Dropping Ink

	http://swiss.csail.mit.edu/~jaffer/Geometry/Marbling-7
	Dropping Ink

With the exception of Textile Designs, all of our marbling has started from concentric color circles. But marblers can drop complicated designs of ink before combing, in between combings, or even skip the combs altogether.

The order in which inks are dropped matters. I take the approach of considering how a point on an ink boundary is moved by each subsequent ink dropped.

The first ink drop in the tank forms a circular spot with area a₁, its size balancing its density against surface tension. If a second drop is put in the center of the first drop, then we should expect the total covered area to increase from a₁ to a₁+a₂. Points at the center will move from radius 0 to radius sqrt(a₁); and boundary points will move from radius sqrt(a₁) to sqrt(a₁+a₂). The transformation of a point at P from radius r ink centered at C is:

C + (P - C) · sqrt ( 1 + r²
||P - C||² )
The image to the right is linked to a PDF showing the sequence of deformations of 5 large regions of ink followed by a random splattering of 5, 25, and 125 smaller ink drops.

The last ink to be dropped will be rendered last and will have a round outline. The second to last drop will have its round outline distorted by the last drop according to the transform given above. Working backwards through the ink droppings, each must be distorted by all the subsequent drops.

The last drop being round, its Minsky-circle epsilon can be rather large. The earlier the drop, the more distorting operations its outline must project through. So earlier drops should have the higher resolution afforded by a smaller epsilon.

A marbling can be represented by a sequence of drops, shifts, and strokes. The shifts and strokes have been described previously. A drop is described by its center coordinate and radius. Contour-rendering processes each ink-drop in order of application:

Step along the circular outline of the current ink drop.
For each shift, stroke, and ink-drop in sequence after the current ink-drop, replace its coordinates as mapped by each operation.
Fill the mapped outline with the ink-color.
Advance to the next ink-drop in sequence.

Ink drops can also be raster-rendered. Given a point at Q and a radius r ink centered at C, if ||Q - C|| < r, then the point Q is within the drop and takes its color. Otherwise the point is mapped to:

C + (Q - C) · sqrt ( 1 - r²
||Q - C||² )
and the underlying step is computed.

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!