.. |newpage| raw:: latex \newpage .. |br| raw:: html
|newpage| Spirals ============================================================== Archimedes' spiral --------------------------------------------------------------------------- .. method:: User.ArchimedesSpiral(a, Resolution, AsPolar = false) The parametric equations of Archimedes' Spiral are .. math:: r(t) = a t, .. math:: x(t) = r \cos(t), .. math:: y(t) = r \sin(t). Changing the parameter `a` controls the distance between loops. See also Wikipedia :cite:p:`Wikipedia2D209`, MathWorld :cite:p:`Wolfram2D209`, MathCurve :cite:p:`MathCurve2D209`. An example in Python .. code-block:: pycon >>> from mpfunlab import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') |Archimedes_Spiral_a| `\quad` |Archimedes_Spiral_b| `\quad` |Archimedes_Spiral_c| .. |Archimedes_Spiral_a| image:: ../_static/ParametricCurves/Spirals/Archimedes_Spiral.* :width: 30 % .. |Archimedes_Spiral_b| image:: ../_static/ParametricCurves/Spirals/Archimedes_Spiral.* :width: 30 % .. |Archimedes_Spiral_c| image:: ../_static/ParametricCurves/Spirals/Archimedes_Spiral.* :width: 30 % **Left figure**: Archimedes Spiral **Middle figure**: Archimedes Spiral **Right figure**: Archimedes Spiral |newpage| Fermat's spiral --------------------------------------------------------------------------- .. method:: User.FermatSpiral(a, Resolution, AsPolar = false) The parametric equations of Fermat' spiral are .. math:: r(t) = \pm a \sqrt{t}, .. math:: x(t) = r \cos(t), .. math:: y(t) = r \sin(t). Changing the parameter `a` controls the distance between loops. See also Wikipedia :cite:p:`Wikipedia2D210`, MathWorld :cite:p:`Wolfram2D210`, MathCurve :cite:p:`Wikipedia2D210`. An example in Python .. code-block:: pycon >>> from mpfunlab import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') |Fermats_Spiral_a| `\quad` |Fermats_Spiral_b| `\quad` |Fermats_Spiral_c| .. |Fermats_Spiral_a| image:: ../_static/ParametricCurves/Spirals/Fermat_Spiral.* :width: 30 % .. |Fermats_Spiral_b| image:: ../_static/ParametricCurves/Spirals/Fermat_Spiral.* :width: 30 % .. |Fermats_Spiral_c| image:: ../_static/ParametricCurves/Spirals/Fermat_Spiral.* :width: 30 % **Left figure**: Fermat's Spiral. **Middle figure**: Fermat's Spiral. **Right figure**: Fermat's Spiral. |newpage| Hyperbolic Spiral --------------------------------------------------------------------------- .. method:: User.HyperbolicSpiral(a, Resolution, AsPolar = false) The parametric equations of the hyperbolic spiral are .. math:: r(t) = \frac{a}{t}, .. math:: x(t) = r \cos(t), .. math:: y(t) = r \sin(t). Changing the parameter `a` controls the distance between loops. See also Wikipedia :cite:p:`Wikipedia2D211`, MathWorld :cite:p:`Wolfram2D211`, MathCurve :cite:p:`MathCurve2D211`. An example in Python .. code-block:: pycon >>> from mpfunlab import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') |Hyperbolic_Spiral_a| `\quad` |Hyperbolic_Spiral_b| `\quad` |Hyperbolic_Spiral_c| .. |Hyperbolic_Spiral_a| image:: ../_static/ParametricCurves/Spirals/Hyperbolic_Spiral.* :width: 30 % .. |Hyperbolic_Spiral_b| image:: ../_static/ParametricCurves/Spirals/Hyperbolic_Spiral.* :width: 30 % .. |Hyperbolic_Spiral_c| image:: ../_static/ParametricCurves/Spirals/Hyperbolic_Spiral.* :width: 30 % **Left figure**: Hyperbolic Spiral. **Middle figure**: Hyperbolic Spiral. **Right figure**: Hyperbolic Spiral. |newpage| Lituus --------------------------------------------------------------------------- .. method:: User.Lituus(a, Resolution, AsPolar = false) The parametric equations of the Lituus are .. math:: r(t) = \frac{a}{\sqrt{t}}, .. math:: x(t) = r \cos(t), .. math:: y(t) = r \sin(t). Changing the parameter `a` controls the distance between loops. See also Wikipedia :cite:p:`Wikipedia2D212`, MathWorld :cite:p:`Wolfram2D212`, MathCurve :cite:p:`MathCurve2D212`. An example in Python .. code-block:: pycon >>> from mpfunlab import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') |Lituus_a| `\quad` |Lituus_b| `\quad` |Lituus_c| .. |Lituus_a| image:: ../_static/ParametricCurves/Spirals/Lituus.