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Coordinate Transformations

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Introduction

Coordinate transformations are nonintuitive enough in 2-D, and positively painful in 3-D. This page tackles them in the following order: (i) vectors in 2-D, (ii) tensors in 2-D, (iii) vectors in 3-D, (iv) tensors in 3-D, and finally (v) 4th rank tensor transforms.

A major aspect of coordinate transforms is the evaluation of the transformation matrix, especially in 3-D. This is touched on here, and discussed at length on the next page.

It is very important to recognize that all coordinate transforms on this page are rotations of the coordinate system while the object itself stays fixed. The "object" can be a vector such as force or velocity, or a tensor such as stress or strain in a component. Object rotations are discussed in later sections.

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2-D Coordinate Transforms of Vectors

The academic potato provides an excellent example of how coordinate transformations apply to vectors, while at the same time stressing that it is the coordinate system that is rotating and not the vector... or potato.

The potato on the left has a vector on it. But without a coordinate system, there is no way to describe the vector. So a coordinate system has been added to the potato as shown on the right, allowing the vector to now be described as \({\bf v} = 2{\bf i} + 9{\bf j}\).

       
So now we introduce a rotated coordinate system shown in blue below, using \(x'\) and \(y'\). The new system is rotated counter-clockwise by an angle, \(\theta\), from the initial coordinate system. Note that the vector itself does not change at all. It is still the very same vector as before. But it is described by different numerical values in the new coordinate system. In this case, the vector is more closely parallel to the new \(x'\) axis than the \(y'\) axis, so the \({\bf i'}\) component will be greater than the \({\bf j'}\) component. The transformation is given below the figure.

The 2-D vector transformation equations are

\[ v'_x = \;\;\; v_x \cos \theta + v_y \sin \theta \] \[ v'_y = -v_x \sin \theta + v_y \cos \theta \]
This can be seen by noting that

• the part of \(v_x\) that lies along the \(x'\) axis is \(v_x \cos \theta\)
• the part of \(v_y\) that lies along the \(x'\) axis is \(v_y \sin \theta\)
• the part of \(v_x\) that lies along the \(y'\) axis is \(-v_x \sin \theta\)
• the part of \(v_y\) that lies along the \(y'\) axis is \(v_y \cos \theta\)

These four factors make up the four terms in the transformation equations. They are easily checked by setting \(\theta = 0^\circ \) and \(\theta = 90^\circ \). When \(\theta = 0^\circ\), then \(v'_x = v_x\) and \(v'_y = v_y\). When \(\theta = 90^\circ\), then \(v'_x = v_y\) and \(v'_y = -v_x\).

Transformation Matrix

It is more convenient to write (and work with) transformation equations using matrices.

\[ \left\{ \matrix {v'_x \\ v'_y} \right\} = \left[ \matrix {\;\;\;\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta} \right] \left\{ \matrix {v_x \\ v_y} \right\} \]
The \(\cos \theta\) terms are on the matrix diagonal while the \(\sin \theta\) terms are off-diagonal. The only potential gotcha is remembering which \(\sin \theta\) term has the minus sign on it. It is always the lower-left term.

The above equation is written in matrix notation as

\[ {\bf v'} = {\bf Q} \cdot {\bf v} \]
where \({\bf Q}\) is the usual letter chosen for the transformation matrix.
A general method exists for formulating transformation matrices based on the cosines of the angles between the axes of the two coordinate systems, i.e., direction cosines. (This also applies to 3-D transforms.) The transformation matrix can be written as

\[ {\bf Q} = \left[ \matrix { \cos(x',x) & \cos(x',y) \\ \cos(y',x) & \cos(y',y) } \right] \]
where \((x',x)\) represents the angle between the \(x'\) and \(x\) axes, \((x',y)\) is the angle between the \(x'\) and \(y\) axes, etc.

The angle between \(x'\) and \(y\) is \((90^\circ - \theta)\), and \(\cos(x',y) = \cos(90^\circ - \theta) = \sin \theta\).

Likewise, the angle between \(y'\) and \(x\) is \((90^\circ + \theta)\), and \(\cos(y',x) = \cos(90^\circ + \theta) = -\sin \theta\).

Tensor Notation

The coordinate transform is written in tensor notation as

\[ v'_i = \lambda_{ij} v_j \]
where \(\lambda_{ij}\) is the transformation matrix \({\bf Q}\). (I don't know why \({\bf Q}\) is used in matrix notation, but \(\lambda_{ij}\), not \(q_{ij}\), is used in tensor notation.) \(\lambda_{ij}\) is defined as

\[ \lambda_{ij} = \cos(x'_i,x_j) \]
For example, if \(i = 1\) and \(j = 2\), then

\[ \lambda_{12} = \cos(x'_1,x_2) = \cos(x',y) \]
\(\lambda_{ij}\) is the direction cosine of the angle between the \(x'_i\) axis and the \(x_j\) axis. Again, this is equally applicable to 3-D transformations as well.

