Introduction to PoVRay I

Most surfaces in the repository are accompanied by PoVRay files to render them, so I think it is time to give a little introduction. You will need the software, available for free. This is a text based ray tracing program. Essentially, you need to describe a scene with objects in it, including light sources and camera. A good start is the following:

global_settings { assumed_gamma 2.2 }
#include "colors.inc"

camera {
	location <0,0,-4>
 	right x*image_width/image_height    
	angle 40 look_at <0,0,0>
 }


light_source { 
	 <5,5,-10>
	color  White
 }
 light_source { 
 	<-5,5,-10>
	color  White
 }

This tells the software a sensible gamma calibration, loads names for colors, and places camera and light sources. Important here is the coordinate system. I like to center my objects at the origin <0,0,0>, and place the camera four units (meters) behind it. The angle 40 is a mild wide angle. The x and y coordinate are horizontal and vertical (screen) coordinates, and z is for depth. I have also placed two light sources up to the right and left behind the camera.

#macro surf(x0,y0) 
 .13*<-(x0*(-3*cos(y0) + x0*x0*cos(3*y0))),
 -(x0*(3*sin(y0) + x0*x0*sin(3*y0))),
  3*x0*x0*cos(2*y0)>
#end

#declare x0=0;         
#declare x1=sqrt(3);

#declare y0=0;				
#declare y1=2*pi;		

Next we need some formula to parametrize a surface. Up above is the code for the Enneper surface. We also need to define the range of the parameters. We do this using variables so that it is easier to change the code for something else.

Today, the plan is to make something like the images above: Placing small colorful beads at the grid points of a parameter grid. The code for this is a simple loop:

#declare eps=0.0001;
			

#macro make_sphere_carpet(x0,x1,xd,y0,y1,yd,rad)
  object {
    union {
      #local xc=x0;
      #while (xc < x1+eps)
        #local yc=y0;
        #while (yc<y1+eps)
          object {
	    sphere {surf(xc,yc), rad*xc}
	    pigment {rgb <(xc-x0)/(x1-x0), (sin(yc)+1)/2,.4>}
            finish {
	      specular 1
	      roughness .1
	      metallic
            }  
	  }
	  #local yc = yc+yd;
	#end
	#local xc = xc+xd;
      #end
    }
 }
#end

The macro make_sphere_carpet takes the previously defined ranges of the parameter values as arguments, and in addition parameters xd and yd that determine how many spheres will be placed. rad controls the size of the spheres. Here, they also increase with the radius xc for dramatic effect. For each grid point (xc,yc) a sphere is placed at the coordinates surf(xc,yc), with a color that is also determined by the coordinate.

#declare xm=80; 			
#declare ym=200;			

#declare sphthick = 0.01;	


object {
   make_sphere_carpet(x0,x1,
      (x1-x0)/xm,y0,y1,(y1-y0)/ym,sphthick)	
   rotate <-90,-30,0>
   translate <-.1,0,0>
} 

The rest is simple: We need to decide on how many spheres to place, and how big to make them, and call the macro. The last bit rotates and translates the surface in a decent position. The rotate command is a bit strange, the arguments are rotations in degrees about the x-, y-, and z-axis, in this order. Next time we will learn how to add a polygon mesh to all of this. Below is a sphere carpet for Catalan’s surface.

Björling Surfaces II

The Björling problem is in its nature a Cauchy problem, and its solution enjoys benefits and side effects of such problems: We have, for an analytic curve and normal, the uniqueness and existence of a minimal surface with these Cauchy data.

Things are even better than in general, because the solution can be computed as an integral. The side effect is that we have little flexibility, and little knowledge about the global appearance of the solution far away from the curve.

It turns out that one can do something about it, if one is willing to sacrifice the precise shape of the curve. Instead of prescribing the complete space curve (x(t), y(t), z(t)), we might be content prescribing just the planar projection (x(t), y(t)). Then there is a way to explicitly determine a third coordinate z(t) so that one can obtain more global control over the Björling surfaces, like finite total curvature, completeness, and flexible speed of a rotating normal.

