.. |newpage| raw:: latex \newpage .. |br| raw:: html
|newpage| Surfaces of revolution: spheres and related ====================================================== See also: https://mathworld.wolfram.com/SurfaceofRevolution.html A surface of revolution is a surface generated by rotating a two-dimensional curve about an axis. The resulting surface therefore always has azimuthal symmetry. Examples of surfaces of revolution include the apple surface, cone (excluding the base), conical frustum (excluding the ends), cylinder (excluding the ends), Darwin-de Sitter spheroid, Gabriel's horn, hyperboloid, lemon surface, oblate spheroid, paraboloid, prolate spheroid, pseudosphere, sphere, spheroid, and torus (and its generalization, the toroid). The standard parameterization of a surface of revolution is given by .. math:: x(u,v) = \phi(v) \cos(u) .. math:: y(u,v) = \phi(v) \sin(u) .. math:: z(u,v) = \psi(v) Prism and cylinder ----------------------------------------------------- See also: http://www.3d-meier.de/tut3/Seite84.html, Cylinder See also: https://mathworld.wolfram.com/Cylinder.html, Cylinder |01a_ROT_Prism3| `\quad` |01b_ROT_Prism4| `\quad` |01c_ROT_Cylinder| .. |01a_ROT_Prism3| image:: ../_static/On2DCurves/RevolutionOpen/01a_ROT_Prism3.3D.xml.jpg :width: 30 % .. |01b_ROT_Prism4| image:: ../_static/On2DCurves/RevolutionOpen/01b_ROT_Prism4.3D.xml.jpg :width: 30 % .. |01c_ROT_Cylinder| image:: ../_static/On2DCurves/RevolutionOpen/01c_ROT_Cylinder.3D.xml.jpg :width: 30 % **Left figure**: Prism3 **Right figure**: Prism4 **Right figure**: Cylinder |newpage| Pyramid and cone ------------------------------------------------------ |02a_ROT_Pyramid3_surface| `\quad` |02b_ROT_Pyramid4_surface| `\quad` |02c_ROT_Cone_surface| .. |02a_ROT_Pyramid3_surface| image:: ../_static/On2DCurves/RevolutionOpen/02a_ROT_Pyramid3_surface.3D.xml.jpg :width: 30 % .. |02b_ROT_Pyramid4_surface| image:: ../_static/On2DCurves/RevolutionOpen/02b_ROT_Pyramid4_surface.3D.xml.jpg :width: 30 % .. |02c_ROT_Cone_surface| image:: ../_static/On2DCurves/RevolutionOpen/02c_ROT_Cone_surface.3D.xml.jpg :width: 30 % **Left figure**: Pyramid3 **Right figure**: Pyramid4 **Right figure**: Cone |newpage| Frustum and cone frustum ------------------------------------------------------ |03a_ROT_Frustum3_surface| `\quad` |03b_ROT_Frustum4_surface| `\quad` |03c_ROT_ConeFrustum_surface| .. |03a_ROT_Frustum3_surface| image:: ../_static/On2DCurves/RevolutionOpen/03a_ROT_Frustum3_surface.3D.xml.jpg :width: 30 % .. |03b_ROT_Frustum4_surface| image:: ../_static/On2DCurves/RevolutionOpen/03b_ROT_Frustum4_surface.3D.xml.jpg :width: 30 % .. |03c_ROT_ConeFrustum_surface| image:: ../_static/On2DCurves/RevolutionOpen/03c_ROT_ConeFrustum_surface.3D.xml.jpg :width: 30 % **Left figure**: Frustum3 **Right figure**: Frustum4 **Right figure**: Cone Frustum |newpage| Insolator --------------------------------------------------- See also: http://www.3d-meier.de/tut3/Seite86.html, Insolator |04a_ROT_Insolator_surface| .. |04a_ROT_Insolator_surface| image:: ../_static/On2DCurves/RevolutionOpen/04a_ROT_Insolator_surface.3D.xml.jpg :width: 30 % **Left figure**: Insolator |newpage| Gabriels Horn -------------------------------------------- See also: https://mathworld.wolfram.com/GabrielsHorn.html See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. |05a_ROT_GabrielHorn_surface| .. |05a_ROT_GabrielHorn_surface| image:: ../_static/On2DCurves/RevolutionOpen/05a_ROT_GabrielHorn_surface.3D.xml.jpg :width: 30 % **Left figure**: Gabriels Horn |newpage| Funnel ------------------------------------------------- // see https://mathworld.wolfram.com/Funnel.html // see