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| http://swiss.csail.mit.edu/~jaffer/Geometry/Marbling-1 |
Mathematical Marbling |
The essential feature I am trying to produce is that when a tine moves
through regions of different color inks, it stretches the boundaries
between the inks to bend around it. If a blue ink region is
surrounded by a red ink region, in the deformation induced by the
movement of the tine through it, the blue region is still surrounded
by the red region.
A mapping taking each point in the plane to its destination after the tine has moved through the area will be represented by a vector-valued function F from a plane (R×R) to a plane. Further, F will be a homeomorphism, a continuous bijection.
The functional composition of homeomorphisms is also a homeomorphism. So if we have a homeomorphism for a single stroke, we can compose the functions representing each movement together to create a more complicated homeomorphism where F1 will be the first stroke and Fn will be the last stroke of the sequence:
| F = Fn ° Fn-1 ° · · · ° F1 |
Ink boundary curves which are continuous in the plane are mapped to continuous curves by homeomorphisms, no matter how much they fold and stretch.
When an area is dense with folds, any low-degree differentiable
function which detours around the lines traced by comb points should
render reasonably well for this purpose. In the treatment below it
turns out that deformations parallel to infinitely long lines are more
tractable than tines which start and stop. To avoid dragging the
lines to infinity we set a maximum displacement z for the
deformation each tine introduces.
The boundaries between virtual ink rings will be traversed using the Minsky circle algorithm; although walking the circles using coordinates generated by sin and cos would work as well. The angular step size is made inversely proportional to the ring radius, making the distance between successive points uniform.
[These images are linked to the
PostScript files which generated them.]
| Fv(x, y) = | ( | x, y + |
z · c
| x - xL | + c | ) |
Fv displaces each point vertically by an amount inversely proportional to its horizontal distance from the tine's x-coordinate xL.
The transform for straight, horizontal strokes is to displace each point horizontally by an amount inversely proportional to its vertical distance from the tine's y-coordinate yL:
| Fh(x, y) = | ( |
z · c
| y - yL | + c | + x, y | ) |
After verifying that a combing and its reverse (eg. North and South)
undid each other, the next combing sequence to try was a
commutator.
Where operators do not commute, composing operators and their inverses
in an asymmetrical sequence often reveals quintessential properties of
their mathematical interactions.
The image to the right is the result of straight combings (10 tines) in the North, East, South, then West directions.
[These images are linked to the PostScript files which generated them.]
The image to the left was combed North, East, North, then East.
Although the depth of combing was the same as the previous picture,
here the circularity of the initial inks is evident.
The parallel displacements with unlimited extent do not shrink or grow
regions of the plane, just shear them. So the linear strokes are
compatible with the (liquid) inks being incompressible.
| d = | (P - B) · N | |
| FL(P) = P + |
z · c
d + c | · M |
Circular tine tracks are also compatible with incompressible flow. In
this case points are displaced along arcs around a center point
C.
| d = | ||P - C|| - r | |
| l = | z · c
d + c | a = | l
|| P - C || |
| FC(P) = C + (P - C) · | ( | cos a -sin a | sin a cos a | ) |
The figure to the left shows the paths of the virtual tines moving
through the field of view. Those clockwise circular motions result in
the marbling shown to the right.
Arcs of increasing radius whose centers track away from the field of
view create a fan like that at the top of this article.
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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! | |