The construction also comes with a parameter that allows to keep a planar curve closed after lifting it into space.

This raises the intriguing question whether there are similar phenomena for other Cauchy problems: Does forgetting one dimension of the Cauchy data allow for some global control?

Resources

Repository Overview: Björling Surfaces

R. López, M. Weber: Explicit Björling Surfaces with Prescribed Geometry

Björling Surfaces I

Plateau’s problem asks to find a minimal surface that spans a given closed curve. This is a global question, and the answers are delicate. In contrast, there is a much simpler local problem, posed by Emanuel Gabriel Björling in 1844:

For a given space curve and normal field along the curve, find a minimal surfaces that contains the curve and has the given normal as surface normal. For instance, for a circle with outer normal we expect a catenoid, and for a straight line with a normal that rotates with constant speed a helicoid.

In his 1844 paper, Björling proved that for any given real analytic space curve and real analytic normal field along the curve, there is a unique local solution. Later, Hermann Amandus Schwarz gave an explicit formula for the Weierstrass data of the solution.

The fact that the solution of Björling’s problem is rather simple has been quite fertile. For instance, one can let a normal rotate about a circle and get Pablo Mira’s circular helicoids. However, one quickly finds oneself in the default situation of the wizard’s apprentice who has learned how to use spell but not acquired knowledge about the consequences.

For instance, who would have thought that if one starts with a planar cycloid as a curve and takes as normal the normal vector to the curve, the resulting surface (already known to Eugène Charles Catalan) contains parabolas as geodesics? Also, the formulas that Schwarz gave us are not always easy to integrate. The helicoids winding along logarithmic helices below (due to Christine Breiner and Stephen Kleene) can be explicitly described, but the formulas span an entire page.

So it would be desirable to have some control about the global nature of the solutions, and also some insight when to expect explicit formulas. We will earn about this next week.

Resources

E.-G. Björling: In integrationem aequationis Derivatarum partialium superficiei, cujus in puncto unoquoque principales ambo radii curvedinis aequales sunt signoque contrario, Archiv der Mathematik IV (1844), 290-315

Mathematica Notebook with Examples

Björling Surfaces Repository Page

Eccentricity

In his thesis, Peter Connor discusses doubly periodic minimal surfaces with parallel top and bottom ends that are cut by vertical planes into simply connected pieces. Here are two examples:

These surfaces can be systematically described using polygonal domains like the one below.

The left and right vertical edges where the curves end correspond to the ends of the surface, and the corners to the points with vertical Gauss map, recording whether it points up or down by a left or right turn. There are only finitely many possibilities for each genus, and they are easy to enumerate. What is not so easy is to find out whether the corresponding surfaces actually exist, either numerically or theoretically. For genus 3 and higher, existence proofs usually rely on understanding the limits of these surfaces, and this is what this post is about. When approaching a limit, a minimal surface necessarily decomposes into fundamentally simpler surfaces.

Above are two examples of Connor’s experimental doubly periodic surfaces of genus 3, near one of their limits. On the left hand side we see 8-ended singly periodic Scherk surfaces emerging, on the right hand side 10-ended Scherk surfaces. There is a simple count that helps to predict what can arise. This count is based on the fact that the total curvature of these surfaces is an integral multiple of 4π, and this multiple stays the same while deforming and approaching limit. Let’s call this multiple the degree of the surface (it is, in fact, nothing but the degree of its Gauss map). The Catenoid and the single and doubly periodic classical Scherk surfaces have all degree 1. More generally, a 2n-ended singly periodic Scherk surface of genus 0 has degree n-1, and a doubly periodic surface with four ends and of genus g has degree g+1. So the two surfaces above have all degree 4. For the left, the limit is a singly periodic Scherk surface of degree 3, but the components are stitched together using a catenoidal neck, adding 1 to the degree. The surface on the right limits in a Scherk surface of degree 4, so doesn’t need stitches.

Above are two more examples of genus 3 from Connor’s list. The left decomposes into two doubly periodic Karcher-Scherk surfaces of degree 2, the right one into two KMR surfaces, which have degree 2 each.