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2-D Coordinate Transforms of Tensors

This section will present the what and how of tensor transforms. The why will have to wait until later.

Coordinate transformations of 2nd rank tensors involve the very same \({\bf Q}\) matrix as vector transforms. A transformation of the stress tensor, \(\boldsymbol{\sigma}\), from the reference \(x-y\) coordinate system to \(\boldsymbol{\sigma'}\) in a new \(x'-y'\) system is done as follows.

\[ \boldsymbol{\sigma'} = {\bf Q} \cdot \boldsymbol{\sigma} \cdot {\bf Q}^T \]
Writing the matrices out explicitly gives

\[ \left[ \matrix{\sigma'_{xx} & \sigma'_{xy} \\ \sigma'_{xy} & \sigma'_{yy} } \right] = \left[ \matrix{\;\;\; \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta } \right] \left[ \matrix{\sigma_{xx} & \sigma_{xy} \\ \sigma_{xy} & \sigma_{yy} } \right] \left[ \matrix{\cos \theta & -\sin \theta \\ \sin \theta & \;\;\; \cos \theta } \right] \]
(Note that the stress tensor is always symmetric, even following transformations.)

Multiplying the matrices out gives

\[ \begin{eqnarray} \sigma'_{xx} & = & \sigma_{xx} \cos^2 \theta + \sigma_{yy} \sin^2 \theta + 2 \sigma_{xy} \sin \theta \cos \theta \\ \\ \sigma'_{yy} & = & \sigma_{xx} \sin^2 \theta + \sigma_{yy} \cos^2 \theta - 2 \sigma_{xy} \sin \theta \cos \theta \\ \\ \sigma'_{xy} & = & (\sigma_{yy} - \sigma_{xx}) \sin \theta \cos \theta + \sigma_{xy} (\cos^2 \theta - \sin^2 \theta) \end{eqnarray} \]
These three equations are exactly the 2-D transform of a stress tensor resulting from summing forces on a differential element and imposing equilibrium. This is also represented by Mohr's circle.

Tensor Notation

The coordinate transform is written in tensor notation as

\[ \sigma'_{mn} = \lambda_{mi} \lambda_{nj} \sigma_{ij} \]
As usual, tensor notation provides extra insight into the process. This time, the insight comes from the subscripts on the lambdas. Each lambda effectively pairs up a subscript on \(\boldsymbol{\sigma'}\) with one on \(\boldsymbol{\sigma}\). This is true regardless of the rank of the tensor.

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3-D Coordinate Transforms of Vectors

Many of the general equations used in 2-D transformations are also applicable in 3-D. Examples include

\[ {\bf v'} = {\bf Q} \cdot {\bf v} \qquad \qquad \qquad \qquad v'_i = \lambda_{ij} v_j \qquad \qquad \qquad \qquad \lambda_{ij} = \cos(x'_i,x_j) \]
Only now the details are different. The vectors have z-components and the transformation matrices are 3x3 instead of 2x2.

\[ {\bf v} = \left\{ \matrix { v_x \\ v_y \\ v_z } \right\} \qquad \qquad \text{and} \qquad \qquad {\bf Q} = \left[ \matrix { \cos(x',x) & \cos(x',y) & \cos(x',z) \\ \cos(y',x) & \cos(y',y) & \cos(y',z) \\ \cos(z',x) & \cos(z',y) & \cos(z',z) } \right] \]
\[ {\bf Q} = \left[ \matrix { \left( \matrix{\text{x-comp} \\ \text{of } {\bf i'}} \right) & \left( \matrix{\text{y-comp} \\ \text{of } {\bf i'}} \right) & \left( \matrix{\text{z-comp} \\ \text{of } {\bf i'}} \right) \\ \left( \matrix{\text{x-comp} \\ \text{of } {\bf j'}} \right) & \left( \matrix{\text{y-comp} \\ \text{of } {\bf j'}} \right) & \left( \matrix{\text{z-comp} \\ \text{of } {\bf j'}} \right) \\ \left( \matrix{\text{x-comp} \\ \text{of } {\bf k'}} \right) & \left( \matrix{\text{y-comp} \\ \text{of } {\bf k'}} \right) & \left( \matrix{\text{z-comp} \\ \text{of } {\bf k'}} \right) } \right] \]
\[ \left\{ \matrix { v'_x \\ v'_y \\ v'_z } \right\} = \left[ \matrix { \cos(x',x) & \cos(x',y) & \cos(x',z) \\ \cos(y',x) & \cos(y',y) & \cos(y',z) \\ \cos(z',x) & \cos(z',y) & \cos(z',z) } \right] \left\{ \matrix { v_x \\ v_y \\ v_z } \right\} \]

