RAY TRACING OF NONLINEAR FRACTALS

Peter Wonka Michael Gervautz

Technical University ViennaInstitute for Computer Graphics

Karlsplatz 13/186/2, A-1040 Vienna, Austriaemail: {wonka|gervautz}@cg.tuwien.ac.at

ABSTRACT

We present a new kind of parametric Lindenmayer-systems called nonlinear CSG-pL-Systems,which are useful for the modeling of nonlinear fractals and fractal like objects. Nonlinear CSG-pL-systems describe cyclic CSG-Graphs, which can include several nonlinear transformations.To render the given objects a ray tracing algorithm is introduced, that is independent of thetransformations and the under laying CSG-primitives. We use tapering, twist and bend asnonlinear transformations for our implementation. The new modeling possibilities and theirvisualization for abstract fractals and natural phenomena are investigated.

KEYWORDS: nonlinear fractals, nonlinear ray tracing, CSG-pL-systems, natural phe-nomena.

1 INTRODUCTION

Fractals are a powerful and memory saving ap-proach to model complex three dimensional ob-jects. With the short description of a fractal itis possible to generate a geometric structure ofan arbitrary amount of detail, that could hardlybe described with an enumerative approach. Tomake fractals a useful modeling tool, it is neces-sary to use a flexible, parameter controlled classof fractals, which makes it possible to have suf-ficient control over the outcome. Since nonlineartransformations have been proven to be intuitiveand convenient aids in the modeling process, it isour goal to develop a modeling tool that combinesthe advantages of nonlinear transformations andfractal modeling.

One observation that can be made when studyingnatural phenomena like plants is the highly com-plex structure and the strong degree of nonlinear-ity of the geometry. For the modeling of naturalphenomena procedural modeling techniques arethe most adequate and produced the best resultsup until now. Therefore we want to use the non-linear CSG-pL-Systems for the modeling of plants.In order to get a high quality picture the objectswill be rendered with ray tracing.

In the past Demko, Hodges and Naylor [DHN85]used iterated function systems(IFS) for the defi-

nition of complex objects. Although they limitedtheir approach to two dimensional objects their re-sults looked very promising. Ray tracing of threedimensional IFS was investigated by Hart and De-Fanti [HD91] using the concept of object instanc-ing to construct the fractals dynamically duringrendering. Nonlinear transformed IFS were inves-tigated by Zair [ZT96] using free form techniquesand Groller [Gro94] presented a method for mod-eling and rendering nonlinear IFS. The problemwith IFS modeling is that the constructed objectsare strictly self similar, which is too restrictive tomake them useful for the modeling of real worldobjects. Another concept which is adequate for thedescription of these objects are L-Systems whichwere introduced 1968 by A. Lindenmayer [Lin68].Their use for the modeling of plants was shownby Prusinkiewicz and Lindenmayer [PLH+90] withseveral extensions like parameter passing, stochas-tic behavior and context sensitive rules. The limi-tation of their approach is the rendering techniquewhich uses huge amounts of memory, because thescene description is built up in advance, which alsomakes nonlinear extensions difficult. A solutionto overcome the memory limitation are CSG-pL-Systems [GT96], which provide similar modelingpossibilities and provide a useful basis for nonlin-ear extensions. Therefore we use CSG-pL-systemsas a basis for our approach to include nonlineartransformations in fractal modeling.

2 BACKGROUND

CSG-pL-systems [GT96] are parametric stringrewriting systems that can be represented bycyclic CSG-graphs, which need much less mem-ory than the explicit CSG-expressions. The pa-rameters are treated like global variables, so thatany node can access all the parameters it needs. Agraph consist of the following elements:

• The leaves of the CSG-graphs are geometricprimitives like cylinder, box or sphere.

• Sub-graphs are combined with the CSG-operators op ∈ {\,∪,∩}.

• Affine transformations are described intransformation-nodes (T-nodes), so thatthey can be applied to primitives and wholesub-graphs as well. A T-Node consists of atransformation list, so that several transfor-mations can be stored in one node. The trans-formations can depend on certain parameters.e.g. scale (a,0,2b).