The big surprise of Connor’s series of examples of genus 3 was the surface to the left. It has the same combinatorial polygon description is the (known) RTW-surface to the right, but the handle has become eccentric. The limit in either case is a toroidal Scherk surface with 8 ends, which has degree 4. In the right case, this is again a known surface, but the left case is new. Proving the existence of either the doubly or the singly periodic version would be highly desirable.

A Tale of Two Squares

Today, we are pushing things a little. We begin with a minimal heptagon with two horizontal straight edges, and the other five edges lying in symmetry planes parallel to the vertical coordinate planes. We also want this heptagon arranged so that extension by reflections/rotations creates an embedded triply periodic surface.

Above is a first example, showing a translational fundamental piece of a triply periodic surface of genus 6 that is tiled by eight such heptagons. We see that the horizontal straight segments are replicated into edges of squares, so that all surfaces we will look at today will sit one way or another between two squares.

A way to encode the possibilities for such heptagons is through the picture above. It represents the divisor of the square of the Gauss map on the quotient torus under the order 2 rotation about a vertical axis. Less technically, the grayed rectangle represents the heptagon, with the two vertical blue edges corresponding to the horizontal straight segments. The red dots are the seven heptagon vertices, and the symbols 0 and ∞ tell whether the Gauss map points up or down.

There are 12 possibilities that lead to different surface candidates. For each of these, a 3-dimensional period problem needs to be solved, leading in the case of success to 1-parameter families. In 8 of these cases, I was able to solve this period problem numerically. No existence or non-existence proof is known in any of these cases at this point.

There are several motivations behind this exploration. For one, I would like to know what limits can occur, and how the limit surfaces are combined. This suggests possibilities for general gluing constructions that would establish the existence of these and many other surfaces, Secondly, some of the limits are surfaces that are either only very difficult to obtain (like the Callahan-Hoffman-Meeks surface of higher genus that emerges above), or have also only been established numerically (like the exotic doubly periodic Karcher-Scherk surface of genus 3 below).

Finally, there is always the chance that an exhaustive enumeration leads to examples with unexpected properties, like the surface below, that in a diagonal view reveals an appealing tunnel system that indicates that we are probably having a special pair of skeletal graphs in front of us.

Resources

Main Page for all 8 surfaces

To Be or Not To Be

Existence is an interesting concept. Even in the most rigorous of sciences, it is not free of dispute.

Above is an image of the Horgan surface, a minimal surface that does not exist, at least not with all the glorious property it appears to have. There are less convincing attempts to convey the existence of a non-existing object, like the catenoid with a handle:

In Mathematics, pictures provide evidence and enjoyment. Existence or non-existence requires proof, and provides certainty. Still, the membrane between to be and not to be can be very thin, as in the example at the top.

The pattern made by the catenoidal necks in the top image – two up and two down, at the corners of a square – does actually occur. One of the first examples I have seen is the Callahan-Hoffman-Meeks surface with an added handle as above. Its existence is not so easy to prove – does it therefore exist a little less?

The triply periodic surface above is the simplest version of the Catenoid-Square pattern I know of, and much more recent. So, in a sense, all this research is to an unconfessed extent concerned with probing the border of existence, from either side.

The Upper Half Disk (Parametrizations 2)

One of the text book examples of conformal maps is given by the complex function f(z) = z+1/z which maps the upper half disk bijectively to the lower half plane. This is the main ingredient for the following problem:

We would like to have a conformal map from a rectangle to a half disk (say of radius r) so that the left and right sides of the rectangle are mapped to half circular arcs around two points a and b.

The inverse of this map can be obtained by composing the text book map above with a Möbius transformation that sends the images of a and b to 0 and infinity, and then use polar coordinates on the upper half plane. Details are in this notebook.

What is this good for? Examples are 4-ended surfaces like the ones above that have sufficient symmetry that one can place the ends as four punctures on the real line, symmetric with respect to a circle.

This can also be useful more generally when existing symmetries cut the surface into pieces with four ends along the boundary of the pieces, as above.