PARAMETRIC MODELING OF TRUNK-ROOT JUNCTIONS

USING ASTROIDAL EXPANSION GEOMETRY

Karthik Mahesh Varadarajan

ACIN, Technical University of Vienna, Vienna, Austria

Keywords: Geometric modeling, Rendering foliage, Trunk-root junctions, L-systems, Astroid, 3D parametric modeling.

Abstract: In the field of foliage or vegetation modeling for computer graphics, algorithms for parametric modeling of

trees have largely focused on branching mechanisms with little emphasis on modeling the base of the trunk,

especially the trunk-root junctions. Roots and trunk-root junctions that appear prominently above the ground

in the case of many species of trees such as the Ficus macrophylla and the Picea sitchensis as a result of age

and soil erosion are usually completely neglected by traditional trunk modeling methods. In this paper, we

introduce a novel parametric modeling scheme to build such trunk-root junctions, while providing for an

elegant framework to construct the bases of trunks and branches in order to provide better characterization

for a variety of flora species. The paper also describes novel schemes for generating branch junctions and

for rendering tree barks.

1 INTRODUCTION

There have been significant developments in the

area of foliage or vegetation modeling in recent

years (Federl and Prusinkiewicz’ 04, Lefebvre and

Neyret ’02). The availability of multi-core GPU

systems has widened the possibilities for realistic

rendering of foliage. While template or model multi-

resolution rendering systems enable fast

visualization of a variety of plant and tree species,

they are nevertheless restricted due to the lack of

uniqueness of each rendered entity. Parametric

models on the other hand, can create a variety of

shapes and unique entities within a single tree sub-

species or across a broad range of possible species.

Parametric L-systems have long been used for

modeling tree branches and trunks. However, little

attention has been dedicated to the parametric

modeling of roots of trees and more importantly

trunk-root junctions that are visible above the

ground.

Algorithms for parametric modeling of trees

have largely focused on branching mechanisms with

little emphasis on modeling the base of the trunk,

especially the trunk-root junctions. Roots and trunk-

root junctions that appear prominently above the

ground in the case of many species of trees such as

the Ficus macrophylla and the Picea sitchensis as a

result of age and soil erosion are usually completely

neglected by traditional trunk modeling methods.

Figures 1 and 2 depict sample images of trunk-root

junctions that are visible above the ground. In this

paper, we introduce a novel parametric modeling

scheme to build such trunk-root junctions, while

providing for an elegant framework to construct the

bases of trunks and branches in order to provide

better characterization for a variety of flora species.

This work is largely focused on modeling

structural components of a generic tree-like

structure, with particular emphasis on root-trunk

junctions. The algorithm is based on a number of

user-modifiable parameters that can be used to

generate a variety of plant species. Fundamentally,

the structural components, angular limitations, and

branching mechanisms may be varied to more

accurately model different kinds of tree growth, soil

and environmental effects.

Modeling of branches and root-trunk junctions,

together with a bio-inspired algorithm for rendering

barks and branches form specific contributions of

this effort.

94

Varadarajan K..

PARAMETRIC MODELING OF TRUNK-ROOT JUNCTIONS USING ASTROIDAL EXPANSION GEOMETRY.

DOI: 10.5220/0003324900940099

In Proceedings of the International Conference on Computer Graphics Theory and Applications (GRAPP-2011), pages 94-99

ISBN: 978-989-8425-45-4

Copyright

c

2011 SCITEPRESS (Science and Technology Publications, Lda.)

Figure 1: A Moreton Bay Fig (Ficus macrophylla) tree in

Australia [Src: DO'Neil, Wikimedia].

Figure 2: A Sitka Spruce (Picea sitchensis) tree in Canada

[Src: Cowichan Bay Journal].

2 RELATED WORK

Extensive work has been carried out on rendering

algorithms for foliage, originating from ground

breaking work by Przemyslaw Prusinkiewicz. The

majority of these algorithms can be classified into

(a) Mono-scale representation – with branching

structure based on (i) cylinders (Prusinkiewicz and

Lindenmayer ‘90) (ii) cone-spheres (iii) generalized

cylinders (Bloomenthal ’95) (iv) implicit surfaces

(Hart and Baker ’96) (iv) sub-division surfaces, bark

modeling based on (i) bump mapping (ii)

displacement mapping (iii) volumetric textures

(Neyret ’98) (Lefebvre and Neyret ’02) (iv)

polygons (iv) texture mapping, (b) Global

representations such as billboards, slices (c)

Structure based and Spatial based multi-scale

representation systems using hierarchical billboards,

volumetric texture slices (Federl and Prusinkiewicz

’04), particle systems, volumetric point

representation systems etc.

While the focus of the method presented in this

paper caters to mono-scale representation, it

specifically focuses on modeling of trunk-root

junctions, which has hardly been modeled

parametrically in the past. The only work similar to

the one in this paper can be attributed to Jijoon Kim

2006. The method presented in this paper for

modeling of barks is similar to methods such as that

of volumetric textures and polygons. L-Systems

based foliage rendering software L-Studio and Xfrog

can use algorithms developed in this paper as

plugins for realistic geometric modeling.

