Serpentine Marbling

	http://swiss.csail.mit.edu/~jaffer/Geometry/Marbling-2
	Serpentine Marbling

In Mathematical Marbling, we explored simple linear and circular deformations and produced images which are clearly marbling, but do not look like the marbled forms artists create.

http://langmuir.chem.utsunomiya-u.ac.jp/colloid/photo/echizen2

Serpentine

This "Serpentine Marble" looks like a horizontal ten-tine comb was stroked upward while wiggling left and right in a sinusoidal path.

Although the distance function to a sinusoid curve can be computed, it is easier to stroke tines vertically; and then add a horizontal displacement proportional to the sin(degrees) of each point's y-coordinate:
W_h(x, y) = (x - 20 · sin(2.5 · y), y)
The dark tone of the work is here produced by interposing a very dark double-wide band between each pair of the other inks.

[images are linked to PostScript files]

Marbling Backward

A nice feature of these particular straight and circular deformations is that the induced translations leave the parameter which controls that motion unaffected. In the linear case, the motion of a point is perpendicular to the controlling distance (point to line). In the circular case the rotational movement around the center does not affect the radius on which it depends.

For a given point Z, if we apply a deformation followed immediately by the same deformation but with negative Z, the two motions are in opposite directions and leave the points where they started. F(-Z) is thus the inverse transform of F(Z). The inverse-composite-map is simply the composite-map functions in reverse order with negated Z:
F^-1 = F₁^-1 _° F₂^-1 _° · · · _° F_n^-1
Where the previous algorithm took ink-circle coordinates and mapped them to their destinations, we can as easily take coordinates in the viewable area, transform them through the inverse-composite-map, and identify which ink band they come from. Since my ink bands are concentric circles, there is no need to explicitly enumerate them. I just calculate the distance from the virtual ink center point, and assign a color (from a palette sequence of 12) with modular arithmetic.

This method eliminates restrictions on the number and thinness of ink bands. Only the viewable area is rendered; one no longer needs to experiment with circle increments and the mostly unseen outer ink-circles.

But unlike the contoured images anti-aliased by GhostScript, assigning the color to a pixel based just on the mapped coordinates' truncated distance from the origin results in aliasing artifacts where multiple bands squeeze through one pixel cell. Computing colors for sub-sampled points would be slow; but the color-number's fractional part can be used to interpolate between the colors of the two bands whose border is closest.

The resulting image is perhaps too soft. Using an S-shaped function to sharpen the distinction between the bands reaches a happy medium.

To the left is the circular marble rendered as nested contours. To the right is the circular marble raster-rendered with the S-shaped function.

The next chapter attempts more complicated effects acheived by physical ink marbling.

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!