Adaptive Ray Tracing of Subdivision Surfaces - Computer Graphics ...

Technical Report TUBS-CG-2003-06 

Adaptive Ray Tracing of Subdivision Surfaces 

Kerstin Müller, Torsten Techmann, Dieter W. Fellner 

{kerstin.mueller,t.techmann,d.fellner}@tu-bs.de 

This TR has been published with The Eurographics Association and Blackwell Publishers 2003, 

Proceedings of Eurographics 2003, CGF 22(3), pages 535–562 and is available at 

www.eg.org/EG/CGF/volume22/issue3/paper175.pdf. 

Institute of Computer Graphics 

University of Technology 

Mühlenpfordtstr. 23, D-38106 Braunschweig 

http://graphics.tu-bs.de 

c○ Computer Graphics, TU Braunschweig, 2003

EUROGRAPHICS 2003 / P. Brunet and D. Fellner 

(Guest Editors) 

Adaptive Ray Tracing of Subdivision Surfaces 

Kerstin Müller, Torsten Techmann and Dieter Fellner 

Volume 22 (2003), Number 3 

Abstract 

Subdivision Surfaces as well as (interactive) ray tracing have become an important issue in computer graphics. 

But ray tracing of subdivision surfaces has received only little attention. We present a new approach for ray 

tracing of subdivision surfaces. The algorithm uses a projection of the ray onto the surface and works mainly in 

two dimensions along this projection. While proceeding from patch to patch, we examine the bounding volume of 

their borders: the lower the distance between ray and subdivision surface, the more refinement steps are adaptively 

applied to the surface but only along the projection of the ray. The adaptive refinement of a patch is controlled by 

curvature, size, its membership to the silhouette, and its potential contribution to the light transport. The algorithm 

is simple and mainly consists of elementary geometric computations. Hence it is fast and easy to implement 

without the need for elaborate preprocessing. The algorithm is robust in the sense that it deals with all features of 

subdivision surfaces like creases and corners. 

Categories and Subject Descriptors (according to ACM CCS): I.3.7 [Computer Graphics]: Raytracing 

1. Introduction 

The increasing interest in subdivision surfaces – due to their 

potential in robust freeform modeling – implies the demand 

for efficient rendering algorithms, both for interactive 

and photorealistic display. While interactive rendering has 

been addressed by several authors 10, 11 and near-interactive 

ray tracing 13, 12 has come within reach, efficient ray tracing 

of subdivision surfaces has only received little attention. 

Firstly, we will have a look at ray tracing of the wellknown 

parametric surfaces besides the subdivision surfaces. 

Woodward 2 proposed a recursive subdivision method carried 

out in the orthogonal viewing coordinate system of the ray. A 

numerical approach is described, e.g, in 4 . Further methods 

for ray tracing parametric surfaces can be found, e.g., in 3, 5 . 

The ray tracing methods for subdivision surfaces existing 

so far can be classified in two different categories: i) The 

surface is tesselated before the ray tracing process and the 

c○ The Eurographics Association and Blackwell Publishers 2003. Published by Blackwell 

Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 

02148, USA. 

resulting polygons are rendered. ii) The intersection test is 

done directly with the subdivision surface. 

For the first category, a tesselator refines the surface uniformly 

or adaptively considering the curvature of the surface. 

The obtained set of polygons is rendered by a ray tracer 

that is able to deal at least with polygons. Consequently a 

variety of freeform surfaces is easy to display via its tesselation 

without the need for special intersection algorithms. 

In order to achieve a high quality image (e.g., visible at reflections 

and at silhouettes of objects and their shadows), a 

sufficiently fine tesselation is necessary. Because the refinement 

is performed independently from the scene description, 

the complete object must be refined. That means also 

parts not necessary for the ray tracing computation are subdivided, 

e.g., parts of the object with no contribution to the 

light transport. This leads to a high number of polygons taking 

into account the exponential growth during a subdivision

step. To get this exponential growth under control, a suitable 

subdivision level for the subdivision object must be chosen 

depending on, e.g., the distance to the camera. 

In the second category, the adaptive refinement of the subdivision 

object is done automatically during the ray tracing 

process. Therefore only parts of the object with contribution 

to the light transport are refined. Within this category, parts 

with a high curvature and members of the silhouette are refined 

further than flat parts in other areas of the object. In 

this way, only the necessary polygons are chosen to obtain a 

high quality image. As a sophisticated example of the second 

class, Kobbelt, Daubert, and Seidel 6 developed a boundingenvelope 

technique for intersecting a ray with a subdivision 

surface. To use this bounding-envelope technique, the minimum 

and maximum values of the basis function of each vertex 

influencing a patch, which is the part of the surface corresponding 

to one face of a control mesh, have to be precomputed. 

This has to be done for every occuring valence. This 

algorithm is fast and has a relatively low memory consumption. 

