An example of <a href="https://en.wikipedia.org/wiki/Joukowsky_transform" class="extiw" title="w:Joukowsky transform">Joukowsky transform</a> of a circle into an airfoil. The circle in complex plane is centered at (-0.08,0.08) and has a radius of 1.08.<br>
The above figure is the complex variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta =\chi +i\eta }"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>ζ<!-- ζ --></mi><mo>=</mo><mi>χ<!-- χ --></mi><mo>+</mo><mi>i</mi><mi>η<!-- η --></mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \zeta =\chi +i\eta }</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/638a942b3b57872612e8e544ae1a2b97ac6c59db" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:10.461ex; height:2.676ex;" alt="\zeta =\chi +i\eta "></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\zeta \right|={\sqrt {(\chi +0.08)^{2}+(\eta -0.08)^{2}}}=1.08}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow><mo>|</mo><mi>ζ<!-- ζ --></mi><mo>|</mo></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><msqrt><mo stretchy="false">(</mo><mi>χ<!-- χ --></mi><mo>+</mo><mn>0.08</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><mi>η<!-- η --></mi><mo>−<!-- − --></mo><mn>0.08</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></msqrt></mrow><mo>=</mo><mn>1.08</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \left|\zeta \right|={\sqrt {(\chi +0.08)^{2}+(\eta -0.08)^{2}}}=1.08}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db3bd9795d1c9d31e2792fa373514bf01989fd24" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.671ex; width:40.185ex; height:4.843ex;" alt="\left|\zeta \right|={\sqrt  {(\chi +0.08)^{2}+(\eta -0.08)^{2}}}=1.08"></span> and the transformed variable below is another complex variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\zeta +{\frac {1}{\zeta }}=x+iy}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>z</mi><mo>=</mo><mi>ζ<!-- ζ --></mi><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mi>ζ<!-- ζ --></mi></mfrac></mrow><mo>=</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle z=\zeta +{\frac {1}{\zeta }}=x+iy}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e804f069a14a3dca8dcfb375aa89ff0e69d031b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:19.347ex; height:5.676ex;" alt="z=\zeta +{\frac  {1}{\zeta }}=x+iy"></span>.
<a href="//commons.wikimedia.org/wiki/User:Krishnavedala" title="User:Krishnavedala">Krishnavedala</a>
<span class="int-own-work" lang="en">Own work</span>