(2016-01-12) 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 shalluse any nonzero vector () orthogonal to the plane:
x + y + z = 0
This can be construed as a limiting case of the previous equation.
(2016-01-16) Horizontal line rotating at a rate proportional to its vertical velocity.
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
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 k
An 1842 theorem due toCatalan (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 ofequation 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
(2016-01-16) Surfaces generated by the motion of a straight line.
(2016-01-16) Ruled surfaces generated by a horizontal line.
At a given point on a surface, the normal curvature is extreme alongthe two perpendicular directions of the lines of curvature.
(2016-01-30) Guldin's theorems (1635) use the relevant centroid's circular trajectory.
(2016-01-12) 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
=
d
=
det (v,v' )
=
z' r'' r' z''
ds
||v|| 3
[ (z') 2 + (r') 2] 3/2
(2016-01-16) The shape of soap films separating regions of distinct pressures.
An unduloid is a surface of revolution whose meridian istraced by the focus of aconic section which rollson the axis.
With a parabola, acatenoidis obtained. The mean curvature is zero.
When it's an hyperbola, the surface has negative mean curvature, which correspondsto a soap film surrounding a region of lower pressure.
A rolling ellipse corresponds to a positive mean curvature and/or a higher innerpressure. We obtain the undulatory shape shown below,which has given its name to the whole family.
(2020-05-06) In a threefold orthogonal system pair of surfaces out of different pencilsintersect along a mutual curvature line.
The cartesian parametric equations are:
