Bernoulli lemniscate . In mathematics, a lemniscate is a type of curve described by the following equation in Cartesian coordinates :
Summary
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- 1 Story
- 2 Definition
- 3 equations
- 4 Construction
- 5 See also
- 6 Sources
History
The Lemniscate is currently considered as a particular case of the ovals studied by Jean Dominique Cassini in 1680 , but it is Jackob Bernoulli and Johann Bernoulli who discovered it and made it known independently, trying to solve a problem posed by creator Leibnitz , together with Isaac Newton of infinitesimal calculus, which gave the scientific community of the time the challenge of finding the equation of the paracentric isochrone. The solution found was the Lemniscata which, in honor of the Bernoulli brothers, was renamed Lemniscata de Bernoulli.
Bernoulli called it lemniscus, which in Latin means “hanging ribbon”.
The graphical representation of this equation generates a curve similar to ∞ . The curve has become the symbol of infinity and is widely used in mathematics.
Definition
It is the curve that is defined as the locus of the points such that the sum of the distances from two focal points is a constant.
Equations
- Cartesian coordinate equation:
- The lemniscate can be described by polar coordinatesaccording to the following equation:
r 2 = a 2 cos2θ
- Similarly, with bipolar coordinates its equation is:
rr’ = a 2 /2
Building
Bernoulli lemniscate
A circumference of center F ‘is constructed, tangent to two perpendicular lines that intersect at point O.
Let P be any point on the circle and Q be the point of intersection of the ray OP and the circle. Plot the circumference of center O and radius PQ.
Let M be the point of intersection of the ray and the center circumference O.
The locus described by point M and the symmetric of M with respect to O (M ‘), as P moves in the center circumference F’, is a Bernuolli Lemniscate.