The implementation of features like creases and corners 

in particular is hard work, because the basis functions 

have to be computed for every practically possible combination 

of valences and creases. This leads to a huge, nonpractical 

number of cases and therefore the algorithm is not 

very easy to implement. A further approach in the style of 

Woodward 2 works mainly in the 2D viewing plane orthogonal 

to the ray. Using the property that the resulting surface 

of one face of the control mesh is located inside the convex 

hull defined by the 1-neighbourhood of its control points, the 

1-neighbourhood is parallel-projected orthogonally onto the 

viewing plane. If the origin of the ray is inside the convex 

hull built by the projected 1-neighbourhood, the corresponding 

face is subdivided. The subfaces are recursively examined 

down to the desired subdivision level. Compared to the 

size of the corresponding patch these bounding volumes are 

relatively large. So there are many intersection tests which 

consume a lot of computation time. But this method is easy 

to implement and supports all special features like creases or 

corners. 

Our goal is to devise an algorithm, that automatically and 

adaptively refines only those parts of a surface with potential 

contribution to the light transport based on curvature, projected 

size, and its membership to the silhouette to obtain a 

high quality image. The algorithm deals with all features of 

subdivision surfaces (e.g., creases and corner) and is easy to 

implement. Only the basic subdivision rules are required as 

well as the rules for the computation of limit points and limit 

normals and no preprocessing step is necessary. 

2. Outline of the algorithm 

Initially, each patch with its bounding volume is sorted into 

an arbitrary space-subdivision scheme. If the ray intersects 

the bounding volume of a patch, our algorithm starts with 

the intersection test between patch and ray. Conceptually the 

Müller et al / Adaptive Ray Tracing of Subdivision Surfaces 

algorithm works in a plane defined by the corners of the 

patch. The parallel projection in direction of its normal is 

called casting a shadow. If the ray casts a shadow on the 

patch (B.t.w., this gave the algorithm its name: ShaoLin = 

Shadow of the Line) and the ray is nearby the patch, an 

intersection is possible. Therefore the patch is subdivided 

recursively according to the refinement criteria until an intersection 

is found or the algorithm determines that no intersection 

occurs. 

SHAdow Of the LINe 

shadowed subpatches: 

level 1 

level 2 

level 3 

Figure 1: Basic idea of the ShaoLin algorithm. 

3. Notations 

The following presentation is based on the Loop subdivision 

scheme 7 . Nevertheless, the techniques can easily be transfered 

to other subdivision schemes (see Section 8.2) taking 

into account the properties described in Section 8.1. 

We use capital letters to denote planes, faces and patches. 

The length of a vector is denoted by | v |. We assume that 

normal vectors are always normalised. The superscript describes 

the subdivision level, e.g., cp l n, whereas the subscript 

indicates the numeration. 

4. Subdivision Surfaces 

A subdivision surface is defined by a control mesh M 0 . By 

applying the subdivision rules to the mesh M 0 , we get a finer 

mesh M 1 . The sequence of control meshes converges to the 

limit surface L(M 0 ). 

cp 0 8 

cp 0 9 

cp 0 10 

cp 0 2 

cp 0 7 

cp 0 3 

F 0 

cp 0 1 

N(F 0 ) 

cp 0 5 

cp 0 4 

cp 0 6 

cp 0 11=n 

cp 1 2 

cp 1 3 

cp 1 1 

ln 2 

ln 3 

ln 1 

l p 2 

T (N(F 0 )) 

l p 3 

l p 1 

L(N(F 0 )) 

Figure 2: Example for Loop subdivision 

c○ The Eurographics Association and Blackwell Publishers 2003.

The vertices of a control mesh M l are called control 

points cp l i. The 1-neighbourhood of cp l i consists of the control 

points adjacent to cp l i and the point cp l i itself. The 1neighbourhood 

of a face F l = △cp l 1cpl2 cpl3 is denoted by 

N(F l ) and consists of the control points cp l 1 ,...,cpl n of 

all faces adjacent to any vertex of F l . The Loop scheme 

provides two kinds of subdivision rules (see Fig. 2). One 

rule creates a vertex cp l+1 

i in the refined mesh M l+1 for 

each vertex cp l i of M l as an affine combination of the 1neighbourhood 

of cp l i. The other rule creates a vertex cp l+1 

j 

in M l+1 for each edge of M l as an affine combination of the 

vertices of the two faces adjacent to that edge. The weights 

for the rules can be found, e.g., in 7, 8 . By applying alternative 

subdivision rules8 special features like creases, corners, 

darts, or conicals can be created on the surface. A limit point 

is defined as l pi := liml→∞ cp l i. The limit normal lni denotes 

the surface normal of L(M l ) at the point l pi. The limit 

surface L(N(F l )) = L(cp l 1 ,...,cpl n) is called the patch corresponding 

to F l . This patch is a part of the limit surface 

L(M l ). T (cp l 1 ,...,cpl n) := T (N(F l )) := △l p1l p2l p3 is the 

chord triangle of F l . 