• Calculation nodes (C-nodes) are used tomodify the parameters. New values can be as-signed to any parameter according to math-ematical expressions. This is very similar tothe assignment of a value in a programminglanguage. e.g. a=a*a+2*c.

• Selection-nodes (S-nodes) are used to con-trol the construction of the CSG-expressionand have more than one successor in thegraph. They select the actual successor basedon a boolean expression. The boolean expres-sion depends on given parameters. S-nodesbehave like if-then-else statements: e.g. if (b ≤3 ∗ c) then select node1 else node2.

CSG-pL-systems describing smaller objects can berendered interactively with Z-buffering to obtaina quick result [SG97] during modeling. For highquality pictures they are used directly for ray trac-ing.

Figure 1 shows an simple example graph represent-ing a Sierpinski-tetrahedron. Beside IFS CSG-pL-systems are mainly used to describe complex mod-els like plants, shells and architectural objects.

3 NONLINEAREXTENSION

For our system design we have to consider twomajor points: modeling and rendering. The useof nonlinear transformations should enrich therange of possible models without making the de-sign of new objects too complicated. For the ren-dering part we choose ray tracing, because it pro-duces highly realistic images especially for outdoorscenes. These are our design goals:

• It should be possible to apply nonlinear trans-formations to geometric primitives and wholeCSG-graphs as well.

• The used class of transformations should bean intuitive modeling aid.

• The transformations should fit into the con-cept of ray tracing CSG-graphs.

• The computational effort should stay in anacceptable range.

Tapering, Twist and Bend [Bar84] were proven tobe very useful to describe the structure of nat-ural phenomena like plants. Branches for exam-ple are almost always bended and tapering canbe used to modify the thickness of the branches.Oppenheimer [Opp86] successfully used the con-cept of tapering and twist for the creation of com-plex tree models. The result of tapering and twistapplied to a twig can be seen in figure 2. There-fore we extended CSG-pL-systems to allow thesethree transformations. Nonlinear transformationscan be used in transformation nodes, together withlinear transformations. In this way whole CSG-graphs can be transformed with one nonlineartransformation.

4 RENDERINGALGORITHM

The idea of ray tracing three dimensional fractalsis to use the concept of object instancing to con-struct only parts of the fractals dynamically dur-ing rendering. The major advantage of this ap-proach is to render fractals with very low mem-ory consumptions. Hart [HD91] presented this ap-proach for IFS and Groller [Gro94] extended thisconcept for ray tracing nonlinear IFS.

The ray tracing algorithm for linear CSG-pL-systems uses a ray transformation instead of an

counter=6

(counter = 0) ?

YES

U

NOcounter -1

U U

T1 T2 T3 T4

selection

calculation

primitive

transform-mation

CSG-operator

Figure 1: The CSG-graph on the left represents the Sierpinski-tetrahedron and can be used for raytracing(right picture).

Figure 2: A set of twigs created with tapering andtwist applied to a simple twig model.

object transformation to calculate the ray-objectintersections. To calculate an intersection pointwith an object represented by a CSG-Graph theray traverses the graph. When a transformationis encountered the ray has to be transformed sothat ray-primitive intersections can be calculatedin the local coordinate system of the primitives.

To keep the advantage of object instancing it isnecessary to transform a ray immediately when-ever it encounters a nonlinear transformation.This transforms the linear ray into a three dimen-sional curve, which makes the representation ofthe ray consisting of an eye and direction vectorno longer sufficient. Another representation has tobe found.

The main idea is to use a piece by piece linearapproximation as proposed by Groller [Gro95].

Within a transformation area the ray is split intoN linear segments defined by N+1 points pi. Sinceit is hard to determine how many segments arenecessary for an accurate representation, an errormetric has to be found which guarantees, that themaximum distance of the linear pieces to the ana-lytically correct ray does not cause noticeable er-rors in the picture. To maintain a certain accuracywithout time intensive computations adaptive re-finement is used.