3 COMPONENT MODELING

The structural components of plants that are dealt

with in this paper include stems (trunks, branches),

barks and root-trunk junctions. The generative

modeling presented in this paper is primarily

focused at achieving realistic stems, as this poses

sufficient unsolved challenges for modeling. This

paper bases its algorithms for geometrically

designing a tree by delving into the natural growth

of foliage. Two main natural phenomena or courses

of events in the growth of a tree, namely concentric

trunk growth and root-trunk junction growth have

been incorporated into the local geometry modeling.

These are explained in sections 3.1.1 and 3.2.1.

3.1 Trunk-Root Junction Modeling

Trunk-Root junctions refer to the base of the tree

where the trunk segment extends through the ground

to form the roots of the tree. In the case of several

common species of trees and plants such as the date

palm (Phoenix dactylifera) and the bamboo

(Bambuseae), the diameter of the trunk remains

more or less uniform from the base to the top. More

importantly, the trunk does not exhibit an expansion

at the trunk-root junction. Traditional parametric

branch and trunk modeling algorithms are successful

in modeling such systems. However, in the case of

trees such as in Figures 1 and 2, this is not true. The

trunk expands into several diverging branches at the

trunk-root junction, which cannot be modeled using

a cylindrical geometric framework.

3.1.1 Phenomenon

The expansive trunk-root junction forms the first

growth phenomenon considered in this paper. In the

case of several species of trees, the roots rise up as

they grow old. Since the trunk is cylindrical and the

PARAMETRIC MODELING OF TRUNK-ROOT JUNCTIONS USING ASTROIDAL EXPANSION GEOMETRY

95

roots exhibit semi-random cylindrical explosion

geometry, the base of the tree can be modeled with

hypocycloids such as the deltoid, astroid etc. with

random perturbations (Figure 3).

Figure 3: Family of hypocycloids – the three and four

pronged geometries are called deltoid and astroid

respectively.

3.1.2 Algorithm

Hypocycloids are very convenient to model trunk-

root junctions. The number of limbs in the

hypocycloid can be varied to produce a variety of

branching systems at the trunk-root junction. This,

when combined with the random perturbation of the

surface contour can yield realistic trunk rendering.

Since the trunk of the tree is best modeled as a

cylindrical structure, it is necessary to model the

continuity between the hypocycloidal trunk-root

junction and the cylindrical trunk. This can be done

by modeling the intermediate region as a folium

with multiple limbs. For the case of the astroidal

trunk-root junction, a quadrifolium is best suited for

the transition to the cylindrical trunk. The

quadrifolium is a rose or rhodonea, generated as a

pedal of the astroid. Pedals can be obtained as the

locus of arbitrary points, one each on the tangents of

a curve, such that the line from a given vertex to the

point and the tangent are perpendicular. An

interpolation scheme such as the spline or bi-cubic

method can be used to define intermediate points.

While the above scheme is well suited for the

generation of a wide variety of trunk-root junctions,

we emphasize the use of the astroid for modeling the

trunk-root junction in this paper. This choice of

hypocycloid for the modeling is the result of the

unique property of the astroid, namely its status as

the envelope of co-axial ellipses whose sum of

major and minor axes is constant (Xahlee, Special

Curves). Astroids are also the evolutes of ellipses.

Astroids belong to a class of curves called the

hypocycloids with cusps pointing away from the

vertex. Hypocycloids are a special case of the

Roulette family, the curves that roll upon other

curves to give the locus of circles. While the

fundamental method to create an astroid is as the

trace of a point on a circle of radius ‘r’ rolling inside

a fixed circle of radius 4 ‘r’ or 4/3 ‘r’ (using single

or double generation mechanisms respectively), the

ellipse method enables one to construct an astroid

analytically by locus generation. A simpler way to

construct the astroid based on ellipses makes use of

the Trammel of Archimedes - a mechanical devise

where a fixed bar with endings sliding on two

perpendicular tracks. The envelope of the moving

bar is then the astroid. A fixed point on the bar

traces out an ellipse. The axes of the astroid are

defined to be the two perpendicular lines passing its

cusps. The length of the tangent cut by the axes is

constant. Thus the astroid can be constructed as the

envelope of co-axial ellipses.

By reducing the difference between the lengths

of the semi-major and semi-minor axes of the

ellipse, characterized by the eccentricity, the ellipse

can be made to converge to a circle when the

eccentricity approaches one or the difference

becomes zero. This gives a natural mechanism for

defining and extrapolating the astroid to a circle,

thus rendering a smooth transition from the astroidal

trunk-root junction to the cylindrical trunk (with a

roughly circular cross-section). The algorithm makes

use of this approach to build the trunk of the tree. It

creates an envelope of ellipses of equal semi-major

and semi-minor axes length sums at every z level,

with this sum varying with the level, thus producing

a natural transition from astroid to ellipse (Stand

Curves).

The algorithm also implements a semi-

randomized transition to produce more organic

irregularity in the trunk and branches. The family of

asteroids and quadrifoliums are shown in Figure 4.