Consider two neighbouring faces with a shared edge, A l 

and B l of mesh M l and the subfaces B l+1 

1 ,B l+1 

2 ,B l+1 

3 ,B l+1 

4 of 

B l where B l+1 

1 ,B l+1 

2 ,B l+1 

3 ,B l+1 

4 


are parts of the mesh M l+1 

(see Fig. 3). T (N(B l+1 

1 )) and T (N(B l+1 

2 )) do in general not 

lie at the border of T (N(A l )). Thus a gap may occur between 

the tesselated limit surfaces of A l and B l caused by their different 

subdivision levels. To obtain a gap-less surface, a gap 

polygon has to be inserted. 

A l 

B l+1 

1 

Bl Bl+1 

4 

B l+1 

3 

B l+1 

2 

¡ ¡ ¢¡¢ 

¢¡¢ ¡ ¡ 

¡ ¡ ¢¡¢ 

¢¡¢ ¡ ¡ 

N(B l+1 

1 ) T (N(B l+1 

N(A 

1 )) 

l ) T (N(A l )) 

N(B l ) 

N(B l+1 

N(B 

4 ) 

l+1 

N(B 

3 ) 

l+1 

2 ) 

T (N(B l+1 

T (N(B 

4 )) 

l+1 

T (N(B 

3 )) 

l+1 

2 )) 

Figure 3: Two neighbouring tesselated patches at different 

subdivision levels: a gap (hatched triangle) occurrs between 

the two surfaces. So it is necessary to insert a gap triangle 

to obtain a closed surface. The dashed curves on the 

right side describe the patches L(N(A l )) and L(N(B l+1 

1 )), 

L(N(B l+1 

2 )), L(N(B l+1 

3 )), L(N(B l+1 

4 )). 

Since the application of a subdivision rule is a convex 

combination we can infer that the convex hull of N(F l ) en- 

c○ The Eurographics Association and Blackwell Publishers 2003. 

closes the patch L(N(F l )). Consequently, if a plane Q does 

not intersect the convex hull of N(F l ), it does not intersect 

the patch L(N(F l )) either. Let Q be a plane orthogonal to the 

chord triangle T = △l p1l p2l p3 of a patch S. If Q intersects 

T , Q also intersects at least one chord triangle of the subpatches 

of S, because l p1,l p2 and l p3 do not move during 

further refinements. 

Let us now have a look at the border curves of a patch: In 

the regular case, that means a valence of 6 (see Fig. 4a), the 

border curve is a Bézier curve b(t) = ∑ 4 i=0 B4 i (t)bi with its 

Bézier control points (e.g., deduced from 9 ): 

b0 = 1 

12 (cpl 5 + cp l 8 + cp l 2 + cp l 1 + cp l 3 + cp l 7 + 6cp l 4) 

b1 = 1 1 

( 

12 2 cpl5 + 1 

2 cpl1 + 3 

2 cpl8 + 3 

2 cpl3 + 6cp l 4 + 2cp l 7) 

b2 = 1 

12 (2cpl8 + 2cp l 3 + 4cp l 4 + 4cp l 7) 

b3 = 1 1 

( 

12 2 cpl6 + 1 

2 cpl10 + 3 

2 cpl8 + 3 

2 cpl3 + 6cp l 7 + 2cp l 4) 

b4 = 1 

12 (cpl8 + cp l 3 + cp l 4 + cp l 9 + cp l 10 + cp l 6 + 6cp l 7) 

It follows from the properties of the Bézier curve that b0 is 

the limit point of the control point cp4, as well as b4 is the 

limit point of the control point cp7. b1 lies on the tangentline 

b0 + αb ′ (0) with α > 0. b3 is also on the tangent-line 

b4 + βb ′ (1) with β < 0. Furthermore, the border curve is inside 

the convex hull of bi,i = 0,...,4. Let P be a parallel projection 

onto a plane in direction of the normal vector of the 

plane. Because of the property of the affine invariance, the 

equation ∑ 4 i=0 P(bi)·B 4 i (t) = P(∑ 4 i=0 bi ·B 4 i (t)) holds. These 

properties are essential for the correctness of the algorithm. 

a) 

cp l 

5 

cp l 

2 

cp l 

1 

b0 

cp l 

4 

cp l 

8 

b1 

cp 

b2 

b(t) 

b3 

l 

3 

cp l 

cp 

7 

l 

10 

cp l 

6 

b4 

cp l 

9 

b) 

cp l 

8 

cp l 

9 

cp l 

7 

cp l 

2 

c0 

cp l 

0 

cp l 

6 

cp l 

3 

cp l+1 

01 

c1 c2 c3 

l p01 c(t) 

cp l 

5 

cp l 

1 

cp l 

4 

Figure 4: Border curve. a) regular case: The fat edge on the 

mesh is corresponding to the border curve b(t). b) irregular 

case: The curve c(t) is similar to the border curve (dashed) 

of the grey marked face, corresponding to the fat marked 

edge on the mesh. This edge is subdivided to obtain its limit 

point l p01. 