* :width: 30 % .. |Lituus_b| image:: ../_static/ParametricCurves/Spirals/Lituus.* :width: 30 % .. |Lituus_c| image:: ../_static/ParametricCurves/Spirals/Lituus.* :width: 30 % **Left figure**: Lituus. **Middle figure**: Lituus. **Right figure**: Lituus. |newpage| Logarithmic Spiral ------------------------------------------------------------------------ .. method:: User.LogarithmicSpiral(a, Resolution, AsPolar = false) The parametric equations of the logarithmic spiral are .. math:: r(t) = a \exp(k t), .. math:: x(t) = r \cos(t), .. math:: y(t) = r \sin(t). where `a>0` and `k \ne 0` are real constants. See also Wikipedia :cite:p:`Wikipedia2D213`, MathWorld :cite:p:`Wolfram2D213`, MathCurve :cite:p:`MathCurve2D213`. An example in Python .. code-block:: pycon >>> from mpfunlab import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') |Logarithmic_Spiral_a| `\quad` |Logarithmic_Spiral_b| `\quad` |Logarithmic_Spiral_c| .. |Logarithmic_Spiral_a| image:: ../_static/ParametricCurves/Spirals/Logarithmic_Spiral.* :width: 30 % .. |Logarithmic_Spiral_b| image:: ../_static/ParametricCurves/Spirals/Logarithmic_Spiral.* :width: 30 % .. |Logarithmic_Spiral_c| image:: ../_static/ParametricCurves/Spirals/Logarithmic_Spiral.* :width: 30 % **Left figure**: Logarithmic Spiral. **Middle figure**: Logarithmic Spiral. **Right figure**: Logarithmic Spiral. |newpage| Poinsot's Spiral ------------------------------------------------------------------------ .. method:: User.PoinsotSpiral(a, Resolution, AsPolar = false) The parametric equations of the Poinsot spiral are .. math:: r(t) = \frac{a}{\alpha \cosh(k t) + \beta \sinh(k t)}, \quad \text{with } \alpha^2 + \beta^2 \ne 0. .. math:: x(t) = r \cos(t), .. math:: y(t) = r \sin(t). where `a>0` and `k \ne 0` are real constants. See also Wikipedia :cite:p:`Wikipedia2D219`, MathWorld :cite:p:`Wolfram2D219`, MathCurve :cite:p:`MathCurve2D219`. An example in Python .. code-block:: pycon >>> from mpfunlab import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') |Poinsot_Spiral_1_a| `\quad` |Poinsot_Spiral_2_a| .. |Poinsot_Spiral_1_a| image:: ../_static/ParametricCurves/Spirals/PoinsotSpiral1.* :width: 30 % .. |Poinsot_Spiral_2_a| image:: ../_static/ParametricCurves/Spirals/PoinsotSpiral1.* :width: 30 % **Left figure**: Poinsot's Spiral, type 1. **Middle figure**: Poinsot's Spiral, type 2. |newpage| Cotes's spiral ------------------------------------------------------------------------ .. method:: User.CotesSpiral(a, Resolution, AsPolar = false) The shape of spirals in the family depends on the parameters. The curves in polar coordinates, `(r, \theta), r>0` are defined by one of the following five equations: .. math:: \frac{1}{r} = \begin{cases} A \cosh(k\theta + \varepsilon) \\ A \exp(k\theta + \varepsilon) \\ A \sinh(k\theta + \varepsilon) \\ A (k\theta + \varepsilon) \\ A \cos(k\theta + \varepsilon) \\ \end{cases} `A > 0`, `k > 0` and `\varepsilon` are arbitrary real constants. `A` determines the size, `k` determines the shape, and `\varepsilon` determines the angular position of the spiral. The first and third forms are Poinsot's spirals; the second is the logarithmic spiral; the fourth is the hyperbolic spiral; the fifth is the epispiral. See also Wikipedia :cite:p:`Wikipedia2D220`, MathWorld :cite:p:`Wolfram2D220`. An example in Python .. code-block:: pycon >>> from mpfunlab import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') |Cotes_spiral_case_1_a| `\quad` |Cotes_spiral_case_1_b| `\quad` |Cotes_spiral_case_1_c| .. |Cotes_spiral_case_1_a| image:: ../_static/ParametricCurves/Spirals/Cotes_Spiral.