=== Geometric Definition equivalent to Cartesian Definition === Let $M$ be a lemniscate of Bernoulli by the geometric definition. Then by definition: :{{:Definition:Lemniscate of Bernoulli/Geometric Definition}} Let $P_1 = \tuple {a, 0}$ and $P_2 = \tuple {-a, 0}$. Let $p = \tuple {x, y}$ be an arbitrary point of $M$. We have: {{begin-eqn}} {{eqn | l = P_1 p       | r = \sqrt {\paren {a - x}^2 + y^2}       | c = Distance Formula }} {{eqn | r = \sqrt {a^2 - 2 a x + x^2 + y^2}       | c =  }} {{eqn | l = P_2 p       | r = \sqrt {\paren {a + x}^2 + y^2}       | c = Distance Formula }} {{eqn | r = \sqrt {a^2 + 2 a x + x^2 + y^2}       | c =  }} {{eqn | ll= \leadsto       | l = \paren {P_1 p} \paren {P_2 p}       | r = \sqrt {a^2 - 2 a x + x^2 + y^2} \sqrt {a^2 + 2 a x + x^2 + y^2}       | c = {{Defof|Lemniscate of Bernoulli|subdef = Geometric Definition}} }} {{eqn | r = a^2       | c =  }} {{eqn | ll= \leadsto       | l = a^4       | r = \paren {a^2 - 2 a x + x^2 + y^2} \paren {a^2 + 2 a x + x^2 + y^2}       | c =  }} {{eqn | r = \paren {a^2 + \paren {x^2 + y^2} }^2 - 4 a^2 x^2       | c = Difference of Two Squares }} {{eqn | r = a^4 + 2 a^2 \paren {x^2 + y^2} + \paren {x^2 + y^2}^2 - 4 a^2 x^2       | c = Square of Sum }} {{eqn | r = a^4 + 2 a^2 \paren {y^2 - x^2} + \paren {x^2 + y^2}^2       | c = simplifying }} {{eqn | ll= \leadsto       | l = \paren {x^2 + y^2}^2       | r = 2 a^2 \paren {x^2 - y^2}       | c =  }} {{end-eqn}} Thus $M$ is a lemniscate of Bernoulli by the Cartesian definition. {{qed|lemma}} === Geometric Definition equivalent to Polar Definition === Let $M$ be a lemniscate of Bernoulli by the geometric definition. Then by definition: :{{:Definition:Lemniscate of Bernoulli/Geometric Definition}} Let $M$ be embedded in a polar coordinate plane whose origin is at $O$ and such that $P_1 = \polar {a, 0}$ and $P_2 = \polar {a, \pi}$. 520px Consider an arbitrary point $p = \polar {r, \theta}$. Let $d_1 = \size {P_1 p}$ and $d_2 = \size {P_2 p}$. We have: {{begin-eqn}} {{eqn | l = d_1 d_2       | r = a^2       | c = {{Defof|Lemniscate of Bernoulli|subdef = Geometric Definition}} }} {{eqn | r = \sqrt {r^2 + a^2 - 2 a r \cos \theta} \times \sqrt {r^2 + a^2 - 2 a r \, \map \cos {\pi - \theta} }       | c = Cosine Rule }} {{eqn | ll= \leadsto       | l = \paren {a^2}^2       | r = \paren {r^2 + a^2 - 2 a r \cos \theta} \paren {r^2 + a^2 + 2 a r \cos \theta}       | c = Cosine of Supplementary Angle, and squaring throughout }} {{eqn | r = \paren {r^2 + a^2}^2 - \paren {2 a r \cos \theta}^2       | c = Difference of Two Squares }} {{eqn | r = \paren {r^2}^2 + 2 a^2 r^2 + \paren {a^2}^2 - 4 a^2 r^2 \cos^2 \theta       | c =  }} {{eqn | ll= \leadsto       | l = r^2       | r = 2 a^2 \paren {2 \cos^2 \theta - 1}       | c = simplifying and rearranging }} {{eqn | ll= \leadsto       | l = r^2       | r = 2 a^2 \cos 2 \theta       | c = Double Angle Formula for Cosine }} {{end-eqn}} {{qed|lemma}} === Parametric Definition equivalent to Cartesian Definition === Let $M$ be a lemniscate of Bernoulli by the parametric definition. Then by definition: :{{:Definition:Lemniscate of Bernoulli/Parametric Definition}} We have: {{begin-eqn}} {{eqn | l = x^2 - y^2       | r = \dfrac {2 a^2 \cos^2 t} {\paren {\sin^2 t + 1}^2} - \dfrac {2 a^2 \cos^2 t \sin^2 t} {\paren {\sin^2 t + 1}^2}       | c =  }} {{eqn | r = \dfrac {2 a^2 \cos^2 t \paren {1 - \sin^2 t} } {\paren {\sin^2 t + 1}^2}       | c =  }} {{eqn | n = 1       | r = \dfrac {2 a^2 \paren {1 - \sin^2 t}^2} {\paren {\sin^2 t + 1}^2}       | c = Sum of Squares of Sine and Cosine }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = x^2 + y^2       | r = \dfrac {2 a^2 \cos^2 t} {\paren {\sin^2 t + 1}^2} + \dfrac {2 a^2 \cos^2 t \sin^2 t} {\paren {\sin^2 t + 1}^2}       | c =  }} {{eqn | r = \dfrac {2 a^2 \cos^2 t \paren {1 + \sin^2 t} } {\paren {\sin^2 t + 1}^2}       | c =  }} {{eqn | r = \dfrac {2 a^2 \paren {1 - \sin^2 t} \paren {1 + \sin^2 t} } {\paren {\sin^2 t + 1}^2}       | c = Sum of Squares of Sine and Cosine }} {{eqn | r = \dfrac {2 a^2 \paren {1 - \sin^2 t} } {\sin^2 t + 1}       | c = simplifying }} {{eqn | ll= \leadsto       | l = \paren {x^2 + y^2}^2       | r = 2 a^2 \dfrac {2 a^2 \paren {1 - \sin^2 t}^2 } {\paren {\sin^2 t + 1}^2}       | c = squaring both sides and extracting $2 a^2$ }} {{eqn | r = 2 a^2 \paren {x^2 - y^2}       | c = from $(1)$ }} {{end-eqn}} {{qed}} Category:Lemniscates aw1r82z4f6ntnzfx2ibe9y50vclrvry