In the irregular case, that means the valence is not equal 

to 6, it is generally not possible to find the Bézier control 

points bi,i = 0,...,4 to obtain the border curve. Stam 9 , e.g., 

proposes a parametrisation for Loop subdivision surfaces in 

irregular cases (one irregular point in the patch): The param- 

c4

eter space is partitioned into an infinite set of triangular tiles 

Ω n k (i.e., regular areas) near the extraordinary point. Depending 

on the parameter (u,v) to be calculated and its assigned 

Ω n k , the resulting point on the surface is computed via the 

eigenstructure of the used subdivision matrix depending on 

the valence. Unfortunately this kind of parametrisation is not 

useful for finding the control points bi,i = 0,...,4 of the irregular 

border curve with the desired convex hull property. 

So we choose a curve c(t) similar to the irregular border 

curve with the properties (see Fig. 4b): 

P1: c(0) = l p0, c(1) = l p1, 

P2: the tangent of the border curve at point l p0 has the same 

direction as c ′ (0), and the tangent of the border curve at 

point l p1 has the same direction as c ′ (1), 

P3: c(0.5) is equal to the limit point l p01 of cp l+1 

01 . 

From this similar curve, we can now calculate the Bézier 

control points ci,i = 0,...,4: 

c0 = l p0 (1) 

c1 = l p0 + 1 

12 tangent0 

(2) 

c2 = 8 

3 (l p01 − 1 

16 c0 − 1 

4 c1 − 1 

4 c3 − 1 

16 c4) (3) 

c3 = l p1 − 1 

12 tangent1 

(4) 

c4 = l p1 (5) 

where tangent0 and tangent1 are the tangential vectors of the 

border curve at points cp l 0 and cpl1 , respectively. 

5. Data Structure 

The implementation of subdivision surfaces uses a versatile 

class hierarchy designed to be extensible and maintainable. 

Each of the classes described below includes an according 

iterator class for navigation purposes and for retrieving information 

about the part of the mesh the iterator points to. 

The base class of this hierarchy is an abstract mesh class 

defining a general interface. This class is used to derive specialized 

mesh classes like a triangle or a quadrangle mesh 

implementing the maintenance and navigation for the respective 

mesh. On the next level of the class hierarchy there 

are special subdivision classes adding subdivision functionality. 

This functionality includes local and global refinement, 

navigation from parent to child and back, computation of 

limit points and normals, chord normals and so on. One of 

the key properties of these subdivision classes is the option 

to perform local subdivision steps. This allows adaptive refinement 

with a minimal effort. 

With this architecture we cover most of the needs that occur 

in applications using subdivision surfaces. The modular 

concept enables us to quickly exchange the subdivision 

scheme or modify the rules of one scheme, e.g., inserting 

rules for creases. 


6. ShaoLin Toolbox 

After l subdivision steps of the starting patch S defined 

by L(cp 0 1 ,...,cp0 n), we obtain 4 l subpatches S j. The border 

curves of S j which are part of the border curves of the 

starting patch are called outer border curves. Subpatches S j 

without outer border curves are called inner patches (see 

Fig. 5). Similarly subpatches with outer border curves are 

outer patches. In subdivision level l we get 3 · 2 l outer border 

curves, 3 · 2 l − 3 outer patches and 4 l − 3 · 2 l + 3 inner 

patches. 

l p1 

l p2 

Figure 5: Inner (white) and outer (grey) subpatches of the 

starting patch S after two subdivision steps with the triangle 

plane T P of S. The three outer border curves of S are drawn 

dashed. 

With T P, we denote the plane defined by the three position 

vectors l p1,l p2 and l p3. The normal vector of T P is 

t pn. For the ray r = (rayorg,raydir) with the ray origin rayorg 

and the ray direction raydir let RP be the plane constructed 

by the position rayorg and the two direction vectors t pn and 

raydir. If t pn and raydir are linearly dependent, the second 

direction vector of RP can be chosen arbitrarily. During ray 

tracing we use an RP-coordinate system defined as follows: 

rpy = t pn, rpz = raydir ×t pn t pn × rpz 

, rpx = 

| raydir ×t pn | | t pn × rpz | 

rpz 

l p3 

T P 

rpy 

rayorg 

t pn 

rpx 

t pn T P 

CP 

RP 

raydir 

Figure 6: Planes used by the ShaoLin algorithm 

PS : IR 3 → T P is the projection in direction t pn onto T P, 

where t pn and T P are defined by a starting patch S. We say 

that the ray r casts a shadow on S if PS(S) ∩ PS(r) �= ∅. An 

intersection between patch S and ray r is possible if and only 

if patch S is shadowed. 