* :width: 30 % .. |Cotes_spiral_case_1_b| image:: ../_static/ParametricCurves/Spirals/Cotes_Spiral.* :width: 30 % .. |Cotes_spiral_case_1_c| image:: ../_static/ParametricCurves/Spirals/Cotes_Spiral.* :width: 30 % **Left figure**: Cote's Spirals. **Middle figure**: Cote's Spirals. **Right figure**: Cote's Spirals. |newpage| Tanh Spiral ------------------------------------------------------------------------ .. method:: User.TanhSpiral(a, Resolution, AsPolar = false) This is a spiral proposed by Bourke :cite:p:`Bourke2D221`, and also discussed by Meier :cite:p:`Meier2D221`. It has the following parametric equations: .. math:: x(t) = \frac{\sinh(2t)}{\cos(2at) + \cosh(2t)}, .. math:: y(t) = \frac{\sin(2at)}{\cos(2at) + \cosh(2t)}, with `-\pi/2 \le t \le \pi/2` and `a>0` (eg:2). An example in Python .. code-block:: pycon >>> from mpfunlab import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') |Tanh_Spiral_a| `\quad` |Tanh_Spiral_b| `\quad` |Tanh_Spiral_c| .. |Tanh_Spiral_a| image:: ../_static/ParametricCurves/Spirals/Tanh_Spiral.* :width: 30 % .. |Tanh_Spiral_b| image:: ../_static/ParametricCurves/Spirals/Tanh_Spiral.* :width: 30 % .. |Tanh_Spiral_c| image:: ../_static/ParametricCurves/Spirals/Tanh_Spiral.* :width: 30 % **Left figure**: Tanh Spiral. **Middle figure**: Tanh Spiral. **Right figure**: Tanh Spiral. |newpage| Nielsens Spiral ------------------------------------------------------------------------ .. method:: User.NielsenSpiral(a, Resolution, AsPolar = false) Nielsen's spiral, also called the sici spiral is the spiral with parametric equations .. math:: x(t) = a \cdot \mathrm{ci}(t), .. math:: y(t) = a \cdot \mathrm{si}(t), where `\mathrm{ci}(t)` is the cosine integral and `\mathrm{si}(t)` is the sine integral. See also Wikipedia :cite:p:`Wikipedia2D217`, MathWorld :cite:p:`Wolfram2D217`, MathCurve :cite:p:`MathCurve2D217`. An example in Python .. code-block:: pycon >>> from mpfunlab import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') |Nielsens_Spiral_a| `\quad` |Nielsens_Spiral_b| `\quad` |Nielsens_Spiral_c| .. |Nielsens_Spiral_a| image:: ../_static/ParametricCurves/Spirals/Nielsens_Spiral.* :width: 30 % .. |Nielsens_Spiral_b| image:: ../_static/ParametricCurves/Spirals/Nielsens_Spiral.* :width: 30 % .. |Nielsens_Spiral_c| image:: ../_static/ParametricCurves/Spirals/Nielsens_Spiral.* :width: 30 % **Left figure**: Nielsens Spiral. **Middle figure**: Nielsens Spiral. **Right figure**: Nielsens Spiral. |newpage| Cornu Spiral ------------------------------------------------------------------------ .. method:: User.CornuSpiral(a, Resolution, AsPolar = false) The Cornu spiral, also known as a Euler spiral or clothoid, is the curve generated by a parametric plot of S(t) against C(t). A Cornu spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin. .. math:: x(t) = a \cdot C(t), .. math:: y(t) = a \cdot S(t), where `C(t)` and `S(t)` are the Fresnel integrals. See also Wikipedia :cite:p:`Wikipedia2D218`, MathWorld :cite:p:`Wolfram2D218`, MathCurve :cite:p:`MathCurve2D218`. An example in Python .. code-block:: pycon >>> from mpfunlab import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') |Cornu_Spiral_a| `\quad` |Cornu_Spiral_b| `\quad` |Cornu_Spiral_c| .. |Cornu_Spiral_a| image:: ../_static/ParametricCurves/Spirals/Cornu_Spiral.* :width: 30 % .. |Cornu_Spiral_b| image:: ../_static/ParametricCurves/Spirals/Cornu_Spiral.* :width: 30 % .. |Cornu_Spiral_c| image:: ../_static/ParametricCurves/Spirals/Cornu_Spiral.* :width: 30 % **Left figure**: Cornu Spiral. **Middle figure**: Cornu Spiral. **Right figure**: Cornu Spiral.