image of Cassinian


2D Coordinate Systems


The Cassinian (Cassinian-oval) is two-dimensional coordinate system.

Object definitions


Mapping of Cassinian
\left\{x\to \frac{a \sqrt{e^u \cos (v)+\sqrt{2 e^u \cos (v)+e^{2 u}+1}+1}}{\sqrt{2}},y\to \frac{a \sqrt{-e^u \cos (v)+\sqrt{2 e^u \cos (v)+e^{2 u}+1}-1}}{\sqrt{2}}\right\}
<math> <mrow> <mo>{</mo> <mrow> <mrow> <mi>x</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mfrac> <mrow> <mi>a</mi> <mo>&#8290;</mo> <msqrt> <mrow> <mrow> <msup> <mi>&#8519;</mi> <mi>u</mi> </msup> <mo>&#8290;</mo> <mrow> <mi>cos</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <msqrt> <mrow> <mrow> <mn>2</mn> <mo>&#8290;</mo> <msup> <mi>&#8519;</mi> <mi>u</mi> </msup> <mo>&#8290;</mo> <mrow> <mi>cos</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <msup> <mi>&#8519;</mi> <mrow> <mn>2</mn> <mo>&#8290;</mo> <mi>u</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </msqrt> <mo>+</mo> <mn>1</mn> </mrow> </msqrt> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>,</mo> <mrow> <mi>y</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mfrac> <mrow> <mi>a</mi> <mo>&#8290;</mo> <msqrt> <mrow> <mrow> <mo>-</mo> <mrow> <mrow> <msup> <mi>&#8519;</mi> <mi>u</mi> </msup> </mrow> <mo>&#8290;</mo> <mrow> <mi>cos</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <msqrt> <mrow> <mrow> <mn>2</mn> <mo>&#8290;</mo> <msup> <mi>&#8519;</mi> <mi>u</mi> </msup> <mo>&#8290;</mo> <mrow> <mi>cos</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> <mo>+</mo> <msup> <mi>&#8519;</mi> <mrow> <mn>2</mn> <mo>&#8290;</mo> <mi>u</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </msqrt> <mo>-</mo> <mn>1</mn> </mrow> </msqrt> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> </math>
{x -> (a*Sqrt[1 + E^u*Cos[v] + Sqrt[1 + E^(2*u) + 2*E^u*Cos[v]]])/Sqrt[2], y -> (a*Sqrt[-1 - E^u*Cos[v] + Sqrt[1 + E^(2*u) + 2*E^u*Cos[v]]])/Sqrt[2]}
[x = 1/2*a*(1+exp(u)*cos(v)+(1+exp(2*u)+2*exp(u)*cos(v))^(1/2))^(1/2)*2^(1/2), y = 1/2*a*(-1-exp(u)*cos(v)+(1+exp(2*u)+2*exp(u)*cos(v))^(1/2))^(1/2)*2^(1/2)]


Constants of Cassinian
<math> <mrow> <mo>{</mo> <mi>a</mi> <mo>}</mo> </mrow> </math>

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2D Coordinate Systems: Cassinian from Differential Geometry Library.

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