Translation and Rotation of Coordinate Axes

Consider two sets of rectangular Cartesian frames of reference $O - x_1x_2$ and $O^\prime - x^\prime_1x^\prime_2$ in a plane. If the frame of reference $O^\prime - x^\prime_1x^\prime_2$ is obtained from $O - x_1x_2$ by a shift of the origin without a change in orientation, then, the transformation is a translation.

If a point $P$ has coordinates $(x_1,x_2)$ and $(x^\prime_1,x^\prime_2)$ with respect to $O - x_1x_2$ and $O^\prime - x^\prime_1x^\prime_2$ respectively and $(C_1,C_2)$ are the coordinates of $0^\prime$ with respect to $O - x_1x_2$, then

$$x_1 = x^\prime_1 + C_1,$$

$$x_2 = x^\prime_2 + C_2,$$

or briefly

$$x_i = x^\prime_i + C_i,\ i = 1,2.$$ (2.7.1)

If the origin remains fixed, and the new axes $Ox^\prime_1,\ Ox^\prime_2$ are obtained by rotating $Ox_1$ and $Ox_2$ through an angle $\theta$ in the counter-clockwise direction, then

the transformation of axes is a rotation. Let the point $P$ have coordinates $(x_1,x_2)$ and $(x^\prime_1,x^\prime_2)$ relative to $O - x_1x_2$ and $O - x^\prime_1x^\prime_2$ respectively. Then,

$$x_1 =\ OA = OB - CD,$$

$$=\ OD \cos \theta - PD\sin\theta ,$$

$$=\ x^\prime_1\cos \theta - x^\prime_2\sin\theta;$$

$$x_2 =\ AP = BD + CP,$$

$$=\ x^\prime_1 \sin\theta + x^\prime_2 \cos\theta .$$

We can write $(x^\prime_1,x^\prime_2)$ in terms of $(x_1,x_2)$ as

$$\begin{split}&x^\prime_1 = x_1\cos\theta + x_2\sin\theta,\\ &x^\prime_2 = - x_1 \sin\theta + x_2\cos\theta. \end{split}$$ (2.7.2)

Using the index notation, the set of eqns. (2.7.2) can be written as

$$x^\prime_i = a_{ij}x_j,\hspace{.5in} i =1,2; \ j = 1,2,$$ (2.7.3)

where $a_{ij}$ are elements of the matrix $[a_{ij}]$;

$$\left[\begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22}\end{ar... ...ray}{cc} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta\end{array}\right].$$

Before we generalize (2.7.1) and (2.7.2) to three dimensions we give below an alternate method of arriving at (2.7.2). Let $\mathbf{e}^\prime_1,\ \mathbf{e}^\prime_2$ denote unit vectors along $x^\prime_1$ and $x^\prime_2$-axes and $\mathbf{e}_1,\ \mathbf{e}_2$ unit vectors along $x_1$ and $x_2$-axes. Then

$$\overrightarrow{OP} =\ x_1\mathbf{e}_1 + x_2 \mathbf{e}_2,$$

$$=\ x^\prime_1 \mathbf{e}^\prime_1 + x^\prime_2 \mathbf{e}^\prime_2.$$

Also

$$\mathbf{e}^\prime_1 = \cos\theta\ \mathbf{e}_1 + \sin\theta\ \mathbf{e}_2,$$

$$\mathbf{e}^\prime_2 = -\sin\theta\ \mathbf{e}_1 + \cos\theta \ \mathbf{e}_2.$$

Therefore,

$$x^\prime_1 =\ \overrightarrow{OP} \cdot \mathbf{e}^\prime_1,$$

$$=\ (x_1\mathbf{e}_1 + x_2\mathbf{e}_2)\cdot (\cos\theta \mathbf{e}_1 + \sin\theta\mathbf{e}_2)$$

