phi_R) & \cos(\phi_{R})
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\cos(\phi_{(\vec{V}, \vec{V}_x)}) \\
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\sin(\phi_{(\vec{V}_{yz}, \vec{V}_z)}) \\
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\cos(\phi_{(\vec{V}_{yz}, \vec{V}_z)})
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\|\vec{V}\| \\
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\|\vec{V}_{yz}\| \\
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\|\vec{V}_{yz}\|
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\end{pmatrix}^\top
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\end{equation*}
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$$
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$$
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= \begin{pmatrix}
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1 & 0 & 0 \\
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0 & \cos(\phi_{R}) & \pm \sin(\phi_{R}) \\
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0 & \mp \sin(\Phi_R) & \cos(\phi_{R})
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\cos(\phi_{(\vec{V}, \vec{V}_x)}) \\
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\sin(\phi_{(\vec{V}_{yz}, \vec{V}_z)}) \\
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\cos(\phi_{(\vec{V}_{yz}, \vec{V}_z)})
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\|\vec{V}\| \\
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\|\vec{V}_{yz}\| \\
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\|\vec{V}_{yz}\|
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\end{pmatrix}^\top
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$$
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* Therefore, the rest of rotation operations could be expressed using rotation matrices as follows:
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* Rotation around Y-axis (**X-convention**):
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$$
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\begin{equation*}
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\vec{V}^{'} = R_y(\pm \hat{\Phi_R}) \times \vec{V} \\
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=
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\begin{pmatrix}
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$$
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\vec{V}^{'} = R_y(\pm \hat{\Phi_R}) \times \vec{V} \\
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= \begin{pmatrix}
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\|\vec{V}_{xz}\|(\cos(\phi_{R})\cos(\phi_{(\vec{V}_{xz}, \vec{V}_x)}) \mp
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\sin(\phi_{R})\sin(\phi_{(\vec{V}_{xz}, \vec{V}_x)})) \\
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\|\vec{V}\|\cos(\phi_{(\vec{V}, \vec{V}_y)}) \\
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\|\vec{V}_{xz}\|(\pm \sin(\phi_{R})\cos(\phi_{(\vec{V}_{xz}, \vec{V}_x)}) + \cos(\phi_{R})\sin(\phi_{(\vec{V}_{xz}, \vec{V}_x)}))
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\end{pmatrix}
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\\
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= \begin{pmatrix}
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\cos(\phi_R) & 0 & \mp \sin(\phi_R) \\
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0 & 1 & 0 \\
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\pm \sin(\phi_R) & 0 & \cos(\phi_R)
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\cos(\phi_{(\vec{V}_{xz}, \vec{V}_x)}) \\
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\cos(\phi_{(\vec{V}, \vec{V}_y)}) \\
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\sin(\phi_{(\vec{V}_{xz}, \vec{V}_x)})
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\|\vec{V}_{xz}\| \\
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\|\vec{V}\| \\
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\|\vec{V}_{xz}\|
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\end{pmatrix}^\top
178-
\end{equation*}
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$$
152+
\end{pmatrix}
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$$
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$$
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= \begin{pmatrix}
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\cos(\phi_R) & 0 & \mp \sin(\phi_R) \\
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0 & 1 & 0 \\
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\pm \sin(\phi_R) & 0 & \cos(\phi_R)
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\cos(\phi_{(\vec{V}_{xz}, \vec{V}_x)}) \\
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\cos(\phi_{(\vec{V}, \vec{V}_y)}) \\
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\sin(\phi_{(\vec{V}_{xz}, \vec{V}_x)})
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\|\vec{V}_{xz}\| \\
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\|\vec{V}\| \\
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\|\vec{V}_{xz}\|
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\end{pmatrix}^\top
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$$
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* Rotation around Z-axis (**X-Convention**) using direction cosines:
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$$
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\begin{equation*}
185-
\vec{V}^{'} = R_z(\pm \hat{\Phi_R}) \times \vec{V} \\
