# Finding center of rotation on a plane using complex numbers

Let $$V_1$$ be the anti-clockwise rotation in the plane about origin with $$\theta$$ angle and $$V_2$$ be the anti-clockwise rotation in the plane about (2,0) by $$\theta$$ angle. finding its center of rotation of composition $$V_2(V_1)$$

around origin i have a transformation for $$V_1$$ that is $$f_1(z)=e^{\iota\theta} z$$ similarly for $$V_2$$ i have $$f_2(z)=e^{\iota\theta} (z-2)+2$$ then $$f_2f_1(z)=e^{\iota\theta} ( f_1(z)-2)+2=e^{\iota\theta} (e^{\iota\theta}z-2)+2$$. now how to find center from this without using matrices.

• there is a mistake in your computations for $f_2\circ f_1$. also, what makes you think that the transformation $V_2\circ V_1$ will be a rotation? Commented Jun 15 at 13:39
• i have edited it, it look likes transformation of $2\theta$, Commented Jun 15 at 13:45
• look for example at the points $(0,0)$ and $(1,0)$. does there distance remain $1$ after applying $V_2\circ V_1$? Commented Jun 15 at 13:48
• if i know about center of rotation from there distance should preserve Commented Jun 15 at 13:54
• Solve $f_2(f_1(z))=z$ for $z$. Commented Jun 15 at 14:01

$$f_2 (f_1 (z)) = e^{i \theta } ( e^{i \theta} z - 2 ) + 2 = e^{i 2 \theta} z - 2 e^{i \theta } + 2$$

And this is a rotation by $$2 \theta$$. To find the center, take the general form of rotation by $$2 \theta$$ about $$z_0$$:

$$g(z) = e^{i 2 \theta} (z - z_0 ) + z_0$$

And compare this form to the earlier equation. We deduce that

$$z_0 ( 1 - e^{i 2 \theta } ) = 2 ( 1 - e^{i \theta} )$$

Therefore, the center of rotation is given by

$$z_0 = \dfrac{ 2 (1 - e^{i \theta} ) }{ 1 - e^{i 2 \theta} } = \dfrac{2}{1 + e^{i \theta} }$$

• yes center is $(1,-\tan{\frac{\theta}{2}})$. Commented Jun 15 at 14:30