Cassini ovals . It is the locus of a point where the product of the distances to two fixed points F1, F2 is the constant (b 2 ).
Summary
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- 1 Story
- 2 equations
- 3 Properties
- 4 Construction
- 5 Sources
History
This curve also called Cassinian or Ellipse Cassinian was conceived by Giovanni Domenico Cassini . He studied this family of curves in relation to the relative movements of the Sun and Earth in 1680 . He believed that these curves could represent planetary orbits.
Cassini conceived of these curves fourteen years before Jakob Bernoulli described his lemniscate. Gian Francesco Malfatti studied these curves in 1781 .
Equations
The Cartesian equation for Cassini ovals is:
(x 2 + y 2 ) 2 -2a 2 (x 2 -y 2 ) + a 4 = b 4
The polar equation for Cassini ovals is:
r 4 + a 4 – 2r 2 a 2 cos 2θ = b 4
F1 = (−a, 0) F2 = (a, 0)
The relationship between values a and b determines the type of curve that will be obtained.
- If b> a, the curve will be an oval that does not cut itself.
- If b = a, Bernoulli’s Lemniscatewill appear .
- If b <a, it will represent two separate ovals.
Properties
If b − a is the inner radius of a torus whose generator circle has radius a. The section formed by a plane parallel to the axis of the torus and distant from it is a Cassini oval.
If b = 2a this plane is an interior tangent to the surface and the section is a Lemniscate. Arbitrary cuts of the bull are not Cassini ovals. The intersection of a torus and a plane parallel to the torus axis is called the spiric section .
Building
To construct Cassine ovals, the following steps must be performed:
- Draw a circumferencewhose diameter is the focal length.
- Once the major axis AB is fixed, mark a point P on the circumference.
- Draw from vertex A the raythat passes through P and that will intersect the circumference at point Q.
- With centers in focus F and in F´ draw two circles of radii AQ and AP, respectively.
- The intersection points M and N of these two circles will be points on the curve.
We apply movement to point P to obtain the representation of the curve (Geometric Place represented by points M and N when point P travels the circumference). We can move a focus on the axis to obtain the different types of ovals.
