# What is wrong with my deformation gradient calculation?

I created two ellipses,
$\hspace{150px}$ ,
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?

• 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 Aug 30 '18 at 8:59
• @PaulHK, silly mistake while writing it,i rewrite it, now its in the correct form. – Onur Berk Töre Aug 30 '18 at 9:12
• 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 Aug 30 '18 at 17:16
• @NathanReed isnt them correct ? There is an small error margin, but compared to difference between rotation matrix and F, its nothing. – Onur Berk Töre Aug 30 '18 at 19:07
• 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 Aug 30 '18 at 19:14