Let us now have a look at the tesselated world: Let T be 

the chord triangle of patch S, and H1,2,3 the three convex 


hulls of the three border curves of S. In the very most cases 

(see Fig. 7) 

PS(S) ⊆ � 

i=1,2,3 PS(Hi) ∪ T (6) 

A patch that fulfills Equation (6) is called a valid patch. 

The projected border curve area is the region on T P outside 

of T between the projected border curve and T . If 

Equation (6) holds, the projected border curve area is part 

of � 

i=1,2,3 PS(Hi) ∪ T . If Equation (6) is not satisfied, we 

can make several subdivision steps and start with valid subpatches 

as starting patches. We obtain valid patches because 

the sequence of subpatches converges to a smooth surface. 

The surface of patch S is smooth, due to the fact that creases 

do not exist in the interior of the patch. In Section 8 we will 

treat the special case that Equation (6) is not satisfied. In the 

following, we assume (6) is true. Thus, intersection between 

patch S and ray r is possible if and only if T or H1,2,3 are 

shadowed, that means: 

�� i=1,2,3 PS(Hi) 

� 

∪ T ∩ PS(r) �= ∅ 

RP 

PS : IR 3 → T P 

T P 


T P 

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Figure 7: Projection onto T P: PS(S) (hatched) is part of 

� 

i=1,2,3 PS(Hi) ∪ T (grey marked). 

Let Tj, j = 1,...,4 l , be the sub-chord-triangles of patch S 

at subdivision level l. Hk, k = 1,...,3·2 l are the border curve 

hulls of S at subdivision level l. The projection function PS 

always uses the projection plane and direction defined by its 

starting patch S (see Fig. 8). An intersection between patch 

S and ray r is possible if and only if at least one Tj or Hk is 

shadowed, that means: 

�� 

PS(Tj) 

j=1,...,4l RP 

� �� 

∪ 

PS(Hk) ∩ PS(r) �= ∅ 

k=1,...,3·2l PS : IR 3 → T P 

T P 

Figure 8: Projection onto T P after one subdivision step with 

its shadowed parts (grey marked). 

To perform a shadow test with a chord triangle T , we 

T P 


check if there is a limit point of T in the right half space 

defined by RP and another limit point of T in the left half 

space. The classification costs of one limit point are 2 additions, 

3 subtractions, 3 multiplications and 1 comparison. 

Therefore the testing costs for all 4 sub-chord-triangles Ti, 

i = 1,...,4 from a subdivision step are at most 12 additions, 

18 subtractions, 18 multiplications, and 6 comparisons. 

The plane orthogonal to the plane RP containing the position 

vector rayorg and the direction vector raydir is called CP 

in the following. A further test uses the convex hull property 

of the patch S. Let C(S) be the convex hull of S built by its 

control points cp 0 1 ,...,cp0 n. We say C(S) is crossing CP if 

CP ∩C(S) �= ∅. Similar to the shadow test, we check if there 

are control points located in the right and the left half space 

defined by CP. An intersection between patch S and ray r is 

possible if and only if C(S) is crossing CP. Let us now consider 

the situation after l subdivision steps: S j, j = 1,...4 l 

are the subpatches of S and C(S j) its corresponding convex 

hulls. Again an intersection between patch S and ray r is 

possible if and only if at least one C(S j) is crossing CP, that 

means CP ∩ ( � 

j=1,...,4 l C(S j)) �= ∅. 

7. Algorithm 

7.1. Description of the Algorithm 

It is typically not trivial to find out whether r casts a shadow 

upon the patch S. For this reason we first test the chord triangle 

T of S plus the hulls of its three border curves, which 

can be done very fast. If T or a hull of its border curves are 

crossed by the shadow, an intersection is possible. A second 

test is applied to check if C(S) is crossing CP. We use these 

two tests to decide in advance, if an intersection is possible 

at all. 

If S passed both tests, S is divided into four subpatches 

S1,...,S4. Then it is checked onto which chord triangle of 

those subpatches Si or onto which hull of the corresponding 

outer border curves the shadow of r is cast. Because we deal 

with a closed tesselated part of the surface, the hulls of the 

inner border curves do not have to be considered. So it is 

sufficient to examine the hulls of the outer border curves, to 

ensure that the transition to the neighbouring patches of S is 

without any gap. 

T P 

S2 

S3 

S4 

S1 

Shadowlist: 

1. S3 

2. S4 

3. S2 

shadowed subpatches: 

level 1 

Figure 9: Shadowlist after one subdivision step 

The shadowed subpatches of S (see Figure 9) are gathered 

in a shadowlist sorted by their distance to rayorg. Beginning 

with the first subpatch in this shadowlist the following pro

cedure is recursively applied to each subpatch in that list until 

an intersection is found or the end of the list is reached. 