http://www.3d-meier.de/tut3/Seite28.html // see http://www.3d-meier.de/tut3/Seite27.html See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. |06a_ROT_Funnel_surface| .. |06a_ROT_Funnel_surface| image:: ../_static/On2DCurves/RevolutionOpen/06a_ROT_Funnel_surface.3D.xml.jpg :width: 30 % **Left figure**: Funnel |newpage| One-Sheeted Hyperboloid ------------------------------------------ See also: https://mathworld.wolfram.com/One-SheetedHyperboloid.html See also: https://en.wikipedia.org/wiki/Hyperboloid |07a_ROT_Hyperboloid_surface| .. |07a_ROT_Hyperboloid_surface| image:: ../_static/On2DCurves/RevolutionOpen/07a_ROT_Hyperboloid_surface.3D.xml.jpg :width: 30 % **Left figure**: One-Sheeted Hyperboloid |newpage| Catenoid ----------------------------------------- See also: https://mathworld.wolfram.com/Catenoid.html See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. |08a_ROT_Catenoid_surface| .. |08a_ROT_Catenoid_surface| image:: ../_static/On2DCurves/RevolutionOpen/08a_ROT_Catenoid_surface.3D.xml.jpg :width: 30 % **Left figure**: Catenoid |newpage| Gauss Cylinder --------------------------------------------- See also: http://www.3d-meier.de/tut3/Seite157.html Gauss Cylinder See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. |09a_ROT_GCylinder_surface| `\quad` |09b_ROT_GCylinder_surface| .. |09a_ROT_GCylinder_surface| image:: ../_static/On2DCurves/RevolutionOpen/09a_ROT_GCylinder_surface.3D.xml.jpg :width: 30 % .. |09b_ROT_GCylinder_surface| image:: ../_static/On2DCurves/RevolutionOpen/09b_ROT_GCylinder_surface.3D.xml.jpg :width: 30 % **Left figure**: Gauss Cylinder **Right figure**: Gauss Cylinder |newpage| Paraboloid ------------------------------------- |10a_ROT_Paraboloid_surface| .. |10a_ROT_Paraboloid_surface| image:: ../_static/On2DCurves/RevolutionOpen/10a_ROT_Paraboloid_surface.3D.xml.jpg :width: 30 % **Left figure**: Paraboloid |newpage| Vase ----------------------------------- |11a_ROT_Vase| `\quad` |11b_ROT_Vase| .. |11a_ROT_Vase| image:: ../_static/On2DCurves/RevolutionOpen/11a_ROT_Vase.3D.xml.jpg :width: 30 % .. |11b_ROT_Vase| image:: ../_static/On2DCurves/RevolutionOpen/11b_ROT_Vase.3D.xml.jpg :width: 30 % **Left figure**: Vase **Right figure**: Vase |newpage| Sphere ------------------------------------------------------------------------------------------ .. method:: User.ParametricSphere(a, Resolution, Revolution = true) See also: http://www.3d-meier.de/tut3/Seite86.html, Sphere |TestSphere_a| .. |TestSphere_a| image:: ../_static/On2DCurves/RevolutionClosed/01a_ROT_Sphere_surface.3D.xml.jpg :width: 30 % **Left figure**: Sphere |01b_ROT_Sphere_lower_half| `\quad` |01c_ROT_Sphere_right_half| `\quad` |01d_ROT_Sphere_quarter| .. |01b_ROT_Sphere_lower_half| image:: ../_static/On2DCurves/RevolutionClosed/01b_ROT_Sphere_lower_half.3D.xml.jpg :width: 30 % .. |01c_ROT_Sphere_right_half| image:: ../_static/On2DCurves/RevolutionClosed/01c_ROT_Sphere_right_half.3D.xml.jpg :width: 30 % .. |01d_ROT_Sphere_quarter| image:: ../_static/On2DCurves/RevolutionClosed/01d_ROT_Sphere_quarter.3D.xml.jpg :width: 30 % **Left figure**: Gear torus surface **Middle figure**: Gear torus surface **Right figure**: Gear torus surface |newpage| Prolate Spheroid ---------------------------------------------------- .. method:: User.ProlateSpheroid(a, Resolution, Revolution = true) See also: http://paulbourke.net/geometry/spherical/ prolate spheroid, or ellipsoid of revolution See also: https://en.wikipedia.org/wiki/Ellipsoid#Parameterization ellipsoid of revolution See also: https://mathworld.wolfram.com/OblateSpheroid.html Oblate Spheroid |ProlateSpheroid_small_a| `\quad` |ProlateSpheroid_small_b| .. |ProlateSpheroid_small_a| image:: ../_static/On2DCurves/RevolutionClosed/03a_ROT_ProlateSpheroid_surface.3D.xml.jpg :width: 30 % .. |ProlateSpheroid_small_b| image:: ../_static/On2DCurves/RevolutionClosed/03b_ROT_ProlateSpheroidSqrt.3D.xml.jpg :width: 30 % **Left figure**: Prolate Spheroid. **Right figure**: Prolate Spheroid via Sqrt. |newpage| Oblate Spheroid ---------------------------------------------------- .. method:: User.OblateSpheroid(a, Resolution, Revolution = true) See also: http://paulbourke.net/geometry/spherical/ prolate spheroid, or ellipsoid of revolution See also: https://en.wikipedia.org/wiki/Ellipsoid#Parameterization ellipsoid of revolution See also: https://mathworld.wolfram.com/OblateSpheroid.html Oblate Spheroid |OblateSpheroid_small_a| `\quad` |OblateSpheroid_small_b| .. |OblateSpheroid_small_a| image:: ../_static/On2DCurves/RevolutionClosed/04a_ROT_OblateSpheroid_surface.3D.xml.jpg :width: 30 % .. |OblateSpheroid_small_b| image:: ../_static/On2DCurves/RevolutionClosed/04b_ROT_OblateSpheroidSqrt.3D.xml.jpg :width: 30 % **Left figure**: Prolate Spheroid. **Right figure**: Prolate Spheroid via Sqrt. |newpage| Drop surface ------------------------------------------------- // see http://www.3d-meier.de/tut3/Seite44.html An example in Python .. code-block:: pycon >>> from mpfebnet import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') An example in C\# .. code-block:: csharp using User; Curve = User.Cardioid(a: 1, Resolution: 200, AsPolar: true); User.Chart2D.Show(Curve, Template = "PolarCurve", Title = "Cardioid"); |07a_ROT_Drop_Surface| .. |07a_ROT_Drop_Surface| image:: ../_static/On2DCurves/RevolutionClosed/07a_ROT_Drop_Surface.3D.xml.jpg :width: 30 % **Left figure**: Drop surface |newpage| Apple surface -------------------------------------------------------------------- Examples in Bourke, towards the end of the page // see http://www.3d-meier.de/tut3/Seite54.html https://paulbourke.net/geometry/spherical/ For the apple surface, also see Meier |09a_ROT_Apple_Surface| .. |09a_ROT_Apple_Surface| image:: ../_static/On2DCurves/RevolutionClosed/09a_ROT_Apple_Surface.3D.xml.jpg :width: 30 % **Left figure**: Apple surface |newpage| Spinning top ------------------------------------------------------------------------- .. method:: User.SpinningTop(a, Resolution, Revolution = true) See also http://www.3d-meier.de/tut3/Seite150.html Kreisel See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. |11a_ROT_Spinningtop_surface| .. |11a_ROT_Spinningtop_surface| image:: ../_static/On2DCurves/RevolutionClosed/11a_ROT_Spinningtop_surface.3D.xml.jpg :width: 30 % **Left figure**: Spinning top |newpage| Pseudosphere ----------------------------------------------------------- .. method:: User.Pseudosphere(a, Resolution, Revolution = true) See also: https://mathworld.wolfram.com/Pseudosphere.html // see http://www.3d-meier.de/tut3/Seite28.html An example in Python .. code-block:: pycon >>> from mpfebnet import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') An example in C\# .. code-block:: csharp using User; Curve = User.Cardioid(a: 1, Resolution: 200, AsPolar: true); User.Chart2D.Show(Curve, Template = "PolarCurve", Title = "Cardioid"); |12a_ROT_Pseudosphere_ROT0b_surface| `\quad` |12b_ROT_Pseudosphere| .. |12a_ROT_Pseudosphere_ROT0b_surface| image:: ../_static/On2DCurves/RevolutionClosed/12a_ROT_Pseudosphere_ROT0b_surface.3D.xml.jpg :width: 30 % .. |12b_ROT_Pseudosphere| image:: ../_static/On2DCurves/RevolutionClosed/12b_ROT_Pseudosphere.3D.xml.jpg :width: 30 % **Left figure**: Pseudosphere **Right figure**: Pseudosphere |newpage| Wave sphere Surface -------------------------------------------------------------------------------------- // see