next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

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 1909-11)

Cartesian equation:

Non-rational sextic.
 

....................
Raphaël Laporte heart (1993)

Curve discovered by the author at the age of 16 for his girlfriend...
Cartesian parametrization: 
Rational biquartic.

Cartesian equation:

x^8-x^6+27*x^4-27*x^2+12*y*x^6-12*y*x^4+42*y^2*x^4
+42*y^2*x^2+2*y^3*x^4+26*y^3*x^2+8*y^3+12*y^4*x^2+12*y^4+6*y^5+y^6=0
 

 

.......
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 eight-like quartics and .
Therefore, two eights = two hearts!
Pierre Daniel heart (2013)

Cartesian equation (non-rational biquartic):
-108-36*y^2-448*x^2*y+112*x^2*y^2+405*x^4-112*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^8-328*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 heart-like curves:
www.mathematische-basteleien.de/heart.htm
mathworld.wolfram.com/HeartCurve.html
Ehrhart egg (1957)

Punctual equation: 
I do not know the original parameters chosen by Ehrahrt (who said that with these parameters, the curve coincides, up to the thickness of the pencil stroke, with the shape of an egg obtained by a statistical study).

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: .
Cartesian equation: 
or 
Asymptotic parabolas (in green): .
Variation: , i.e. , the eight-like portion being the lemniscate of Gérono.
 

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: 
Closer:

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.
 
next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

© Robert FERRÉOL  2017