Developable surface - What represents the picture below?

Question

Can someone explain me what represents Figure 2?

Answer 1

Fig 2 represents unwinding of a taut string ( tangents) around the helix of stiction (a cuspidal edge), its loci forming a net of tangents and involutes on a developable helicoid (which is a singly curved surface of Gaussian curvature $K=0$). The second part of Fig 2 is development showing constant width of strip $PQ$

The involutes/tangents are not continuous but discretized from lumped smooth differential area elements as skewed quadrilateral meshed elements with straight boundaries.

In curvature product $ K= k_1\cdot k_2=0 $ one line (1) is always straight in the originating plane and curved when bending along direction (2) as parallel circles of cone/cylinder. In developable helicoid curves intersecting straight tangents are 3D involutes.

Just as for the planar involutes of circle the curved width along involute between tangents is always a constant, being equal to the starting helical arc length between between successive involutes, it is true here also, a feature I believe should be helpful in discretization with tesselations.

Googling suggests an article " Discrete surfaces for Architectural Design" by Helmut Pottman etal, Institute of Geometry, Report 2007- (30 Jan 2007).

From differential geometry we have that every developable surface is an envelope of one parameter family of contacting planes along their generators or rulings, which are surface developments of the two/three types.

A Ruled surface is generated by differentiating with respect to first parameter with one term added of the derivative multiplied by ruling parameter $v$. A picture of this e.g. can be also found on D.J. Struik text book on "Lectures on Classic Differential Geometry".

Some $(u,v)$ parameterizations.

1) Cone ( concurrent rulings) $$ x= u \sin\alpha \sin v, y= u \sin \alpha \cos v, z = u \cos\alpha $$

2) Cylinder (parallel rulings) which is a special case of Cone $( \,\alpha = \pi/2, u=a) $

$$ x= a \sin v, y= a \cos v, z = 0 $$

3) Developable Helicoid. Only the third case ( rulings tangent to a cuspidal edge) is included for illustration given here (cone and cylinder are not shown presumably due to their simplicity). However its inclusion imho would have helped identify the surface mesh with subdivision of each of the two parameters.

$$ x = a (\cos u - v \sin u), y = a (\sin u + v \cos u), z = b (u + v) $$

A physical paper model can be made/demonstrated by a simple experiment with its development: Cut out annular area (between two radii) of a circle, next cut along a radius and pull apart the cut ends. The inner helical edge so formed is the Line of stiction, the curve to which all rulings (generators) are tangent to, shown thickened as a Tube in magenta. It is a cuspidal edge of form $z= r^{3/2}=(x^2+y^2)^{3/4}.$

Developable Helicoid Reference is a good reference showing an involute among meshes that are quadrangular rulings and also related:

Stiction line for K = 0

al = Pi/3;
x = u Sin[al] Sin[t]; y = u Sin[al] Cos[t]; z = u Cos[al];
Cone = ParametricPlot3D[{x, y, z}, {t, 0, 2 Pi}, {u, 0, 0.5}, 
  Mesh -> {19, 4}]
a = 1; b = 1.25;
x = a (Cos[t] - v Sin[t]); y = a (Sin[t] + v Cos[t]); z = b (t + v);
DevHelicoid = 
  ParametricPlot3D[{x, y, z}, {t, -Pi, Pi}, {v, -2.5, 5.5}, 
   Mesh -> {19, 4}];
x = a Cos[t]; y = a Sin[t]; z = b t;
StictionLine = 
  ParametricPlot3D[{x, y, z}, {t, -5, 5}, 
   PlotStyle -> {Tube[.2], Magenta}];
Show[{DevHelicoid, StictionLine}, Boxed -> False, Axes -> False]

In the above Mathematica program Mesh fineness can be set as needed. The meshes when discretized as quadrangles become suitable as QUAD elements in manufacturing or architectural modeling software. Pictured at right is part of a surface with narrow $v$ interval, here called PQ strip.