4.1 Adaptive Refinement

Given a nonlinear transformation fnl, the ray de-fined by pi, 1 ≤ i ≤ N + 1 is transformed withthe inverse transformation f−1

nl . After this trans-formation the accuracy of the representation doesnot hold the error bound in general and differs fordifferent ray segments. To improve the approxi-mation for every three points pi, pi+1 and pi+2

the maximum error is estimated with the dis-tance from the point f−1

nl (pi+1) to the chordf−1

nl (pi)f−1nl (pi+2). This is the height of the trian-

gle defined by the three points. If this distance ex-ceeds a defined threshold new points are insertedbetween pi, pi+1 and between pi+1, pi+2. The re-finement takes place until the condition is met forall points. The process of the adaptive refinementis illustrated in figure 3. As the ray traverses thegraph down to a primitive it passes many non-linear transformations. Therefore the process ofadaptive refinement has to be repeated after eachtransformation.

In our implementation a T-node consists of a listof m linear or nonlinear transformations fi. The

transformation of a whole T-node is defined byfT−node = (f1◦f2 · · ·◦fm). Adaptive refinement isdone once for the transformation fT−node insteadof calculating each transformation fi separately,gaining speed.

h > hmax insert points

h > hmax ?

Figure 3: Adaptive-refinement of two ray seg-ments. If the height of the triangle defined by thethree points is greater than a certain threshold,the ray segments are refined.

For our final pictures we used a threshold of10−5units for objects which are about 1 unitsbig. This conservative estimation was empiricallyfound because of test measurements. A maximumbound for the occurring error cannot be given withthis method.

4.2 Ray-primitive intersection

As the nonlinear ray consists of linear segmentsthe ray-primitive intersection can be calculated bysuccessively intersecting a single segment with theprimitive using existing linear intersection algo-rithms. This means all primitives of a ray tracercan be used in combination with nonlinear trans-formations. In many scenes only union-CSG op-erators are used. For these cases an additionalspeedup can be achieved. The calculation can stopafter the first intersecting segment, because no fur-ther intersections can be visible.

4.3 Normal vector calculation

To handle nonlinear transformations in ray tracingit is necessary to know not only the transformationfunction but also the inverse transformation func-tion and the normal vector transformation. Ta-pering and twist, in the most general case, use anarbitrary function f : R → R to control the trans-formation. For the calculation of the normal vectortransformation the first derivation of this functionis needed. Numerical calculations during the ren-dering process are much too expensive. Thereforeonly certain transformations are used, where thefirst derivation can be calculated easily in advance.

For the most cases a piece by piece linear functionis sufficient.

5 IMPROVEMENTS

5.1 Bounding Boxes

I)

II)

III)

Figure 4: This figure shows the bounding boxproblem for 2 dimensions: I) A cylinder is trans-formed with linear transformations. The 4 pointsare sufficient for a bounding box calculation af-ter the transformation. II) The cylinder is bendedand the cylinder is no longer surrounded by thebounding box determined by the four points. III)The solution is to use additional points for thebounding box calculation.

Bounding boxes are very essential for the perfor-mance of the ray tracing algorithm. They providetwo advantages:

• Bounding Boxes reduce significantly the in-tersection calculation for hierarchical struc-tures.

• Bounding boxes guarantee a finite transfor-mation area. The nonlinear ray can be clippedwith each bounding box reducing the numberof ray segments to be considered for furthercalculations.

The bounding box calculation for linear CSG-pL-systems was investigated by Traxler et al. [TG95].Their approach used hyper bounding boxes foreach node of the CSG-Graph instead for each in-stance of each node. Traxler used eight pointsto approximate the convex hull of primitives.These eight points together with all transforma-tions along the path to the primitive can be used

to calculate the contribution of a primitive tothe bounding box of an intermediate node. Theirmethod calculates bounding boxes which are tightenough for ray tracing.

To adapt this algorithm for nonlinear CSG-graphswe have to consider, that a set of 8 points might nolonger be sufficient to approximate the convex hullfor a nonlinear transformed primitive (see figure4). To solve this problem we extended the pointset to form a mesh defined by the original box.

These points are used as the new approximationof the convex hull. With the new approximationit is very unlikely, that a part of the transformedprimitive lies far outside the transformed point set.However even a fine mesh cannot guarantee a cor-rect bounding; small parts of the object may lieoutside the bended mesh.