The parameters for this part of the algorithm are the patch to 

examine, the ray, and the current subdivision level. 

It is checked whether the control points of the current 

patch cross CP. If they do, an intersection is possible and 

we decide whether to make a further refinement. To obtain 

a high quality image with low consumption of run time and 

memory, three refinement criteria are considered: 

• Silhouette: If the patch belongs to the silhouette, it should 

be refined further than the inner parts of the object. A 

chord triangle with normal n counts as a silhouette triangle, 

if sil :=| n·raydir |≤ εs, 0 ≤ εs ≤ 1 where εs specifies 

the silhouette. 

• Curvature: Parts of the object with a high curvature 

need further refinement to look smooth. Instead of 

expensively computing the exact curvature value of a 

patch, we subdivide the patch initially. Now we can evaluate 

the maximum distance of the three new limit points 

l p4,l p5,l p6, originating from the four sub-triangles, to its 

corresponding edges of the chord triangle △l p1l p2l p3. If 

cv := max 

i=1,2,3 {| ai − bi·ai 

bi·bi · bi |} ≥ εc, 

with ai = l p j − l pi, j = i + 3 

bi = l p (i+1) mod 3 − l pi 

l p 6 

l p 1 

for an εc ≥ 0, then a further refinement is necessary. 

• Size: An additional refinement of the patch is not necessary 

if the resulting sub-chord-triangles projected onto 

the image plane IP are too small with regard to the pixel 

size. In order to achieve smooth reflections, the size of the 

chord triangles projected onto IP must not exceed a given 

maximum. 

To take into account all the termination criteria, we use a 

combined ε and check the following: 

if projected size of the patch on IP < minSize then 

stop refinement; 

else 

if projected size of the patch on IP > maxSize then 

further refinement necessary; 

else 

if cv 

sil < ε then 

stop refinement; 

else 

further refinement necessary; 

end if; 

end if; 

end if; 

l p 3 

l p 4 


l p 5 

l p 2 

If a further subdivision step is necessary, it is determined 

which of the new subpatches is crossed by the shadow of the 

ray. Those patches are gathered in a new shadowlist, again 

ordered by their distance to rayorg. At this point the recursion 

of ShaoLin starts again for each patch of that list. 

7.2. W-Case 

Special attention is required for the following: Let A and 

B be subpatches of the start patch S. In the case depicted 

in Figure 10a only B is put into the shadowlist, even if 

PS(A) ∩ PS(r) �= ∅, because T (A) is not shadowed. Since the 

tesselation of S forms a closed surface on each subdivision 

level, no ray intersection can get lost. 

a) b) B3 c) B3 

B 

B4 �� 

B4 

B2 ©�©�©�©�©�©�©�©�©�©�©�©�©�©�©�© B1 B2 B1 

�� 

A2 

©�©�©�©�©�©�©�©�©�©�©�©�©�©�©�© 

A1 A4 

A 

A3 

shadowed limit triangles: 

�� 

level 1 level 2 

undefined area 

Figure 10: W-Case: A gap occurs at subdivision level 2 and 

is closed by the reactivation of A. 

A recursive call for element B in the shadowlist follows. 

Again, it is tested if C(B) is crossing CP and further refinements 

are necessary. If the tests are successful, B is 

subdivided into B1,B2,B3,B4. In general, it is PS(T (B)) �= 

� 

i=1,...,4 PS(T (Bi)), which means, the intersection between 

G := PS(T (B)) \ � 

i=1,...,4 PS(T (Bi)) and the projection of 

the neighbouring patch PS(A) may be non-empty. This is the 

case, if PS(r) leaves and re-enters � 

i=1,...,4 PS(T (Bi)) at the 

outer edge path of the face pair (B1,B2) (fat marked in Fig. 

10b). We call this a w-case. 

Whenever this happens, the shadow crosses an undefined 

area between B1 and B2 – a so-called gap (the hatched area 

in Fig. 10b). This gap belongs to the projected area of the 

neighbour patch A. To close the gap, patch A gets ‘reactivated’ 

and is added to the shadowlist. In the shadowlist A is 

tagged as a w-case. Now the shadowlist contains B1, A (wcase), 

B2. When the recursive call for A (w-case) is reached, 

the tag w-case leads to a special treatment for A: The crossing 

test with CP and the evaluation of the three termination 

criteria are skipped, hence a refinement of A is enforced. 

Now the check for the shadow crossing the subfaces and the 

processing of possible w-cases are applied as usual. As a result 

of this algorithm the ray traverses a closed surface. In 

the above example the patches B1, A1, A4, A3, and B2 are 

processed (Fig. 10c). In the same way possible w-cases at 

the other two outer edge paths of the face pairs (B2,B3) and 

(B3,B1) have to be examined. 