$$=\ x_1 \cos \theta + x_2 \sin\theta ;$$

$$x^\prime_2 =\ \overrightarrow{OP}\cdot \mathbf{e}^\prime_2,$$

$$=\ -x_1 \sin\theta + x_2 \cos\theta.$$

This latter approach can more easily be adopted to the 3-dimensional case. If the primed axes $O - x^\prime_1x^\prime_2x^\prime_3$ are obtained from the unprimed axes $O-x_1x_2x_3$ just by a translation, then the coordinates of a point with respect to the two sets of axes are related by (2.7.1) wherein the index $i$ ranges from $1$ to $3$. Now let us assume that the primed axes are obtained from the unprimed axes by a rotation only. Let us denote unit vectors along $x_1,x_2,x_3$ by $\mathbf{e}_1,\ \mathbf{e}_2$, and $\mathbf{e}_3$ respectively and those along $x^\prime_1,\ x^\prime_2,\ x^\prime_3$ by $\mathbf{e}^\prime_1,\ \mathbf{e}^\prime_2$ and $\mathbf{e}^\prime_3$ respectively. If

$$a_{1j} = \ \mathbf{e}^\prime_1\ \ \mathbf{e}_j,$$

then $a_{11},\ a_{12}$ and $a_{13}$ are the direction cosines of $\mathbf{e}^\prime_1$ with respect to the unprimed axes. We can write

$$\mathbf{e}^\prime_1=\ a_{11}\mathbf{e}_1+a_{12}\mathbf{e}_2+a_{13} \mathbf{e}_3 ,$$

$$=\ a_{1i}\mathbf{e}_i.$$

Similarly,

$$\mathbf{e}^\prime_2 = a_{2i}\mathbf{e}_i,\ \mathbf{e}^\prime_3 = a_{3i} \mathbf{e}_i.$$

Or

$$\mathbf{e}^\prime_i = a_{ij}\mathbf{e}_j.$$ (2.7.4)

Note that the matrix $a_{ij}$ is $3\times 3$. Since

$$\mathbf{e}^\prime_i \cdot \mathbf{e}^\prime_j = \delta_{ij},$$ (2.7.5)

therefore,

$$\delta_{ij} =\ a_{ik}\mathbf{e}_k\cdot a_{jp}\mathbf{e}_p = a_{ik}a_{jp}\mathbf{e}_k\cdot \mathbf{e}_p,$$

$$=\ a_{ik}a_{jp}\delta_{kp} = a_{ik}a_{jk}.$$ (2.7.6)

Equations (2.7.6) are equivalent to the following six equations.

$$\begin{split}&a^2_{11} + a^2_{12} + a^2_{13} = 1,\\ &a^2_{21}... ...&a_{31} a_{11} + a_{32} a_{12} + a_{33} a_{13} = 0. \end{split}$$ (2.7.7)

The first three equations are equivalent to the statement that $\mathbf{e}^\prime_1,\ \mathbf{e}^\prime_2$ and $\mathbf{e}^\prime_3$ are unit vectors; the last three equations are equivalent to the statement that $\mathbf{e}^\prime_1,\ \mathbf{e}^\prime_2,\ \mathbf{e}^\prime_3$ are mutually orthogonal. Of course, we can write $\mathbf{e}_i$'s in terms of $\mathbf{e}^\prime_i$'s. Since

$$a_{j1} = \ \mathbf{e}_1\ \ \mathbf{e}^\prime_j,$$

therefore,

$$\mathbf{e}_1 = a_{j1}\mathbf{e}^\prime_j\ \ \mathbf{e}_i = a_{ji}\mathbf{e}^\prime_j.$$ (2.7.8)

From the point of view of the solution of a set of simultaneous linear equations, the matrix $a_{ji}$ in (2.7.8) must be identified as the inverse of the matrix $a_{ij}$:

$$[a_{ij}]^{-1} = [a_{ji}] = [a_{ij}]^T.$$ (2.7.9)

Here $[a_{ij}]^T$ is the transpose of the matrix $[a_{ij}]$. A matrix $[a_{ij}]$, which satisfies eqn. (2.7.9) is called an orthogonal matrix. That is, the transpose of an orthogonal matrix equals its inverse. A transformation is said to be orthogonal if the associated matrix is orthogonal. The matrix $[a_{ij}]$ in (2.7.4) defining a rotation of coordinate axes is orthogonal.