The mathematical formulation of these geometric

entities is depicted in Table 1.

Figure 4: (A) Family of astroids; (B) Astroid as envelope

of ellipses; (C) Quadrifolium.

GRAPP 2011 - International Conference on Computer Graphics Theory and Applications

96

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

97

execution of the program required the development

of optimization parameters. These include the

Triangle Reduction Factor, the factor by which the

number of triangles is reduced for rendering (by

absolute or relative levels), the Curve Speed-up

Factor that controls the number of points for the

ellipse generator, the Generator Ellipse Count, that

controls the number of ellipses that form the Astroid

or Circle and the Base Randomizer Factor that

determines the number of triangles in the complex

astroidal base.

Various tree type characterizing parameters

include the Base Spread Factor which determines

the ratio of the base to the trunk of the tree, the

Height of Trunk parameter, the Chisel mode that

allows one to specify if the outer bark is to be

shrunk, randomly chiseled out or both, two Chisel

factors to control the chiseling in the two modes, the

Base and the Top Radii, the Bark Depth Parameter,

that determines the depth of the inner bark from the

outer one and the Tilt spread to create a tilted tree.

After the generation of the inner and outer bark

geometrical models as envelope of ellipses with

varying major and minor axes and necessary tilting

of both the inner and outer bark cylinders, the

resulting geometry is Delaunay triangulated and

converted to patches. The Delaunay triangulation of

a point set is a collection of edges satisfying an

"Empty Circle" property (i.e., for each edge we can

find a circle containing the edge's endpoints but not

containing any other points (UCI – Delaunay)). The

outer bark triangles, after performing necessary

limiting by count, are scaled, and some triangles are

eliminated based on chisel ratios, thus rendering the

natural bark effect.

Normals for the triangles are generated by simple

cross-product of the edges. In order to produce an

elegant look with Phong shading, the normal values

at each vertex are calculated as the average of the

normals of the different triangles that share the

vertex.

Figure 5: Trunk-Root Modeling.

Implementation of the local geometric modeling of

the branches was done similar to that of the trunk,

but with a single bark structure and the addition of

functionality to solve the problem of the varying

junctions. Additional parameters include junction

type, previous radii, initial and final center

coordinates.

Figure 6: Double-Bark Modeling.

Figure 7: Branch Modeling.

Figure 8: Branch Modeling with Color Mapping.

5 RESULTS

Figures 5 and 6 demonstrate the modeling of trunk

using the trunk-root junction algorithm and the

double bark algorithm for a variety of parameters.

While the generated visualizations presented in this

paper for the trunk-root geometry largely consists of

convex boundaries, the system can also be used to

generate junctions with concave boundaries. The

concave boundaries can be obtained by reducing the

bounds on the extreme values used for the ellipse

axes lengths in the envelope generation process.

This is shown in Figure 4B. Alternatively,

quadrifoliums (Figure 4C) can be used, depending

upon the tree species to be rendered. Figure 7 shows

the modeling of branches using the randomized

cylinder approach and Figure 8 after color mapping.

Figure 9 demonstrates the final rendering of the

entire tree structures using a basic tree rendering

program taking input from the modeled trunk-root

junctions and branches as templates for the

rendering. It can be seen that the final rendering,

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98

though limited in terms of visual quality due to

simplicity of the tree renderer, provides realistic

visualization of the geometry of trees similar to fig

and spruce, that posses trunk-root junctions above

the ground.

Figure 9: Complete Tree Rendering.

6 FUTURE WORK

The next step involves the optimization of the code

for real-time rendering of forests and other foliage

using GPU systems. Development of plugins for

usage with Xfrog, Maya and L-Studio will enable

practical testing of the developed algorithms on a

larger scale. The developed algorithms can also be

extended to help render other types of trees with

unique geometry characterization requirements for

modeling.

REFERENCES

Jijoon Kim, A Growth Model for Root Systems of Virtual

Plants with Soil and Moisture Control, IEICE Trans,

2006.

P. Prusinkiewicz and A. Lindenmayer. The algorithmic

beauty of plants. Springer Verlag, 1990.

J. Bloomenthal. Skeletal Design of Natural Forms. PhD

thesis, University of Calgary, 1995.

J. C. Hart and B. Baker. Implicit modeling of tree

surfaces. In Implicit Surfaces, 1996.

P. Federl and P. Prusinkiewicz. Finite element model of

fracture formation on growing surfaces, ICCS 2004.

S. Lefebvre and F. Neyret. Synthesizing bark.

Eurographics Workshop on Rendering, 2002.

F. Neyret. Modeling, animating, and rendering complex

scenes using volumetric textures. IEEE Transactions

on Visualization and Computer Graphics, 1998.

Xahlee, Special Curves http://www.xahlee.org/

SpecialPlaneCurves_dir/specialPlaneCurves.html

Stand Curves, http://www-groups.dcs.st-

and.ac.uk/~history/Curves/Curves.html

UCI Delaunay, http://www.ics.uci.edu/~eppstein/gina/

delaunay.html

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