http://people.csail.mit.edu/jaffer/Marbling/Dropping-Ink | |

## Dropping Ink |

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.

Consider the tank as being completely covered with a film of "empty
ink" initially. The first ink drop in the tank forms a circular spot
with area *a*. If a second drop with area *b* is put in the
center of the first drop, then the total covered area increases
from *a* to *a*+*b*. Points very near the center will
move from infinitesimally small radius to radius
sqrt(*a*/*π*); and boundary points will move from radius
sqrt(*a*/*π*) to radius
sqrt((*a*+*b*)/*π*).

The movements of a point on the surface do not depend on its ink
color; so its motion is the same as for the concentric ink-drop case.
Therefore, given a point *P* and a new ink drop of
radius *r* centered at *C*, move the point radially
from *P* to:

C + (P − C) · sqrt | ( | 1 + | r^{2}|| P − C||^{2} | ) |

One correspondent complains that, because the divergence of this transform is not 0, it can't be incompressible.

Divergence is defined for a continuously differentiable vector field. But this transform is not continuously differentiable around

C; thus its divergence isn't well-defined. The common definition of incompressible is in terms of the divergence of a vector field. For a vector-field where the divergence isn't well-defined, the definition is silent.However, I can show that this transform preserves the area of all neighborhoods not containing

C. Consider the annulus centered onChaving inner radius sqrt(a/π) and outer radius sqrt((a+b)/π). Its area isb.If ink is injected at

Cforming a new circular region centered onChaving areae, the annulus having areabwill expand to have an inner radius of sqrt((a+e)/π) and an outer radius of sqrt((a+b+e)/π). The area of the expanded annulus is stillb. Note that the expanded annulus is thinner than the original.Because this transform is radially symmetric, any annulus slice with sides forming angles with

Cofθandηwill map to the expanded annulus slice between the angles ofθandη. Because the annulus and slices can be made arbitrarily small, all neighborhoods not containingCmap to regions having the same area.

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 − | r^{2}|| Q − C||^{2} | ); |

Copyright © 2006, 2010 Aubrey Jaffer

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! |