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=
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\begin{pmatrix}
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$$
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\vec{V}^{'} = R_z(\pm \hat{\Phi_R}) \times \vec{V} \\
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= \begin{pmatrix}
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\|\vec{V}_{xy}\|(\cos(\phi_{R})\cos(\phi_{(\vec{V}_{xy}, \vec{V}_x)}) \mp
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\sin(\phi_{R})\sin(\phi_{(\vec{V}_{xy}, \vec{V}_x)})) \\
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\|\vec{V}_{xy}\|(\pm \sin(\phi_{R})\cos(\phi_{(\vec{V}_{xy}, \vec{V}_x)}) + \cos(\phi_{R})\sin(\phi_{(\vec{V}_{xy}, \vec{V}_x)})) \\
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\|\vec{V}\|\cos(\phi_{(\vec{V}, \vec{V}_z)})
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\end{pmatrix}
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\\
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= \begin{pmatrix}
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\cos(\phi_R) & \mp \sin(\phi_R) & 0 \\
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\pm \sin(\phi_R) & \cos(\phi_R) & 0 \\
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0 & 0 & 1
198-
\end{pmatrix}
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\times
200-
\begin{pmatrix}
201-
\cos(\phi_{(\vec{V}_{xy}, \vec{V}_x)}) \\
202-
\sin(\phi_{(\vec{V}_{xy}, \vec{V}_x)}) \\
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\cos(\phi_{(\vec{V}, \vec{V}_z)}) \\
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\end{pmatrix}
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\times
206-
\begin{pmatrix}
207-
\|\vec{V}_{xy}\| \\
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\|\vec{V}_{xy}\| \\
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\|\vec{V}\|
210-
\end{pmatrix}^\top
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\end{equation*}
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$$
184+
\end{pmatrix}
185+
$$
186+
187+
$$
188+
= \begin{pmatrix}
189+
\cos(\phi_R) & \mp \sin(\phi_R) & 0 \\
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\pm \sin(\phi_R) & \cos(\phi_R) & 0 \\
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0 & 0 & 1
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\cos(\phi_{(\vec{V}_{xy}, \vec{V}_x)}) \\
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\sin(\phi_{(\vec{V}_{xy}, \vec{V}_x)}) \\
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\cos(\phi_{(\vec{V}, \vec{V}_z)}) \\
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\end{pmatrix}
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\times
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\begin{pmatrix}
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\|\vec{V}_{xy}\| \\
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\|\vec{V}_{xy}\| \\
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\|\vec{V}\|
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\end{pmatrix}^\top
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$$
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* A set of rotations could be achieved by multiplying the rotation matrices to get the final rotation matrix, and then multiply it with the vector coordinates.
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@@ -218,7 +211,7 @@ $$
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> * The rotation matrix could be derived from the result of the addition trigonometric identities retrieved after adding the angle $\phi_{R}$ to the initial angle $\phi_{0}$.
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> * The resulting vector (v') could be obtained by other means:
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> 1) Adding the angle of rotation to the initial angle and constructing the rectangular coordinates from the polar vector.
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> 2) Deriving the final vector (v') by evaluating the addition trigonometeric identities (i.e., $\cos(\phi_0 \pm \phi_R)$ and $\sin(\phi_0 \pm \phi_R)$).
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> 2) Deriving the final vector (v') by evaluating the addition trigonometeric identities (i.e., $\cos(\phi_0 \pm \phi_R)$ and $\sin(\phi_0 \pm \phi_R)$ ).
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* Mathematical Modelling: Errors of Rotation and Error Handling Techniques.
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@@ -243,4 +236,4 @@ $$
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4) [Applied Linear Algebra, Springer](https://link.springer.com/book/10.1007/978-3-319-91041-3)
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5) [Handbook of Mathematics, Springer](https://link.springer.com/book/10.1007/978-3-662-46221-8)
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6) [Spinal Cord Injuries](https://www.ncbi.nlm.nih.gov/books/NBK560721/)
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7) [Medical and Biological Engineering and Computing: Use of cardan angles to locate rigid bodies in three-dimensional space](https://link.springer.com/article/10.1007/BF02441745)
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7) [Medical and Biological Engineering and Computing: Use of cardan angles to locate rigid bodies in three-dimensional space](https://link.springer.com/article/10.1007/BF02441745)

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