Therefore the point set of the original boundingbox is scaled up a bit before the fine mesh is con-structed. In this way we reduce the probabilitythat parts of the object lie outside the boundingmesh after transformation.

This approach satisfies two important conditions:The calculation works exact enough, so that nobounding errors occurred in our tests and the com-putation is inexpensive compared to the renderingtime.

5.2 Strip Trees

The straightforward approach to successively in-tersect each segment of the ray with a boundingbox works well, but can further be improved. Afaster solution would be a hierarchical handling ofthe ray. We use strip trees to subdivide nonlinearrays and create hierarchical bounding boxes forthem.

The strip tree was introduced by Ballard [Bal81]and first used by Kajiya for ray tracing [Kaj83].The strip tree used in our approach is quite simi-lar to the strip tree proposed by Groller [Gro95],who uses axis aligned bounding boxes for a threedimensional ray described by an explicit mathe-matical function.

We will also use axis aligned boxes for our ray.Since our ray is represented by linear segmentsthe construction of the strip tree is straightforwardand very fast.

The leaves of the tree are linear ray segmentsbound by a box. The inner nodes of the tree com-bine the boxes of their successors. To construct the

Figure 5: The figure shows a strip tree for 8 raysegments.

tree the bounding boxes are combined bottom-upfrom the leaves up to the top.

For the ray-object bounding box intersection testit is important to note, that we do not need theexact intersection point. It is enough to deter-mine which segments of the nonlinear ray are com-pletely outside the object bounding box and clipthem. For this calculation the strip tree is testedagainst the object bounding box. Starting with theroot node, the bounding box of the strip tree nodeis intersected with the object bounding box. Threeconfigurations are possible:

• The bounding box of the strip tree node is in-side the object bounding box. In this case allsegments of the ray are considered for furthercalculations.

• The boxes have no overlapping area. Thismeans that the parts of the ray bound by thestrip tree node can be clipped.

• The boxes partly overlap. In this case the ob-ject bounding box is tested against the twosuccessors of the strip tree node. If the con-sidered strip tree node is a leaf node the raysegment can not be clipped .

Traversing down the graph this way, only ray seg-ments reach a primitive which lie inside the prim-itive bounding box.

6 MODELING

We used nonlinear CSG-pL-systems for the mod-eling of abstract fractals and plants. Since nonlin-

ear IFS are a subset of nonlinear CSG-pL-systemswe can render nonlinear IFS and nonlinear trans-formed IFS. The use of nonlinear transformationsgave us several advantages:

• The modeling of complex transformations ap-plied to a whole graph is possible. We can takeany object represented through a CSG-Graphand apply a nonlinear transformation to it.

• For complicated nonlinear geometry it issometimes too expensive to approximate itwith linear transformed primitives. A stronglytwisted stem for example would need a largeamount of primitives for an adequate repre-sentation.

• A great variety of similar objects can be pro-duced by applying different nonlinear trans-formations to them. An example can be seenin picture 2.

7 RESULTS

Nonlinear CSG-pL-systems were implemented asan extension to the well known free-ware ray tracerPovray. The implementation is very modular anduses the core functions of POV 3.0 so that nearlyall primitives and features offered by Povray canbe used. Nonlinear fractals can be seen in the pic-tures 6 and 7. They show the use of nonlineartransformations in each level of recursion. Bothfractals were produced by a variation of the Sier-pinski tetrahedron definition described in figure1. Nonlinear tapering is applied within the recur-sive description of the fractal. In figure 6 an ad-ditional tapering is used to transform the wholeobject. Picture 8 shows the construction of palmsand the hut in figure 9 demonstrates the use oftwist for the construction of climbing plants. Theplants are constructed with linear transformationsand a twist is applied to obtain the final geometry.The memory requirements for all scenes are under100 Kbyte.

8 CONCLUSION

Nonlinear transformations were used to create anew kind of pL-Systems called nonlinear CSG-pL-systems. We showed that the given system is auseful modeling tool for abstract fractals and nat-ural phenomena. Our experience was, that the ray

tracing algorithm and the bounding box calcula-tions are very stable and even extreme transfor-mations do not result in visible errors in the finalpicture. But one has to be cautious using nonlineartransformations for the modeling of real world ob-jects because transformations applied to a wholeCSG-Graph might result in unwanted side effects.The leaves in picture 9 for example are all twistedaround the stem. This is not a problem for theshown picture, but for close views the modelingartifacts become visible.

