Nephroid

Nephroid

Nephroid ( Greek. Νεφρός - kidney, Greek. Εἶδο вид - view) is a 6th-order flat algebraic curve that is described by a fixed point on a circle that rolls outside of a twice as large circle. It is a special case of epicycloids with $k=2$ ${\ displaystyle k = 2}$ $k = 2$ . It is so named from dr. Greek. νεφρός - “kidney” and εἶδος - “appearance, figure” because of its shape resembling a kidney .

Nephroid as a caustic circle

Equations

Parametric equations :

{\begin{cases}x=3r\cos \varphi -r\cos(\alpha +3\varphi )\\y=3r\sin \varphi -r\sin(\alpha +3\varphi )\end{cases}}

{\ displaystyle {\ begin {cases} x = 3r \ cos \ varphi -r \ cos (\ alpha +3 \ varphi) \\ y = 3r \ sin \ varphi -r \ sin (\ alpha +3 \ varphi) \ end {cases}}}

If a $\alpha$ ${\ displaystyle \ alpha}$ equals zero:

{\begin{cases}x=6r\cos \varphi -4r\cos ^{3}\varphi \\y=4r\sin ^{3}\varphi \end{cases}}

{\ displaystyle {\ begin {cases} x = 6r \ cos \ varphi -4r \ cos ^ {3} \ varphi \\ y = 4r \ sin ^ {3} \ varphi \ end {cases}}}

Implicit Cartesian equation:

$(x^{2}+y^{2}-4a^{2})^{3}=108a^{4}y^{2}$ ${\ displaystyle (x ^ {2} + y ^ {2} -4a ^ {2}) ^ {3} = 108a ^ {4} y ^ {2}}$

Nephroid

Equations

See also

Literature

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