Marbling the Torus

	http://swiss.csail.mit.edu/~jaffer/Geometry/Marbling-4
	Marbling the Torus

Although ink circles are finite in number in the contour marbling case, the entire R×R plane is affected by the comb homeomorphisms. How would we marble some other two-dimensional manifold, or create a marbled wallpaper design which matches itself at the edges?

The Torus

Imagine a rectangular sheet rolled so that one edge meets its opposite edge. In order for a pattern drawn on the sheet to blend invisibly, the colors and boundary-line slopes must match at the joined edges.

Now imagine that tube stretched and rolled so that its top and bottom circles meet. As before, to look continuous the colors and boundary lines must match.

This is the torus. To a 2-dimensional observer in the torus surface, the universe looks infinite, but repetitive; looking directly north or south multiple copies of each feature are evenly spaced; looking directly east or west the copies of each feature are evenly spaced -- but not necessarily with the same spacing as appears north-south.

Flatland

If we unbend the torus sheet and tile the plane with it (all with the same orientation), the 2-dimensional perception of the universe is unchanged; features repeat at uniform intervals both horizontally and vertically. The seams between copies of the sheet will be invisible. Thus the creation of a marbled torus and marbling wallpaper are equivalent.

Many of the deformations discussed so far are not well-defined on the torus.

Since each feature appears infinitely many times, the total displacement will be an infinite sum. For the vertical stroke case the combined displacement is:
∑
j z · c
| x - x_L+ j · r | + c

But this sum does not converge. We would need displacement functions with exponents of | x - x_L+ j · r | larger than 1.
A straight tilted tine-line can wrap through the rectangular sheet multiple times depending on its slope and the rectangle's repeat dimensions.
Large-diameter circles will be self intersecting, which invalidates their treatment in chapter 1.
Determining the interior of contours looks to be complicated; it may be beyond PostScript's graphical abilities.

Regroup

At this point it is worth rethinking strategy. We can do quite a bit of marbling using just horizontal and vertical strokes and sinusoidal (or other) parallel displacements. The good news is that horizontal and vertical displacments work in the torus.

A given point always lies in one half-open interval between a straight stroke and one of its (parallel) images. Instead of summing the contributions from every stroke image, sum only the contributions from these two closest lines.

Because the contributions are symmetrical to both sides of the stroke, it doesn't matter which side of the interval is half-open (when the point lies on the stroke).

Execution

Creating homeomorphisms from the toroid patch onto itself requires discipline:

The period of a sinusoid (or other) parallel displacement must be an integer harmonic of the period of the whole image rectangle in that direction.
The wraparound of coordinates is accomplished by calling fmod of any coordinate which changes in the Composite-map sequence.
```
/fmod 	% real modulus --> real
{
    /X2 exch def /X1 exch def
    X1 X2 X1 X2 div floor mul sub
} bind def
```

The new stroke function determines which side of its stroke the incoming point is. In addition to calculating the displacement due to the primary stroke, it adds in the displacement due to the closest image.

Because the repeat lengths xrep and yrep may be different, I returned to using separate routines for horizontal and vertical strokes.

/Vertical       % px py xc dy --> px py
{   /dy exch def
    /xc exch def
    /py exch def /px exch def
    /a  xc px sub dy mul abs c add z.c exch div
        xc xrep px xc lt {sub} {add} ifelse
           px sub dy mul abs c add z.c exch div add def
    px py dy a mul add yrep fmod
} bind def

/Horizontal     % px py yc dx --> px py
{   /dx exch def
    /yc exch def
    /py exch def /px exch def
    /a  yc py sub dx mul abs c add z.c exch div
        yc yrep py yc lt {sub} {add} ifelse
           py sub dx mul abs c add z.c exch div add def
    px dx a mul add xrep fmod py
} bind def

/Up {-1 Vertical} bind def
/Down {1 Vertical} bind def
/Left {1 Horizontal} bind def
/Right {-1 Horizontal} bind def

The resulting bouquet image can tile the plane with perfect continuity across seams. A monochrome version of this pattern is the background for these pages.

[image is linked to bouquet tiling]

cylinder.scm generates a 3d marbled cylinder, shown here from the four compass directions.

torus.scm generates a 3d marbled torus

The next chapter explores application for these designs.

I am a guest and not a member of the MIT Computer Science and Artificial Intelligence Laboratory. My actions and comments do not reflect in any way on MIT.
		Topological Computer Graphics
	agj @ alum.mit.edu	Go Figure!