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Final Answers
© 2000-2016   Gérard P. Michon, Ph.D.

Surfaces  in  3D

Related articles on this site:

Related Links (Outside this Site)

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

Wikipedia :   Surfaces

 
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Surfaces in Three Dimensions

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.


(2016-01-12)   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.


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

 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

 Come back later, we're
 still working on this one...

Jean-Baptiste Meusnier (1754-1793)
 
Wikipedia :   Helicoid   |   Generalized helicoid       MathWorld :   Circular helicoid


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

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 still working on this one...

Lines of curvature


(2016-01-16)   Catalan Surfaces
Ruled surfaces generated by a horizontal line.

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 still working on this one...

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


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

The term  torse  is considered archaic.

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 still working on this one...

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


 Gaspard Monge (2016-01-12)   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.

 Come back later, we're
 still working on this one...

Lines of curvature


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

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 still working on this one...

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


 Leonhard Euler (2016-01-12)   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

 Come back later, we're
 still working on this one...

Lines of curvature


(2016-01-16)   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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