http://www.3d-meier.de/tut3/Seite63.html |13a_ROT_WaveSphere| `\quad` |13b_ROT_WaveSphere| .. |13a_ROT_WaveSphere| image:: ../_static/On2DCurves/RevolutionClosed/13a_ROT_WaveSphere.3D.xml.jpg :width: 30 % .. |13b_ROT_WaveSphere| image:: ../_static/On2DCurves/RevolutionClosed/13b_ROT_WaveSphere.3D.xml.jpg :width: 30 % **Left figure**: Wave sphere Surface **Right figure**: Wave sphere Surface Low resolution torus surface, part1 ----------------------------------------------------------------------------------------- // see https://mathworld.wolfram.com/Torus.html See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. |91a_ROT_Torus_surface| `\quad` |91b_ROT_Torus_surface| `\quad` |91c_ROT_Torus_surface| .. |91a_ROT_Torus_surface| image:: ../_static/On2DCurves/RevolutionTori/91a_ROT_Torus_surface.3D.xml.jpg :width: 30 % .. |91b_ROT_Torus_surface| image:: ../_static/On2DCurves/RevolutionTori/91b_ROT_Torus_surface.3D.xml.jpg :width: 30 % .. |91c_ROT_Torus_surface| image:: ../_static/On2DCurves/RevolutionTori/91c_ROT_Torus_surface.3D.xml.jpg :width: 30 % **Left figure**: Low resolution torus surface, part1 **Middle figure**: Low resolution torus surface, part1 **Right figure**: Low resolution torus surface, part1 |newpage| Low resolution torus surface, part2 ----------------------------------------------------------------------------------------- See also: http://www.3d-meier.de/tut3/Seite165.html See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. |92a_ROT_Torus_surface| `\quad` |92b_ROT_Torus_surface| `\quad` |92c_ROT_Torus_surface| .. |92a_ROT_Torus_surface| image:: ../_static/On2DCurves/RevolutionTori/92a_ROT_Torus_surface.3D.xml.jpg :width: 30 % .. |92b_ROT_Torus_surface| image:: ../_static/On2DCurves/RevolutionTori/92b_ROT_Torus_surface.3D.xml.jpg :width: 30 % .. |92c_ROT_Torus_surface| image:: ../_static/On2DCurves/RevolutionTori/92c_ROT_Torus_surface.3D.xml.jpg :width: 30 % **Left figure**: Low resolution torus surface, part1 **Middle figure**: Low resolution torus surface, part1 **Right figure**: Low resolution torus surface, part1 |newpage| Low resolution torus surface, part3 ----------------------------------------------------------------------------------------- See also: http://www.3d-meier.de/tut3/Seite165.html See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. |93a_ROT_Torus_surface| `\quad` |93b_ROT_Torus_surface| `\quad` |93c_ROT_Torus_surface| .. |93a_ROT_Torus_surface| image:: ../_static/On2DCurves/RevolutionTori/93a_ROT_Torus_surface.3D.xml.jpg :width: 30 % .. |93b_ROT_Torus_surface| image:: ../_static/On2DCurves/RevolutionTori/93b_ROT_Torus_surface.3D.xml.jpg :width: 30 % .. |93c_ROT_Torus_surface| image:: ../_static/On2DCurves/RevolutionTori/93c_ROT_Torus_surface.3D.xml.jpg :width: 30 % **Left figure**: Low resolution torus surface, part1 **Middle figure**: Low resolution torus surface, part1 **Right figure**: Low resolution torus surface, part1 |newpage| Torus ----------------------------------------------------------------------------------------- .. method:: User.Torus(a, Resolution, Revolution = true) See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. An example in Python .. code-block:: pycon >>> from mpfebnet import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') An example in C\# .. code-block:: csharp using User; Curve = User.Cardioid(a: 1, Resolution: 200, AsPolar: true); User.Chart2D.Show(Curve, Template = "PolarCurve", Title = "Cardioid"); |01a_ROT_Torus_surface| `\quad` |01b_ROT_Torus_lower_half| `\quad` |01c_ROT_Torus_right_half| .. |01a_ROT_Torus_surface| image:: ../_static/On2DCurves/RevolutionTori/01a_ROT_Torus_surface.3D.xml.jpg :width: 30 % .. |01b_ROT_Torus_lower_half| image:: ../_static/On2DCurves/RevolutionTori/01b_ROT_Torus_lower_half.3D.xml.jpg :width: 30 % .. |01c_ROT_Torus_right_half| image:: ../_static/On2DCurves/RevolutionTori/01c_ROT_Torus_right_half.3D.xml.jpg :width: 30 % **Left figure**: Torus **Middle figure**: Torus **Right figure**: Torus |newpage| Horn Torus ---------------------------------------------------------------------------------------------- See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. An example in Python .. code-block:: pycon >>> from mpfebnet import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') An example in C\# .. code-block:: csharp using User; Curve = User.Cardioid(a: 1, Resolution: 200, AsPolar: true); User.Chart2D.Show(Curve, Template = "PolarCurve", Title = "Cardioid"); |02a_ROT_HornTorus_surface| `\quad` |02b_ROT_HornTorus_lower_half| `\quad` |02c_ROT_HornTorus_right_half| .. |02a_ROT_HornTorus_surface| image:: ../_static/On2DCurves/RevolutionTori/02a_ROT_HornTorus_surface.3D.xml.jpg :width: 30 % .. |02b_ROT_HornTorus_lower_half| image:: ../_static/On2DCurves/RevolutionTori/02b_ROT_HornTorus_lower_half.3D.xml.jpg :width: 30 % .. |02c_ROT_HornTorus_right_half| image:: ../_static/On2DCurves/RevolutionTori/02c_ROT_HornTorus_right_half.3D.xml.jpg :width: 30 % **Left figure**: Horn Torus **Middle figure**: Horn Torus **Right figure**: Horn Torus |newpage| Spindle Torus ---------------------------------------------------------------------------------------------- See also: https://mathworld.wolfram.com/EllipticTorus.html See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. An example in Python .. code-block:: pycon >>> from mpfebnet import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') An example in C\# .. code-block:: csharp using User; Curve = User.Cardioid(a: 1, Resolution: 200, AsPolar: true); User.Chart2D.Show(Curve, Template = "PolarCurve", Title = "Cardioid"); |03a_ROT_SpindleTorus_surface| `\quad` |03b_ROT_SpindleTorus_lower_half| `\quad` |03c_ROT_SpindleTorus_right_half| .. |03a_ROT_SpindleTorus_surface| image:: ../_static/On2DCurves/RevolutionTori/03a_ROT_SpindleTorus_surface.3D.xml.jpg :width: 30 % .. |03b_ROT_SpindleTorus_lower_half| image:: ../_static/On2DCurves/RevolutionTori/03b_ROT_SpindleTorus_lower_half.3D.xml.jpg :width: 30 % .. |03c_ROT_SpindleTorus_right_half| image:: ../_static/On2DCurves/RevolutionTori/03c_ROT_SpindleTorus_right_half.3D.xml.jpg :width: 30 % **Left figure**: Spindle Torus **Middle figure**: Spindle Torus **Right figure**: Spindle Torus |newpage| Elliptic Torus ---------------------------------------------------------------------------------------------- .. method:: User.EllipticTorus(a, Resolution, Revolution = true) See also: https://mathworld.wolfram.com/EllipticTorus.html See also Wikipedia :cite:p:`Wikipedia2D101`, MathWorld :cite:p:`Wolfram3D101`. An example in Python .. code-block:: pycon >>> from mpfebnet import User >>> Curve = User.Cardioid(a = 1, Resolution = 200, AsPolar = true) >>> User.Chart2D.Show(Curve, Template = 'PolarCurve', Title = 'Cardioid') An example in C\# .. code-block:: csharp using User; Curve = User.Cardioid(a: 1, Resolution: 200, AsPolar: true); User.Chart2D.Show(Curve, Template = "PolarCurve", Title = "Cardioid"); |04a_ROT_EllipticTorus| `\quad` |04b_ROT_EllipticTorus| .. |04a_ROT_EllipticTorus| image:: ../_static/On2DCurves/RevolutionTori/04a_ROT_EllipticTorus.3D.xml.jpg :width: 30 % .. |04b_ROT_EllipticTorus| image:: ../_static/On2DCurves/RevolutionTori/04a_ROT_EllipticTorus.3D.xml.jpg :width: 30 % **Left figure**: Spindle Torus **Right figure**: Spindle Torus