CS 563
Advanced Topics in
Computer Graphics
Butterfly Patterns
Guy Mann
Butterfly
wings are made up of 4 wings, 2 forewings and 2 hindwings. They attached at the second and third
thoracic segments (the meso- and meta-thorax) where strong
muscles in the thorax move the wings up and down in a figure-eight pattern
during flight.
A
wing consists of two membranes that are nourished and supported by tubular
veins which connect them. The front and
back wing membranes are flat sheets of cells, all alike, which at a certain
point begin to make yellow pigment in some locations, and others black, yet
others white or orange. From these pigmentations the butterfly pattern is
created. The veins in the wing function to
give support to the wing and to provide an oxygen exchange mechanism,
breathing.
Butterfly wing patterns serve a number of evolutionary benefits for the butterfly. The patterns provide camouflage in some cases; such as the Australian leafwing butterfly which is shaped and colored like a leaf. Other butterflies have evolved warning colorations so that in most cases brightly-colored butterflies and moths are either bad-tasting or mimic of similar-looking bad-tasting butterflies. Other butterflies protect themselves from predators by using large eyespots to appear larger and more dangerous. The patterns of the wings also allow the butterflies to attract mates. Butterflies also use their wings to soak up heat because they are cold-blooded.
Butterfly wings
are of great interest to scientist, biologists particularly, because at the
genetic level, the processes that initiate arms on humans and spots on
butterfly wings are almost completely identical. "Butterfly wing patterns are a very
amenable system for studying morphological evolution because they develop on a
two-dimensional, epidermal surface made up of tiny scales, each of which
produces only one pigment. A single sheet of cells is not nearly as complex as
a three-dimensional structure." Development and evolution on the Wing,
by Antonia Monteiro, Ph.D., University at Buffalo
assistant professor of biological sciences, and W. Owen McMillan and Durell Kapan of the University of
Puerto Rico
Computer Scientist
are not as concerned with the biological reasons for existence
and genetics. So, why study butterfly
patterns? The answer to this is mostly
that they look pretty and we want to make things that look pretty. The long answer is that the wing patterns of
butterflies are complex patterns that are not ‘solved’ and therefore
interesting. As well as being an
interesting problem any good model of the pattern creation of a butterfly wing
could give insight into the morphological evolution of wing patterns.
The pattern of a
wing resides in scales. Each scale is an
extension of a single cell in the wing membrane, and each scale is a single
color. The overall pattern is a very
fine 
mosaic
of these scales. The front and back of
the wings usually have different patterns.
Patterns are
derived from a system of parallel bands that run from the front edge to the
rear margin of the wings, occurring generally in pairs of bands called symmetry
systems. In a symmetry system a
particular sequence of colors in one band is matched by colors in the opposing
or complementary sequence in the second band with the two bands together
forming a single whole.
The
wing is compartmentalized, so that each color band is broken up and segmented. The pattern that develops within one of these
areas that is bounded by wing veins is almost completely independent from the
pattern developed in adjoining wing cells.
Separate sets of genes regulate pattern development from area to area. Genes act independently on each pattern
element to control its size, shape, color and position.
Specialized groups of cells have point-like organizing centers (eye-spots). Species differ in the placement of these organizing centers or in their selective elimination. Organizing centers send out chemicals that diffuse across the wing, interacting with each other to determine precisely where the pigments will be synthesized. When the organizing centers are perfectly aligned the pigments that they induce create banding patterns. When some organizing centers are suppressed, or shifted, the result is highly dislocated patterns.
Although not much seems to be written on the subject of butterfly pattern creation in relation to computer graphics biologists; who seem extremely interested in the topic; have created a frame work which we can use to think about the wing pattern. Above we have mentioned eye-spots, banding, and vein bounding of the pattern. These are the important ideas to take with us into the next section because they are the effects which must be simulated in a model of butterfly wing patterns.
The Naïve Method discussed below consists of
procedurally creating a wing shape. Veins
are created within the wing shape from the shoulder to the wing edge. With the wing shape and the veins specified
the pattern is created by combining images and placed within the geometry.
Wing broken down into fundamental control
points each with an angle and length from the shoulder (polar coordinates). After specification, the NURBS outline is
evaluated into 
a long string of line segments. Intersection
with the wing outline is determined by first testing ray intersection with the
finite plane in which the wing is located, and then testing whether this point
of intersection is located within the segmented outline.
The veins are important in the wing structure
for this method. They are used for
texture mapping and they divide the wing into sections. Veins are created by specifying the number of
veins at the shoulder edge on the wing and the number of veins at the outer
edge of the wing. These two values
determine the need for branching veins. NURBS curves are used to model the
veins. From the vein definitions the
method enables a [u,v] coordinatizing of each section, so that spots, stripes,
etc. will follow the curves and also align with the spots in adjacent sections.
Once the wing shape and vein structure have
been created it is time to make the pattern. First a wing background is created either by a
randomly generated color and/or repeated scale texture. Then each section of a wing is identified and
mapped from sector space to wing space and spots are placed so that they curve
inside the veins and textures are placed in rows along different vein sections. Larger spots are then placed outside of vein
sections with their position based on the vein representation. Finally the upper and lower wing textures are
duplicated and flipped to create horizontal symmetry.
Even though we call this
the Naïve Method it has some points which make interesting enough for us to
have looked at it. The method includes
an interesting and flexible wing creation procedure and it creates bands and
spots in sensible locations. This wing
shape definition algorithm is something we will not find in our next method. However the draw-backs of this method are that
lots of arbitrary values are used in the entire procedure, the control of the
patterns is very limited, and the method really only mimics butterfly patterns
without modeling them.
In the Fractal Method the pattern for the wing is created by generating a fundamental pattern from a specific fractal equation. From this fundamental pattern a few variation are created. These patterns are combined into a pattern and then mapped onto a defined wing shape. In this pattern eye-spots and banding are modeled by using fractals. Eyespots are modeled using Basins of Attraction while the colored bands are created from the chaotic structures between these eyespots.
Fractals are created by using the (x,y) coordinate system as a complex plane; (Real, Imaginary). Each point in the plane is iterated on to determine if it converges to zero based on some function; f(z). The color of each point in the plane is based on the number of iterations the point goes through before converging to zero or reaching the maximum number of iteration if it does not converge.
Basins of
Attraction are defined by the roots of f(z). Around the roots f(z)
will converge to zero more quickly than far away from a root. This means that the areas around 
the
roots will be colored similiarly and thus appear as
eye-spots.Bands occur where
the basins of attraction meet. By
keeping the roots in a line with each other eye-spots and bands perpendicular
to the roots are created. This allows
for making different patterns by the manipulation of the location of the basins
of attraction. To add randomness to the
fractal a convergence test is picked at random from a set of valid convergence
tests.
This is a very
interesting method because it tries to use a mathematical repeatable method to
create the wing patterns, it can create bands and spots with a realistic look
and the wing shape is easily customizable.
However the problems with this method is that it does not take vein
structure into account and to create any realistic pattern you need to know the
desired effect of the output and then define roots for your equation which will
give you that output.
References:
- Wen-Kai Dai, Ruei-Chuan
Chang, and Zen-Chung Shih Fractal pattern for a butterfly wing. The
Visual Computer, Springer-Verlag :1995
- Adrian
Porter, Newton's Method,
- ZoomSchool, Butterfly Wing Anatomy, [Online]
Available: http://www.enchantedlearning.com/subjects/butterfly/