7.3. Intersection Test 

If the termination criteria determine that the current patch 

does not need further refinement, a possible ray intersection 

with the current patch’s chord triangle is computed. If the 

current patch is an outer subpatch of the original patch S, 

also the concerned border curve areas have to be tested for 

ray intersection. To examine these regions, the Bézier control 

points of the border curve are projected onto a plane V P 

defined by the point rayorg and the normal raydir. Ray r intersects 

the border curve area if and only if rayorg is in the 

convex hull of the control points projected onto V P. 

If an intersection is found, the surface normal of the intersection 

point is calculated using the limit normals at the 

corners of the patch and the barycentric coordinates of the 

intersection point with respect to the patch’s chord triangle. 

T P 


parts of the patch 

tested for intersection: 

level 1 

level 2 

gap triangle 

Figure 11: Tested part of the patch: chord triangles, convex 

hulls of the outer border curve areas and a gap triangle. 

If the above tests do not find an intersection for the current 

patch, the current subdivision level is stored and the 

algorithm continues with the next patch (in the appropriate 

shadowlist). If the intersection test of the next patch occurs 

at another subdivision level than the last stored level, the ray 

may possibly hit a gap between those two chord triangles. To 

prevent such a gaping of the surface, intermediate polygons 

are created between the chord triangles of different subdivision 

levels (see Figure 11). These intermediate polygons are 

then checked for ray intersection. This and the inspection of 

the outer border curve areas guarantee a closed surface while 

finding a ray intersection. 

7.4. ShaoLin in Pseudocode 

The ShaoLin algorithm takes a start patch, a ray, and the 

initial subdivision level (i.e. 0) as initial parameters. 

bool ShaoLin (patch, ray, level) 

if control points of patch cross CP then 

if further refinement necessary then 

Subdivide patch; 

Compute shadow and shadowlist; 

for each shadow-patch i in the shadowlist do 

if shadow-patch i was activated by w-case then 

// shadow-patch i is not a subpatch of patch! 

nextlevel=level; 

else 

nextlevel=level + 1; 


end if; 

if ShaoLin (shadow-patch i , ray, nextlevel) then 

return true; 

end if; 

end for; 

return false; 

else 

// intersection possible 

if last tested level �= level then 

Insert gap polygon between 

the different subdivision levels 

if ray intersects gap polygon then 

return true; 

end if; 

end if; 

if ray intersects chord triangle then 

return true; 

end if; 

if patch is outer patch ∧ 

ray intersects its outer border curve area(s) then 

return true; 

end if; 

end if; 

else 

// intersection not possible 

return false; 

end if; 

8. Discussion 

8.1. Prerequisites 

To guarantee the correctness of the algorithm the following 

requirements must be fulfilled: 

a) The projection of the border curve area onto T P lies completely 

inside the projection of the border curve’s bounding 

volume. 

b) PS(S) ⊆ � 

i=1,2,3 PS(Hi) ∪ T 

Regarding a): In the regular case the convex hull of the 

control points of the border curve forms the desired bounding 

volume (Fig. 4). In the irregular case five fix control 

points to construct the border curve do not exist, so we use 

a similar curve c(t) that meets the requirements P1, P2, and 

P3 (see Section 4). The control points of c(t) are used to 

build the bounding volume of such a border curve. Since the 

algorithm does not need the exact shape of that curve but 

merely the projection of its bounding volume, this part of 

the algorithm is a well working heuristics. It is also possible 

to use other bounding volumes for the border curve (e.g., an 

enlarged sampling of the border curve). 

Regarding b): Equation (6) is true for common geometric 

models. Figure 12 shows a case where (6) is not true. For 

clarity, the left parts of Figures 12 and 13 show a cross section 

of the patch in question. The extreme positioning of the

control points of patch S leads to S overlapping itself. PS(Hi) 

is inside PS(S) and PS(S) ⊃ � 

i=1,2,3 PS(Hi) ∪ T . 

TP PS : IR 3 → T P 


��TP �� 

�� 

�� 

�� 

�� 

Figure 12: Special case PS(S) �⊆ � 

i=1,2,3 PS(Hi) ∪ T . 

Left: cross section. Right: projected view onto T P, 

� 

i=1,2,3 PS(Hi) ∪ T (hatched) is inside PS(S) (grey marked). 

Since there are no creases inside of S, any further refinement 

of S provides patches with smaller curvature. This 

means, beginning with a certain subdivision level l and its 

associated subpatches A j, j = 1,...,4 l , for each of these A j 

no self-overlapping occurs, when projecting them onto T PA j 

(see Fig. 13). Hence PA j (A j) ⊆ � 

i=1,2,3 PA j (Hi) ∪ Tj is true 

for each A j. So the algorithm can start with those A j as start 

patches. 

