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DECORATIVE CURVES
This is an index of some curves, the interest of which is less mathematical than merely reproducing existing shapes.
Start with the water drop, the fish, the torpedo,
the mouth, the pinochoid,
the Maltese cross, the
multicardioidds,
the Rosillo curves, the basins.
Habenicht trefoil (1895, Brocard
p. 100)
Polar equation: Opposite, n = 3 and 4. See other trefoils at the bottom of the page dedicated to the quadrifolium. 

Eugen Beutel heart (1909)
(Eugen Beutel: Algebraische Kurven, G.J. Göschen, Leipzig 190911) Cartesian equation:


Raphaël Laporte heart (1993)
Curve discovered by the author at the age of 16 for his girlfriend...
Cartesian equation: x^8x^6+27*x^427*x^2+12*y*x^612*y*x^4+42*y^2*x^4


Dwight Boddorf heart (2008)
Polar equation: (portion of a curve that can in fact be extended on both sides of the sharp point) 

Jurjen
Boss heart
i.e. This heart is composed of one half of the reunion of two eightlike quartics: and . Therefore, two eights = two hearts! 

Pierre Daniel heart (2013)
Cartesian equation (nonrational biquartic): 10836*y^2448*x^2*y+112*x^2*y^2+405*x^4112*y^3+12*y^4+66*x^4*y^2+224*x^4*y+128*x^2*y^3+24*y^4*x^2+ 30*x^6+24*y^5+4*y^6+x^8328*x^2+216*y+8*x^6*y+2*x^6*y^2+8*x^4*y^3+y^4*x^4=0 For an index of heartlike curves: www.mathematischebasteleien.de/heart.htm mathworld.wolfram.com/HeartCurve.html 

Ehrhart egg (1957)
Punctual equation:
Here, A = (0 ; 0), B = (0 ; 0,2), C = (0 ; 1), cte = 2,2. Ehrhart called these curves "hyperellipses with 3 foci"; they are referred to on this site as triellipses. See other eggs at ovoid.


Bow tie
Polar equation: .


T. Fay butterfly (1989)
Polar equation: (on the right ). 

L. Sautereau butterflies (2010 ?) See also the Cundy and Rollett butterfly. 

Swastika (Cundy and Rollet)
Polar equation: Cartesian equation: On the right:


Yin Yang curve
Polar equation: for (opposite with a = p/2), plus the circle . Variation with the asymptotic circle on the right: , i.e. , with a = 1/2. R. Ferréol (2006)


Flying saucer
Take two curves like , for example and rotate around the symmetry axis. See also here. 

See also the meander curve and the elastic curve.
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© Robert FERRÉOL 2017