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

\[x(u,v) = \phi(v) \cos(u)\]

\[y(u,v) = \phi(v) \sin(u)\]

\[z(u,v) = \psi(v)\]

Prism and cylinder

See also: http://www.3d-meier.de/tut3/Seite84.html, Cylinder

See also: https://mathworld.wolfram.com/Cylinder.html, Cylinder

\(\quad\) \(\quad\)

Left figure: Prism3

Right figure: Prism4

Right figure: Cylinder

Pyramid and cone

\(\quad\) \(\quad\)

Left figure: Pyramid3

Right figure: Pyramid4

Right figure: Cone

Frustum and cone frustum

\(\quad\) \(\quad\)

Left figure: Frustum3

Right figure: Frustum4

Right figure: Cone Frustum

Insolator

See also: http://www.3d-meier.de/tut3/Seite86.html, Insolator

Left figure: Insolator

Gabriels Horn

See also: https://mathworld.wolfram.com/GabrielsHorn.html

See also Wikipedia [1484], MathWorld [1160].

Left figure: Gabriels Horn

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 [1484], MathWorld [1160].

Left figure: Funnel

One-Sheeted Hyperboloid

See also: https://mathworld.wolfram.com/One-SheetedHyperboloid.html

See also: https://en.wikipedia.org/wiki/Hyperboloid

Left figure: One-Sheeted Hyperboloid

Catenoid

See also: https://mathworld.wolfram.com/Catenoid.html

See also Wikipedia [1484], MathWorld [1160].

Left figure: Catenoid

Gauss Cylinder

See also: http://www.3d-meier.de/tut3/Seite157.html Gauss Cylinder

See also Wikipedia [1484], MathWorld [1160].

\(\quad\)

Left figure: Gauss Cylinder

Right figure: Gauss Cylinder

Paraboloid

Left figure: Paraboloid

Vase

\(\quad\)

Left figure: Vase

Right figure: Vase

Sphere

User.ParametricSphere(a, Resolution, Revolution=true)

See also: http://www.3d-meier.de/tut3/Seite86.html, Sphere

Left figure: Sphere

\(\quad\) \(\quad\)

Left figure: Gear torus surface

Middle figure: Gear torus surface

Right figure: Gear torus surface

Prolate Spheroid

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

\(\quad\)

Left figure: Prolate Spheroid.

Right figure: Prolate Spheroid via Sqrt.

Oblate Spheroid

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

\(\quad\)

Left figure: Prolate Spheroid.

Right figure: Prolate Spheroid via Sqrt.

Drop surface

// see http://www.3d-meier.de/tut3/Seite44.html

An example in Python

>>> 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#

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

Left figure: Drop surface

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

Left figure: Apple surface

Spinning top

User.SpinningTop(a, Resolution, Revolution=true)

See also http://www.3d-meier.de/tut3/Seite150.html Kreisel

See also Wikipedia [1484], MathWorld [1160].

Left figure: Spinning top

Pseudosphere

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

>>> 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#

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

\(\quad\)

Left figure: Pseudosphere

Right figure: Pseudosphere

Wave sphere Surface

// see http://www.3d-meier.de/tut3/Seite63.html

\(\quad\)

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 [1484], MathWorld [1160].

\(\quad\) \(\quad\)

Left figure: Low resolution torus surface, part1

Middle figure: Low resolution torus surface, part1

Right figure: Low resolution torus surface, part1

Low resolution torus surface, part2

See also: http://www.3d-meier.de/tut3/Seite165.html

See also Wikipedia [1484], MathWorld [1160].

\(\quad\) \(\quad\)

Left figure: Low resolution torus surface, part1

Middle figure: Low resolution torus surface, part1

Right figure: Low resolution torus surface, part1

Low resolution torus surface, part3

See also: http://www.3d-meier.de/tut3/Seite165.html

See also Wikipedia [1484], MathWorld [1160].

\(\quad\) \(\quad\)

Left figure: Low resolution torus surface, part1

Middle figure: Low resolution torus surface, part1

Right figure: Low resolution torus surface, part1

Torus

User.Torus(a, Resolution, Revolution=true)

See also Wikipedia [1484], MathWorld [1160].

An example in Python

>>> 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#

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

\(\quad\) \(\quad\)

Left figure: Torus

Middle figure: Torus

Right figure: Torus

Horn Torus

See also Wikipedia [1484], MathWorld [1160].

An example in Python

>>> 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#

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

\(\quad\) \(\quad\)

Left figure: Horn Torus

Middle figure: Horn Torus

Right figure: Horn Torus

Spindle Torus

See also: https://mathworld.wolfram.com/EllipticTorus.html

See also Wikipedia [1484], MathWorld [1160].

An example in Python

>>> 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#

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

\(\quad\) \(\quad\)

Left figure: Spindle Torus

Middle figure: Spindle Torus

Right figure: Spindle Torus

Elliptic Torus

User.EllipticTorus(a, Resolution, Revolution=true)

See also: https://mathworld.wolfram.com/EllipticTorus.html

See also Wikipedia [1484], MathWorld [1160].

An example in Python

>>> 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#

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

\(\quad\)

Left figure: Spindle Torus

Right figure: Spindle Torus