Related Links (Outside this Site)

Gallery of Surfaces. Virtual Math Museum.
Classic Curves and Surfaces. "National Curve Bank" of Gustavo Gordillo.

Wikipedia : Surfaces

The most explicit way to specify a surface is to give the 3 cartesian coordinates of an arbitrary point as functions of 2 parameters, traditionally denoted u and v. The equation of a surface is the relation satisfied by the coordinates of every point.

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(2016年01月12日) Cartesian Equation of a Plane
Let (a,b,c) be the point of the plane that's closest to the origin.

When (a,b,c) is not (0,0,0) the plane's cartesian equation is:

a x + b y + c z = a² + b² + c²

Otherwise, we're dealing with a plane going through the origin and shall use any nonzero vector (a,b,g) orthogonal to the plane:

a x + b y + g z = 0

This can be construed as a limiting case of the previous equation.

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(2016年01月16日) Helicoid (Euler 1774, Meusnier 1776)
Horizontal line rotating at a rate proportional to its vertical velocity.

[画像: Helicoid ] The cartesian parametric equations are:

• x = u cos v
• y = u sin v
• z = k v

The equation in cylindrical coordinates is just:

z = k q

For a right-handed helicoid (as depicted above) the constant k is positive. It's negative for a left-handed one. The plane is an helicoid (with k = 0).

The constant k is homogeneous to a length per unit of angle. It's related to the wavelength a (the constant signed vertical displacement between two consecutive sheets) by the following relation, if angles are in radians:

a = 2 p k

An 1842 theorem due to Catalan (1814-1894) states that helicoids (planes included) are the only ruled minimal surfaces. [ Proof ]

Generalized Helicoids :

A generalized helicoid is generated by helical rotation of an abitrary curve of equation z = f (x). Its cartesian parametric equations are:

• x = u cos v
• y = u sin v
• z = f (u) + k v

The cylindrical equation is: z = f (r) + k q

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Jean-Baptiste Meusnier (1754-1793)

Wikipedia : Helicoid | Generalized helicoid MathWorld : Circular helicoid

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Gaspard Monge (2016年01月16日) Ruled Surfaces (French: surfaces réglées )
Surfaces generated by the motion of a straight line.

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Lines of curvature

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(2016年01月16日) Catalan Surfaces
Ruled surfaces generated by a horizontal line.

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Catalan surfaces | Eugène Catalan (1814-1894; X1833)

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Gaspard Monge (2016年01月16日) Developable surfaces (zero Gaussian curvature).
They are a special type of ruled surfaces.

The term torse is considered archaic.

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Gaussian curvature (K) | Wikipedia : Developable surface (torse)

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Gaspard Monge (2016年01月12日) Surfaces of Revolution
The meridians and the parallels are lines of curvature.

At a given point on a surface, the normal curvature is extreme along the two perpendicular directions of the lines of curvature.

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Lines of curvature

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(2016年01月30日) Surface area and volume of a solid of revolution.
Guldin's theorems (1635) use the relevant centroid's circular trajectory.

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Pappus of Alexandria (c. AD 290-350) | Paul Guldin (1577-1643)

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Leonhard Euler (2016年01月12日) Euler's Catenoid (1741)
Surface of revolution of minimal surface area.

Because the plane of a meridian is orthogonal to the surface, the normal curvature of the meridian is equal to its curvature given by the formula:

1 = dj = det ( v, v' ) = z' r'' - r' z''

Vinculum Vinculum Vinculum Vinculum

r ds ||v|| ³ [ (z' ) ² + (r' ) ² ] ^3/2

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Lines of curvature

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(2016年01月16日) Surfaces of Constant Mean Curvature (CMC)
The shape of soap films separating regions of distinct pressures.

An unduloid is a surface of revolution whose meridian is traced by the focus of a conic section which rolls on the axis.

With a parabola, a catenoid is obtained. The mean curvature is zero.

When it's an hyperbola, the surface has negative mean curvature, which corresponds to a soap film surrounding a region of lower pressure.

A rolling ellipse corresponds to a positive mean curvature and/or a higher inner pressure. We obtain the undulatory shape shown below, which has given its name to the whole family.

Unduloid

Mean curvature | Wikipedia : CMC surface (constant mean curvature) | Wente torus (1984): video.

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Charles Dupin (2020年05月06日) Dupin's theorem
In a threefold orthogonal system pair of surfaces out of different pencils intersect along a mutual curvature line.

Dupin's theorem | Charles Dupin (1784-1873; X1801)