Decorative curves

Decorative curves#

Fish curve#

User.FishCurve(a, Resolution, AsPolar=false)#

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 [1493], MathWorld [1137], MathCurve [325].

An example in Python

>>> from mpfunlab import User
>>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true)
>>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid')

Fish_curve_a \(\quad\) Fish_curve_b \(\quad\) Fish_curve_c

Left figure: Fish curve

Middle figure: Fish curve

Right figure: Fish curve

Heart curve#

User.HeartCurve(a, Resolution, AsPolar=false)#

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 [1495], MathWorld [1138].

An example in Python

>>> from mpfunlab import User
>>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true)
>>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid')

An example in C#

using User;
Curve = User.Cardioid(a: 1, Resolution: 200, AsPolar: true);
User.Chart2D.Show(Curve, Template = "PolarCurve", Title = "Cardioid");

Heart_curve_a \(\quad\) Heart_curve_b \(\quad\) Heart_curve_c

Left figure: Heart curve

Middle figure: Heart curve

Right figure: Heart curve

Chrysanthemum curve#

User.ChrysanthemumCurve(a, Resolution, AsPolar=false)#

The chrysanthemum curve (see Bourke [38]) 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

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

An example in Python

>>> from mpfunlab import User
>>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true)
>>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid')

An example in C#

using User;
Curve = User.Cardioid(a: 1, Resolution: 200, AsPolar: true);
User.Chart2D.Show(Curve, Template = "PolarCurve", Title = "Cardioid");

Chrysanthemum_curve_a \(\quad\) Chrysanthemum_curve_b \(\quad\) Chrysanthemum_curve_c

Left figure: Chrysanthemum curve

Middle figure: Chrysanthemum curve

Right figure: Chrysanthemum curve

Butterfly curve#

User.ButterflyCurve(a, Resolution, AsPolar=false)#

See also Wikipedia [1519], MathWorld [1147].

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

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

An example in Python

>>> from mpfunlab import User
>>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true)
>>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid')

An example in C#

using User;
Curve = User.Cardioid(a: 1, Resolution: 200, AsPolar: true);
User.Chart2D.Show(Curve, Template = "PolarCurve", Title = "Cardioid");

Butterfly_curve_a \(\quad\) Butterfly_curve_b \(\quad\) Butterfly_curve_c

Left figure: Butterfly curve

Middle figure: Butterfly curve

Right figure: Butterfly curve

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