9 ACKNOWLEDGMENTS

Thanks to Christoph Traxler for his useful com-ments on that project and Wolfgang Deutsch forparts of the implementation.

References

[Bal81] Dana H. Ballard. Strip trees: A hierar-chal representation for curves. Commu-nications of the ACM, 24(5):310–321,May 1981.

[Bar84] A. H. Barr. Global and local deforma-tions of solid primitives. In H. Chris-tiansen, editor, SIGGRAPH ’84 Con-ference Proceedings (Minneapolis, MN,July 23-27, 1984), pages 21–31. ACM,July 1984.

[Bar86] A. H. Barr. Ray tracing deformedsurfaces. Computer Graphics (SIG-GRAPH ’86 Proceedings), 20(4):287–296, August 1986.

[Bar89] M. Barnsley. Fractals Everywhere. Aca-demic Press, San Diego, 1989.

[Boa92] OpenGL Architecture Review Board.OpenGL Reference Manual: The Offi-cial Reference Document for OpenGL,Release 1. Addison-Wesley, Reading,MA, USA, 1992.

[Bou85] C. Bouville. Bounding ellipsoids forray-fractal intersection. ComputerGraphics, 19(3):45–52, July 1985.

[DHN85] S. Demko, L. Hodges, and B. Naylor.Construction of fractal objects with it-erated function systems. ComputerGraphics, 19(3):271–278, July 1985.

[DSG96] Christoph Traxler Dieter Schmalstiegand M. Gervautz. Interactive ren-dering of natural phenomena with di-rected cyclic graphs. Technical Re-port TR-186-2-96-16, Institute of Com-puter Graphics 186-2, Technical Uni-versity of Vienna, Vienna, Austria,1996. human contact: technical-report@cg.tuwien.ac.at.

[Gla89] Andrew S. Glassner, editor. An In-troduction to Ray Tracing. AcademicPress, 1989.

[Gro93] E. Groeller. Fraktale Skriptum zur Vor-lesung. Institut fuer Computergrafik,TU Wien, 1993.

[Gro94] E. Groeller. Modeling and renderingof nonlinear iterated function systems.Computer & Graphics, 18(5):739–748,1994.

[Gro95] E. Groeller. Nonlinear ray tracing: vi-sualizing strange worlds. The VisualComputer, 95(11):263–274, 1995.

[GT94] Michael Gervautz and ChristophTraxler. Representation and realis-tic rendering of natural phenomenawith cyclic CSG-graphs. TechnicalReport TR-186-2-94-19, Institute ofComputer Graphics 186-2, TechnicalUniversity of Vienna, Vienna, Austria,December 1994. human contact:technical-report@cg.tuwien.ac.at.

[GT96] M. Gervautz and C. Traxler. Represen-tation and realistic rendering of natu-ral phenomena with cyclic CSG graphs.The Visual Computer, 12(2):62–71,1996. ISSN 0178-2789.

[HB87] Brian Von Herzen and Alan H.Barr. Accurate triangulations of de-formed, intersecting surfaces. Com-puter Graphics (SIGGRAPH ’87 Pro-ceedings), 21(4):103–110, July 1987.

[HD91] John C. Hart and Thomas A. DeFanti.Efficient anti-aliased rendering of 3Dlinear fractals. In Thomas W. Seder-berg, editor, Proceedings of the ACMSIGGRAPH Conference on ComputerGraphics (SIGGRAPH ’91), pages 91–100, Las Vegas, Nevada, USA, July1991. ACM Press.

[Kaj83] James T. Kajiya. New techniques forray tracing procedurally defined ob-jects. Computer Graphics, 17(3):91–102, July 1983.

[KK86] Timothy L. Kay and James T. Kajiya.Ray tracing complex scenes. In Com-puter Graphics (SIGGRAPH ’86 Pro-ceedings), pages 269–78, August 1986.Published as Computer Graphics (SIG-GRAPH ’86 Proceedings), volume 20,number 4.