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3-D Coordinate Transforms of Tensors

Once again, the rules don't change, only the particulars do.

\[ \boldsymbol{\sigma'} = {\bf Q} \cdot \boldsymbol{\sigma} \cdot {\bf Q}^T \qquad \qquad \qquad \qquad \sigma'_{mn} = \lambda_{mi} \lambda_{nj} \sigma_{ij} \qquad \qquad \qquad \qquad \lambda_{ij} = \cos(x'_i,x_j) \]
Writing the matrices out explicitly gives

\[ \left[ \matrix{\sigma'_{xx} & \sigma'_{xy} & \sigma'_{xz} \\ \sigma'_{xy} & \sigma'_{yy} & \sigma'_{yz} \\ \sigma'_{xz} & \sigma'_{yz} & \sigma'_{zz} } \right] = \left[ \matrix { \cos(x',x) & \cos(x',y) & \cos(x',z) \\ \cos(y',x) & \cos(y',y) & \cos(y',z) \\ \cos(z',x) & \cos(z',y) & \cos(z',z) } \right] \left[ \matrix{\sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{xy} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{xz} & \sigma_{yz} & \sigma_{zz} } \right] \left[ \matrix { \cos(x',x) & \cos(y',x) & \cos(z',x) \\ \cos(x',y) & \cos(y',y) & \cos(z',y) \\ \cos(x',z) & \cos(y',z) & \cos(z',z) } \right] \]
This webpage performs coordinate transforms on 3-D tensors. Try it out.

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Coordinate Transforms of 4th Rank Tensors

We will see in the section on Hooke's Law that the stiffness tensor is 4th rank, i.e., 3x3x3x3 (not 4x4). It is written as \(C_{ijkl}\) because it relates any strain component, \(\epsilon_{kl}\), to any stress component, \(\sigma_{ij}\), i.e., \(\sigma_{ij} = C_{ijkl} \epsilon_{kl}\). The coordinate transformation law for the 4th rank stiffness tensor is easily written in tensor notation as

\[ C'_{ijkl} = \lambda_{im} \lambda_{jn} \lambda_{ko} \lambda_{lp} C_{mnop} \]
The tensor equation directs how to write the transformation in matrix notation.

\[ {\bf C'} = {\bf Q} \cdot {\bf Q} \cdot {\bf C} \cdot {\bf Q}^T \cdot {\bf Q}^T \]

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Summary

The coordinate transform of a vector in matrix and tensor notation is

\[ {\bf v'} = {\bf Q} \cdot {\bf v} \qquad \qquad \text{and} \qquad \qquad v'_i = \lambda_{ij} v_j \]
The coordinate transform of a tensor in matrix and tensor notation is

\[ \boldsymbol{\sigma'} = {\bf Q} \cdot \boldsymbol{\sigma} \cdot {\bf Q}^T \qquad \qquad \text{and} \qquad \qquad \sigma'_{mn} = \lambda_{mi} \lambda_{nj} \sigma_{ij} \]
Note that \({\bf Q}\) and \(\lambda_{ij}\) are the same transformation matrix.

In 2-D, \({\bf Q}\) and \(\lambda_{ij}\) are defined as

\[ {\bf Q} = \left[ \matrix {\;\;\;\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta} \right] \]
which is a special case of the more general 3-D form

\[ {\bf Q} = \left[ \matrix { \cos(x',x) & \cos(x',y) & \cos(x',z) \\ \cos(y',x) & \cos(y',y) & \cos(y',z) \\ \cos(z',x) & \cos(z',y) & \cos(z',z) } \right] \]
where \(\cos(x',x)\) is the direction cosine of the angle between the rotated \(x'\) axis and the reference \(x\) axis.

The coordinate transform of a 4th rank tensor is

\[ {\bf C'} = {\bf Q} \cdot {\bf Q} \cdot {\bf C} \cdot {\bf Q}^T \cdot {\bf Q}^T \]
in matrix notation and

\[ C'_{ijkl} = \lambda_{im} \lambda_{jn} \lambda_{ko} \lambda_{lp} C_{mnop} \]
in tensor notation.