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I have an object, $P$, and a 3 DoF transformation matrix, $T$. In my work, I want to say that $T$ can only model pitching, heaving and surging of $P$.

Currently, I write it like this,

\begin{equation} \begin{aligned} & & & NP = \ T \cdot P \\ & \text{subject to} & & T = \begin{bmatrix} cos\ \alpha & 0 & -sin\ \alpha & d_x \\ 0 & 1 & 0 & 1 \\ sin\ \alpha & 0 & cos\ \alpha & d_z \\ 0 & 0 & 0 & 1 \end{bmatrix} \\ \end{aligned} \end{equation}

However, I don't think above formula is formally correct. Any idea how to improve that?

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    $T$ is a rotation about the $y$-axis followed by a translation by $(d_x,1,d_z)$. Is that really what you wanted? – 2017-01-31
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    @amd Yes.. But I don't see anyone write it like that as a constraint. – 2017-02-01

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