[Lin68] Aristid Lindenmayer. Mathematicalmodels for cellular interaction in devel-opment parts i and ii. Journal of The-oretical Biology, 18:280–315, 1968.

[MP96] Radomir Mech and PrzemyslawPrusinkiewicz. Visual models of plantsinteracting with their environment. InHolly Rushmeier, editor, SIGGRAPH96 Conference Proceedings, AnnualConference Series, pages 397–410.ACM SIGGRAPH, Addison Wesley,August 1996. held in New Orleans,Louisiana, 04-09 August 1996.

[Opp86] P. E. Oppenheimer. Real time designand animation of fractal plants andtrees. Computer Graphics, 20(4):55–64,August 1986.

[PJM94] P. Prusinkiewicz, M. James, andR. Mech. Synthetic topiary. Com-puter Graphics, 28(Annual ConferenceSeries):351–358, 1994.

[PLH+90] Przemyslaw Prusinkiewicz, AristidLindenmayer, James S. Hanan, et al.The Algorithmic Beauty of Plants.Springer-Verlag, 1990.

[PT97] POV-Team. Persitence of Vi-sion Ray-Tracer User’s Docu-mentation 3.0.10. POV-Team seehttp://www.povray.org/, 1997.

[RW80] Steven M. Rubin and Turner Whitted.A 3-dimensional representation for fastrendering of complex scenes. ComputerGraphics, 14(3):110–116, July 1980.

[SB87] John M. Snyder and Alan H. Barr.Ray tracing complex models contain-ing surface tessellations. ComputerGraphics (SIGGRAPH ’87 Proceed-ings), 21(4):119–128, July 1987.

[Sch94] J. Schipfer. Iterierte Funktionssystemenichtaffiner Funktionen. Diplomarbeit,Kopierer, 1994.

[SG97] D. Schmalstieg and M. Gervautz. Mod-eling and rendering of outdoor scenesfor distributed virtual enviroments. InProceedings of ACM Symposium onVirtual Reality Software and Tech-nology 1997 (VRST’97) ,Lausanne,Switzerland,Sep. 15-17, 1997), pages209–216. ACM, September 1997.

[Smi84] A. R. Smith. Plants, fractals, and for-mal languages. In H. Christiansen, ed-itor, SIGGRAPH ’84 Conference Pro-ceedings (Minneapolis, MN, July 23-27,1984), pages 1–10. ACM, July 1984.

[Sut63] Ivan E. Sutherland. Sketchpad: Aman-machine graphical communica-tion system. In Proceedings AFIPSSpring Joint Computer Conference,volume 23, pages 329–346, Detroit,Michigan, May 1963.

[TG95] C. Traxler and M. Gervautz. Cal-culation of tight bounding volumesfor cyclic csg-graphs. Proceedings of11th Spring Conference on ComputerGraphics, Bratislava, 11, 1995.

[TG96] C. Traxler and M. Gervautz. Using ge-netic algorithms to improve the visualquality of fractal plants generated withCSG-PL systems. In Winter Schoolof Computer Graphics 1996, February1996. held at University of West Bo-hemia, Plzen, Czech Republic, 12-16February 1996.

[Tra97] C. Traxler. Modelling and Realis-tic Rendering of Natural Scenes withCyclic CSG-Graphs. Dissertation,1997.

[Wer94] J. Wernecke. The Inventor Mentor.Addison-Wesley, 1994.

[ZT96] C. E. Zair and E. Tosan. Fractal model-ing using free form techniques. Compu-ter Graphics Forum, 15(3):C269–C278,September 1996.

Figure 6: Nonlinear IFS that was transformed withone additional tapering. The rendering of this ob-ject took about two hours (A comparable linearversion would need 40-60 minutes).

Figure 7: Nonlinear IFS that was transformed withone additional tapering (120-180 minutes render-ing time).

Figure 8: These palms were created using different twigs from picture 2.

Figure 9: The left picture shows a fence full of climbing plants. The plants and the fence together arerepresented as one nonlinear CSG-pL-system (about 300 minutes rendering time). The right pictureshows a conifer.