Decorative curves

Fish curve

The fish curve has the parametric equation

\[x(t) = a \cos(t) - \frac{a \sin^2(t)}{\sqrt{2}},\]

\[y(t) = a \cos(t) \sin(t).\]

See also Wikipedia [1377], MathWorld [1047], MathCurve [311].

The Python code for the example below can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D01_FishCurve.py, and the XML code in the file D01_FishCurve.2D.xml.

Heart curve

The heart curve has the parametric equation

\[x(t) = 16 \sin^3(t),\]

\[y(t) = 13 \cos(t) - 5 \cos(2t) - 2 \cos(3t) - \cos(4t)\]

See also Wikipedia [1379], MathWorld [1048].

The Python code for the example below can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D02_HeartCurve.py, and the XML code in the file D02_HeartCurve.2D.xml.

Chrysanthemum curve

The chrysanthemum curve (see Bourke [33]) is given in polar coordinates by the following

\[r = 5 (1 + \sin(11 t / 5)) - 4 \sin^4(17 t / 3) \sin^8(2 \cos(3 t) - 28 t)\quad \text{where } 0 \le t \le 21 \pi\]

and in Cartesian coordinates

\[x = r \cos(t)\]

\[y = r \sin(t)\]

The Python code for the example below can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D03_ChrysanthemumCurve.py, and the XML code in the file D03_ChrysanthemumCurve.2D.xml.

Butterfly curve

The butterfly curve is given in polar coordinates by the following

\[r = \exp(\cos(t)) - 2 \cos(4t) - \sin^5(t/12)\]

and in Cartesian coordinates

\[x = r \cos(t)\]

\[y = r \sin(t)\]

See also Wikipedia [1400], MathWorld [1057].

The Python code for the example below can be found online in the DataXlCalcNet repository or in the corresponding local DataXlCalcNet folder in the file D04_ButterflyCurve.py, and the XML code in the file D04_ButterflyCurve.2D.xml.