A function in the Cartesian plane can be transformed into polar coordinates by wrapping one axis around itself and collapsing it to a point. This is illustrated here transforming the Cartesian graph <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 y=\sin(6x)+2}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>y</mi><mo>=</mo><mi>sin</mi><mo>⁡<!-- ⁡ --></mo><mo stretchy="false">(</mo><mn>6</mn><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle y=\sin(6x)+2}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7998dbeb35231ba5a52e174170c268370fb8c41" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:15.414ex; height:2.843ex;" alt="{\displaystyle y=\sin(6x)+2}"></span> into the polar graph <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 r=\sin(6\theta )+2}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>r</mi><mo>=</mo><mi>sin</mi><mo>⁡<!-- ⁡ --></mo><mo stretchy="false">(</mo><mn>6</mn><mi>θ<!-- θ --></mi><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle r=\sin(6\theta )+2}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec6a87747c5f73dd77edda133c3707d1586f54c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:15.068ex; height:2.843ex;" alt="{\displaystyle r=\sin(6\theta )+2}"></span>.  
<p>The general transformation takes the following steps:
</p><p>1. Start with Cartesian graph.  
</p><p>2. Clip the graph to satisfy <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 y&gt;0}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>y</mi><mo>&gt;</mo><mn>0</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle y&gt;0}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c973e3cbfee5d7ab9ca2348b578b6ec19a8c019a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y&gt;0}"></span> (not necessary in the example <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 y=\sin(6x)+2}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>y</mi><mo>=</mo><mi>sin</mi><mo>⁡<!-- ⁡ --></mo><mo stretchy="false">(</mo><mn>6</mn><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle y=\sin(6x)+2}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7998dbeb35231ba5a52e174170c268370fb8c41" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:15.414ex; height:2.843ex;" alt="{\displaystyle y=\sin(6x)+2}"></span>).
</p><p>2. Reflect in the line <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 y=x}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>y</mi><mo>=</mo><mi>x</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle y=x}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0abe2e7da593fb7b41d44e87a97fefdd8998b77" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="y=x"></span>.
</p>
3. Bend it to backwards on itself, as shown in the animation, to obtain the polar graph.
<a href="//commons.wikimedia.org/wiki/User:LucasVB" title="User:LucasVB">Lucas Vieira</a>
6 September 2010 (upload date)
<span class="int-own-work" lang="en">Own work</span>