User:Cornince/circlediff/test1

$$s^2 = \Delta x^2 + \Delta y^2\,$$

$$r^2 = x^2 + y^2\,$$

a is x or y and b is the other variable

$$\Delta a^2 = \dfrac{ (s^4a^2) + (s^4a^2 - r^2s^4 - 4r^2s^2a^2 + 4r^4s^2) \pm 2(-s^2a)\sqrt{s^4a^2 - r^2s^4 - 4r^2s^2a^2 + 4r^4s^2} } {  4r^4  }\,$$

$$\begin{align} s^2 = & \dfrac{ 2s^4a^2 - r^2s^4 - 4r^2s^2a^2 + 4r^4s^2 + 2(-s^2a)\sqrt{s^4a^2 - r^2s^4 - 4r^2s^2a^2 + 4r^4s^2} } {  4r^4  } \\ & + \dfrac{ 2s^4b^2 - r^2s^4 - 4r^2s^2b^2 + 4r^4s^2 - 2(-s^2b)\sqrt{s^4b^2 - r^2s^4 - 4r^2s^2b^2 + 4r^4s^2} } {  4r^4  } \\ = & \left [ 2s^4a^2 - r^2s^4 - 4r^2s^2a^2 + 4r^4s^2 + 2s^4b^2 - r^2s^4 - 4r^2s^2b^2 + 4r^4s^2 \right. \\ & \left. + 2(-s^2a)\sqrt{s^4a^2 - r^2s^4 - 4r^2s^2a^2 + 4r^4s^2} - 2(-s^2b)\sqrt{s^4b^2 - r^2s^4 - 4r^2s^2b^2 + 4r^4s^2} \right ] / 4r^4 \\ = & \left [ 2s^4a^2 - 4r^2s^2a^2 + 2s^4(r^2-a^2) - 4r^2s^2(r^2-a^2) - 2r^2s^4 + 8r^4s^2 \right. \\ & \left. + 2(-s^2a)\sqrt{s^4a^2 - r^2s^4 - 4r^2s^2a^2 + 4r^4s^2} - 2(-s^2\sqrt{r^2-a^2})\sqrt{s^4(r^2-a^2) - r^2s^4 - 4r^2s^2(r^2-a^2) + 4r^4s^2} \right ] / 4r^4 \\ = & \left [ 4r^4s^2 + 2(-s^2a)\sqrt{s^4a^2 - r^2s^4 - 4r^2s^2a^2 + 4r^4s^2} \right. \\ & \left. - 2(-s^2\sqrt{r^2-a^2})\sqrt{-s^4a^2 + 4r^2s^2a^2} \right ] / 4r^4 \\ = & s^2 \dfrac{ 4r^4 - 2a \sqrt{s^4a^2 - r^2s^4 - 4r^2s^2a^2 + 4r^4s^2} + 2 \sqrt{-s^4a^2(r^2-a^2) + 4r^2s^2a^2(r^2-a^2)} } { 4r^4 } \\ = & s^2 \dfrac{ 4r^4 - 2a \sqrt{s^4a^2 - r^2s^4 - 4r^2s^2a^2 + 4r^4s^2} + 2 \sqrt{s^4a^4 - r^2s^4a^2 - 4r^2s^2a^4 + 4r^4s^2a^2} } { 4r^4 } \\ = & s^2 \dfrac{ 4r^4 - 2a \sqrt{s^4a^2 - r^2s^4 - 4r^2s^2a^2 + 4r^4s^2} + 2a \sqrt{s^4a^2 - r^2s^4 - 4r^2s^2a^2 + 4r^4s^2} } { 4r^4 } \\ = & s^2 \dfrac{ 4r^4 } { 4r^4 } \\ = & s^2 \\ \end{align}\,$$