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Table 1: Mathematical modeling of the ellipse, astroid and
the quadrifolium.
Ellipse
Parametric: {a*cos(t), b*sin(t)} 0< t ≤ 2 π
Cartesian: x
2
/a + y
2
/b = 1 b=a*(1-e
2
)
1/2
Astroid
Parametric: {cos(t)
3
, sin(t)
3
}, 0 < t ≤ 2 * π
Cartesian: (x
2
+y
2
-1)
3
+ 27 * x
2
* y
2
= 0
Equivalent equation: x
2/3
+ y
2/3
= 1
Rose
(Rhodonea)
Polar equation: r=cos(p/q*θ); p, q are
integers (typical range: 1 to 11)
Quadrifolium: p/q =2
Cartesian Equation for a 4-pedaled rose
r=cos(2*θ) rotated by 2*π/8 is
(x
2
+y
2
)
3
==4*x
2
*y
2
3.2 Trunk Bark Modeling
Traditional methods for modeling the trunk bark
involve the use of bump maps. In this paper, an
alternate scheme of geometry based parametric
modeling of the bark using concentric layers is
presented.
3.2.1 Phenomenon
Plants generally grow as cylindrical structures above
the ground once their roots have developed. The
local geometry of the trunk can be formulated as
cylinders. As they grow, the bark expands in
concentric circles. The most actively growing part of
the trunk is the innermost ring, which gets pushes
out the outer rings. The cylindrical rings at the outer
layer are the oldest and this phenomenon is used in
the dating of trees. Because the outer layers are the
oldest, the bark eventually loses flexibility and
becomes taut. The pressure from the inner cylinders
causes the external bark to crack up. Crevice-like
structures are formed. Since pressure is directed
radially outward from an equidistant center, the
cracks in the bark appear at points almost equidistant
on the circumference of the bark. The radial pressure
also causes the vertical cleavage lines. Furthermore,
the effects of weathering create random fissures on
the bark.
3.2.2 Algorithm
Incorporating this understanding into our algorithm,
we generate the fissured geometry of the bark: a
superimposition of two cylindrically varying
structures. The face or planes of the tessellated
polygons on the outer structure are shrunk to smaller
triangles to simulate the cracking up of the bark.
This is done by reducing the dimensions of the
triangles along the vertical direction (simulating a
radial bark expansion along the cross-section of the
trunk) in a pseudo-random fashion. A randomized
polygon removal scheme has been employed to
produce the effect of weathered fissures. In other
words, some polygonal along the vertical dimension
of the trunk are arbitrarily removed. These effects
result in the fissured bark geometry demonstrated in
Figure 5. Application of different texture maps to the
two cylinders yields the effect of a chiseled outer
bark on a younger inner bark.
3.3 Branch Modeling
Branches are modeled as cylinders with diameters
constrained by parameters of radii and tilt. Three
specific issues were considered in the generation of
the branches. Combining cylinders at junctions is a
cumbersome process and involves a lot of projection
geometry. In nature, branch junctions can be
geometrically categorized as: T-junctions, Y-
junctions and I-junctions.
T-junctions occur when a branch grows out of
another, which extends much further. Such junctions
are modeled as composing of cylindrical branches
with semi-randomly varying radii and sinusoidally
varying base, embedded at a certain distance into the
parent branch, thereby producing smooth transition
geometry.
For the I-sections or continuation branches,
which chiefly occur when the branches have low
radii of curvature or a high degree of curvature,
smoothing effects are produced by extrapolating the
parent cylinder and semi-randomly varying the
gradient of inclination weighted by the parent
cylinder radius to yield the child cylinder top radius,
rather than create two separate cylinders.
For the Y-sections, the joint is more complex, as
the top radius of the parent cylinder and the bottom
radius of the child are different. Mounting the child
cylinders on the parent would produce huge areas of
discontinuity. This issue has been solved by using a
sinusoidally varying base for the child cylinder, as in
the case of the T-junction, and creating a spherical
geometry atop the parent cylinder.
4 IMPLEMENTATION
ISSUES –GEOMETRY
AND TRIANGULATION
Parameterization and generalization of various
aspects of the double-barked geometric tree model
enables creation of different types of trees. Limits of
feasible space and time complexity involved in the
PARAMETRIC MODELING OF TRUNK-ROOT JUNCTIONS USING ASTROIDAL EXPANSION GEOMETRY
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