<p>I created two ellipses,<br/>$\hspace{150px}$<a href="https://i.sstatic.net/WfzOU.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/WfzOU.png" width="300"/></a>,<br/>where the red ellipsis is as the blue one, except translated to the right and rotated by ${30}^{\circ} .$ Using rotation matrix, </p>

<p>$$
\left[
  \begin{array}{cc}
    x  \\[2px]
    y   \end{array}
\right]
\
\phantom{F} ~=~
\left[
  \begin{array}{cc}
    cos(30^{\circ})  &amp; -sin(30^{\circ})  \\[2px]
    sin(30^{\circ}) &amp; \phantom{}cos(30^{\circ})
  \end{array}
\right]
\
*
\left[
  \begin{array}{cc}
    X  \\[2px]
    Y   \end{array}
\right]
\
$$</p>

<p>Next I tried to calculate deformation gradient by using principal axes(a and b for initial ellipsis,c and d for translated one), and then I decomposed $\vec{c}$ and $\vec{d}$ vectors,$$
\begin{alignat}{7}
\vec{c} &amp;~~=~~ 1.0021 \, \vec{b} &amp;~&amp;+ &amp;~&amp;0.8654 \, \vec{a} \\[5px]
\vec{d} &amp;~~=~~ 0.8654 \, \vec{b} &amp;&amp;- &amp;&amp;0.2505 \, \vec{a}
\end{alignat}
$$and created the deformation gradient as$$
F ~=~
\left[
  \begin{array}{cc}
    \phantom{-} 0.8654  &amp;    1.0021  \\[2px]
             -  0.2505  &amp;    0.8654
  \end{array}
\right]
\,.
$$But it is obvious that, this is not the same as the ${30}^{\circ}$ rotation matrix that I had expected it to be,$$
\phantom{F} {\llap{\textsf{rotation matrix}}} ~=~
\left[
  \begin{array}{cc}
    0.8660  &amp;            -  0.5000  \\[2px]
    0.5000  &amp;   \phantom{-} 0.8660
  \end{array}
\right]
\,.
$$</p>

<p><strong>Question:</strong>  Why aren't the deformation gradient, $F ,$ and and the rotation matrix, $\textsf{rotation matrix} ,$ the same?</p>

CC BY-SA 4.0
2018-08-30T08:59:12.127Z
I'm not sure your rotation matrix is correct, usually the sin/cos are swapped in the second row    e.g  [ cos -sin ] [ sin cos ]
PaulHK
CC BY-SA 4.0
2018-08-30T09:12:20.940Z
@PaulHK, silly mistake while writing it,i rewrite it, now its in the correct form.
onurtore
CC BY-SA 4.0
2018-08-30T17:16:07.850Z
The c and d vectors do not look correct in the diagram. They should be equivalent to a and b but rotated. How are you finding those vectors? Maybe there's a bug there.
Nathan Reed
CC BY-SA 4.0
2018-08-30T19:07:17.413Z
@NathanReed isnt them correct ? There is an small error margin, but compared to difference between rotation matrix and F, its nothing.
onurtore
CC BY-SA 4.0
2018-08-30T19:14:32.973Z
I don't think it's a small error margin. The component of c along b, and the component of d along a, are both off by about a factor of 2. The vectors as shown in the diagram are plainly quite a bit off from the true axes of the red ellipse.
Nathan Reed
<transformations>
What is wrong with my deformation gradient calculation?
onurtore
# What is wrong with my deformation gradient calculation?

(Asked by: username_0 on 2018-08-30)

I created two ellipses,  
$\hspace{150px}$[![](https://i.sstatic.net/WfzOU.png)](https://i.sstatic.net/WfzOU.png),  
where the red ellipsis is as the blue one, except translated to the right and rotated by ${30}^{\circ} .$ Using rotation matrix,

$$
\left[
\begin{array}{cc}
x \\[2px]
y \end{array}
\right]
\
\phantom{F} ~=~
\left[
\begin{array}{cc}
cos(30^{\circ}) & -sin(30^{\circ}) \\[2px]
sin(30^{\circ}) & \phantom{}cos(30^{\circ})
\end{array}
\right]
\
\*
\left[
\begin{array}{cc}
X \\[2px]
Y \end{array}
\right]
\
$$

Next I tried to calculate deformation gradient by using principal axes(a and b for initial ellipsis,c and d for translated one), and then I decomposed $\vec{c}$ and $\vec{d}$ vectors,$$
\begin{alignat}{7}
\vec{c} &~~=~~ 1.0021 \, \vec{b} &~&+ &~&0.8654 \, \vec{a} \\[5px]
\vec{d} &~~=~~ 0.8654 \, \vec{b} &&- &&0.2505 \, \vec{a}
\end{alignat}
$$and created the deformation gradient as$$
F ~=~
\left[
\begin{array}{cc}
\phantom{-} 0.8654 & 1.0021 \\[2px]
- 0.2505 & 0.8654
\end{array}
\right]
\,.
$$But it is obvious that, this is not the same as the ${30}^{\circ}$ rotation matrix that I had expected it to be,$$
\phantom{F} {\llap{\textsf{rotation matrix}}} ~=~
\left[
\begin{array}{cc}
0.8660 & - 0.5000 \\[2px]
0.5000 & \phantom{-} 0.8660
\end{array}
\right]
\,.
$$

**Question:** Why aren't the deformation gradient, $F ,$ and and the rotation matrix, $\textsf{rotation matrix} ,$ the same?

**Tags:** transformations

> **Comments:**
> **username_1**: I'm not sure your rotation matrix is correct, usually the sin/cos are swapped in the second row    e.g  [ cos -sin ] [ sin cos ]  
> **username_0**: @username_1, silly mistake while writing it,i rewrite it, now its in the correct form.  
> **username_2**: The c and d vectors do not look correct in the diagram. They should be equivalent to a and b but rotated. How are you finding those vectors? Maybe there's a bug there.  
> **username_0**: @username_2 isnt them correct ? There is an small error margin, but compared to difference between rotation matrix and F, its nothing.  
> **username_2**: I don't think it's a small error margin. The component of c along b, and the component of d along a, are both off by about a factor of 2. The vectors as shown in the diagram are plainly quite a bit off from the true axes of the red ellipse.  

---

## 0 answers

