Category:Definitions/Lemniscates
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This category contains definitions related to Lemniscates.
Related results can be found in Category:Lemniscates.
Geometric Definition
Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$.
The lemniscate of Bernoulli is the locus of points $M$ in the plane such that:
- $P_1 M \times P_2 M = a^2$
Cartesian Definition
The lemniscate of Bernoulli is the curve defined by the Cartesian equation:
- $\paren {x^2 + y^2}^2 = 2 a^2 \paren {x^2 - y^2}$
Polar Definition
The lemniscate of Bernoulli is the curve defined by the polar equation:
- $r^2 = 2 a^2 \cos 2 \theta$
Parametric Definition
The lemniscate of Bernoulli is the curve defined by the parametric equation:
- $\begin{cases} x = \dfrac {a \sqrt 2 \cos t} {\sin^2 t + 1} \\ y = \dfrac {a \sqrt 2 \cos t \sin t} {\sin^2 t + 1} \end{cases}$
Pages in category "Definitions/Lemniscates"
The following 16 pages are in this category, out of 16 total.
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- Definition:Lemniscate of Bernoulli
- Definition:Lemniscate of Bernoulli/Cartesian Definition
- Definition:Lemniscate of Bernoulli/Cartesian Definition/Also defined as
- Definition:Lemniscate of Bernoulli/Focus
- Definition:Lemniscate of Bernoulli/Geometric Definition
- Definition:Lemniscate of Bernoulli/Lobe
- Definition:Lemniscate of Bernoulli/Major Axis
- Definition:Lemniscate of Bernoulli/Major Semiaxis
- Definition:Lemniscate of Bernoulli/Parametric Definition
- Definition:Lemniscate of Bernoulli/Polar Definition
- Definition:Lemniscate of Bernoulli/Polar Definition/Also defined as
- Definition:Lobe of Lemniscate of Bernoulli