Golden Egg
From Portal
Cartesian Ovals
- Cartesian Ovals Mathworld Wolfram.com
- ChickScope University of Illinois Champaign-Urbana
- "Rene Descartes, a famous French mathematician and philosopher (perhaps best known for the deduction 'I think, therefore I am') invented the Cartesian coordinates that bear his name, and also discovered an interesting way to modify the pins-and-string construction for ellipses to produce more egg-shaped curves. In an ordinary ellipse, there are two fixed points (called the foci) in the midddle [sic] of the figure. The sum of the distances from a point on the ellipse to each of the two foci is the same for all points on the ellipse. In a Cartesian oval, there are still two foci. However now the distance from a point to the one focus plus TWICE its distance to the other focus is what remains the same for all points on the curve:
A Cartesian Oval is the figure consisting of all those points for which the sum of the distance to one focus plus twice the distance to a second focus is a constant.
You can draw such curves using pins and string if you modify the usual method for ellipses as follows: instead of joining the ends of the string to make a loop, tie the one end of the string to one of the pins, and attach the other end to the point of your pen or pencil. Loop the string once around the other pin and use the pencil point to pull the string tight. Now drag the pencil around."
- "Rene Descartes, a famous French mathematician and philosopher (perhaps best known for the deduction 'I think, therefore I am') invented the Cartesian coordinates that bear his name, and also discovered an interesting way to modify the pins-and-string construction for ellipses to produce more egg-shaped curves. In an ordinary ellipse, there are two fixed points (called the foci) in the midddle [sic] of the figure. The sum of the distances from a point on the ellipse to each of the two foci is the same for all points on the ellipse. In a Cartesian oval, there are still two foci. However now the distance from a point to the one focus plus TWICE its distance to the other focus is what remains the same for all points on the curve:
Cassinian Ovals
- Cassinian Ovals XahLee.org
- "Cassinian oval describe a family of curves. Cassinian Oval is defined as follows: Given fixed points F1 and F2. Given a constant c. The locus of points such that distance[P,F1] * distance[P,F2] == c is cassinian oval.
Cassinian oval is analogous to the definition of ellipse, where sum of two distances is replace by product."
- "Cassinian oval describe a family of curves. Cassinian Oval is defined as follows: Given fixed points F1 and F2. Given a constant c. The locus of points such that distance[P,F1] * distance[P,F2] == c is cassinian oval.
Quantum Ovals
Atomic orbital illustration of electron probability zones for the 3_{dz2} electron shells.
- See also The Orbitron Orbitron Website by Mark Winter: a gallery of atomic orbitals and molecular orbitals on the WWW
Notes
The angle of the side of a pyramid to its base is about 52° if the height is equal to the radius of a circle of the same circumference as the sum of the length of the sides of the base.
This angle of 52.84° is also the only angle that a plane may intersect a hyperbolic cone and create an egg-shaped ellipsoid of Golden Ratio proportion between the length and width of the egg-shaped curve.
Coats & Schauberger - Living Energies (2001);^{[1]}
See also
- Special Plane Curves XahLee.org
References
- ↑ Coats & Schauberger - Living Energies (2001) eBook at Scribd.com (http://www.scribd.com/doc/7953873/Coats-Schauberger-Living-Energies-2001)