For an orthogonal matrix we have

$$[\mathbf{a}][\mathbf{a}]^T = \mathbf{1}.$$

Therefore

$${\rm det}([\mathbf{a}][\mathbf{a}]^T) = 1,$$

$$\ \ {\rm det}[\mathbf{a}]\, {\rm det}[\mathbf{a}]^T = 1,$$

$$\ \ {\rm det}[\mathbf{a}]\, {\rm det}[\mathbf{a}] = 1,$$

and thus

$${\rm det}[\mathbf{a}] = \pm 1.$$

An orthogonal matrix whose determinant equals $+1$ is called proper orthogonal and the one whose determinant equals $-1$ is called improper orthogonal. A proper orthogonal matrix transforms a right-handed triad of axes into a right-handed set of axes whereas an improper orthogonal matrix transforms a right-handed set of axes into a left-handed set of axes or vice-versa.

Exercise: Consider a cube formed by the orthonormal vectors $\mathbf{e}^\prime_1,\ \mathbf{e}^\prime_2$ and $\mathbf{e}^\prime_3$. By setting the volume of this cube equal to 1, show that ${\rm det}[a_{ij}] = 1$.

Consider a vector $\mathbf{O}\mathbf{P}$ emanating from the origin $O$ and ending at a point $P$. With respect to the primed and unprimed axes,

$$\mathbf{O}\mathbf{P} =\ x^\prime_1\mathbf{e}^\prime_1 + x^\prime_2\mathbf{e}^\prime_2 + x^\prime_3\mathbf{e}^\prime_3,$$

$$=\ x^\prime_j\mathbf{e}^\prime_j,$$

$$=\ x_j\mathbf{e}_j.$$

Similarly,

$$x^\prime_i =\ \mathbf{O}\mathbf{P} \cdot \mathbf{e}^\prime_i,$$

$$=\ (x_j\mathbf{e}_j)\cdot (a_{ik}\mathbf{e}_k),$$

$$=\ x_ja_{ik}\delta_{jk},$$

$$=\ a_{ij}x_j.$$

Example: The components of a vector $\mathbf{A}$ with respect to unprimed axes are $A_i = (0,1,1)$. Consider a set of primed coordinate axes obtained by rotating the unprimed axes through an angle of $30^\circ$ about the $x_3$-axis (see Fig. ). What are the components, $A^\prime_i$, of this vector with respect to the primed set of axes?

Solution:

$$\mathbf{e}^\prime_3 = \mathbf{e}_3;\ \mathbf{e}^\prime_1 =\ \cos 30 \mathbf{e}_1 + \sin 30\mathbf{e}_2,$$

$$\mathbf{e}^\prime_2 =\ -\sin 30\mathbf{e}_1 + \cos 30\mathbf{e}_2.$$

Therefore

$$[a_{ij}] = [\mathbf{e}^\prime_i\cdot \mathbf{e}_j] = \left[\begin... ...}\cos 30 & \sin 30 & 0\\ -\sin 30 & \cos 30 & 0\\ 0 & 0 & 1\end{array}\right]$$

Now

$$A^\prime_i = a_{ij}A_j$$

can be written as

$$\left[\begin{array}{c}A^\prime_1\\ A^\prime_2\\ A^\prime_3\end{... ...ray}\right] = \left[\begin{array}{c}\sin 30\\ \cos 30\\ 1\end{array}\right] .$$

Hence

$$A^\prime_i = (0.5,\ \ 0.866,\ \ 1).$$

Summarizing our discussion of the transformation of coordinate axes, we note that a general transformation from unprimed to primed axes combines both a translation and a rotation of the axes. This can be written as

$$x^\prime_i = a_{ij}x_j + c_i$$ (2.7.10)

where $a_{ij}$ is an orthogonal matrix and $c_i$ is a constant. Under this transformation, the components of a vector $\mathbf{A}$ in the two sets of axes are related as

$$A^\prime_i = a_{ij}A_j.$$ (2.7.11)