T P A1 T P A2 

PA 1 : IR 3 → T PA 1 

PA 2 : IR 3 → T PA 2 

T P A1 

T P A2 

Figure 13: Special case after one subdivision step: we obtain 

four new triangle planes, T PA1 and T PA2 are illustrated 

on the left side in the cross section view. The projections of 

the subpatches (grey marked, right side) into their triangle 

planes fulfill PA j (A j) ⊆ � 

i=1,2,3 PA j (Hi) ∪ Tj, j = 1,2,3 

To detect this case, the control point mesh of the patch S 

at a given subdivision level l, l is chosen sufficiently big, is 

projected onto T P. If a planar graph results, Equation (6) is 

fulfilled. The reverse implication is not necessarily true. In 

the case of a non planar graph, a subdivision step will be 

done with S and the subpatches of S will be tested recursively 

as new start patches with new T P’s. This procedure 

terminates, because the sequence of control points of the 

subpatches converges to a smooth surface (creases etc. do 

not appear in the interior of the patch). The detection method 

is more of theoretical interest because cases where Equation 

(6) is not true are rather pathological and do normally not 

occur in practice. 

8.2. Other Subdivision Schemes 

The ShaoLin algorithm can easily be adapted to other kinds 

of subdivision surfaces fulfilling the prerequisites of Section 

8.1. Firstly a suitable plane T P has to be defined. For 

Catmull-Clark surfaces1 this plane may for instance be determined 

by 1/4(l p0 + l p1 + l p2 + l p3) as a point on the 

plane and ∑ 3 i=0 ∆l pi × ∆l p (i+1) mod 4 as the plane’s normal, 

where ∆l pi = l pi − l p (i−1) mod 4 and l pi are the limit points 

of the starting patch. The other planes and the hulls of the 

border curves are computed as described in Section 3. The 

algorithm works in the same manner as described in Section 

7 – with the minor difference for Catmull-Clark surfaces 

that we have to deal with shadowed quadrangles in the same 

way we process shadowed triangles for Loop surfaces. The 

implementation of the ShaoLin algorithm for Catmull-Clark 

surfaces is in progress. 

8.3. Optimisations 

We applied two kinds of optimisations concerning a) the algorithm 

and b) the underlying data structure. During the execution 

of the algorithm, we try to avoid unnecessary border 

curve area tests. If, for example, only one subpatch of 

a patch is crossed by the shadow, it is not necessary to examine 

the border curve areas of the subpatches lying on the 

opposite edge of that patch. Furthermore intersection tests 

for border curve areas at the silhouette are needless, because 

this would not have a visible effect due to the fine tesselation 

there. The mesh implementation caches the limit points. Further, 

the valence of the vertices is stored instead of counted 

each time. 

9. Results 

9.1. Visual Results 

First of all, only those parts of the object are refined, which 

potentially contribute to the light transport. Hence, back 

parts and covered parts without contact to, e.g., shadow rays 

are not built. However, parts of the object critical to obtain 

a high quality image are especially fine subdivided (the silhouette, 

high curvature areas, and parts near the camera). We 

avoid refinements in subpixel domains (resulting from too 

small triangles) as well as non smooth reflections (resulting 

from too large triangles). This way, we use the rendering 

time and memory efficiently. 

To visualise the work of the adaptive refinement, we rendered 

a scene as ‘level map’. There the color of each pixel 

indicates the subdivision level where an intersection occurs. 

This can be observed in Figure 14. 

9.2. Run Time and Memory Consumption 

We chose three scenes for measuring the run time and the 

memory consumption: a simple scene with 268 faces in the 

subdivision surface control mesh (Frogship), a scene with 

5756 faces (Col-Henge), and a complex scene with 10328 

faces (Eggs’n’Bunnies). The computer used for this benchmarks 

was a Pentium IV with an Intel 845 chipset, 1.7 GHz 

and 1 GB RAM. The executable was built by a Microsoft 

Visual C++ 6.0 compiler using Windows 2000. We use the 

ray tracer integrated in the MRT14 rendering package (Modular 

Rendering Tool), an extensible library of rendering algorithms 

and data structures. To assess the results of the 


Subdivision level: 

1 2 3 4 5 6 


7 

(no hit) 

Figure 14: Level map and original image of Frogship: 

adaptively refined subdivision surface by illumination and 

shadow rays. 

ShaoLin algorithm, the scenes were also rendered with two 

other approaches: i) Uniform: If the bounding volume of a 

subdivision object is hit by the ray, the object is tesselated 

once in a given subdivision level. ii) Adaptive: we implemented 

an algorithm following Woodward’s approach (see 

Section 1) extended by the same refinement criteria like the 

ShaoLin algorithm. To adapt and optimise it for subdivision 

surfaces, the control as well as the limit points are cached 

and all intersection tests are done in 2D. For simplicity we 

call it A3 (Another Adaptive Algorithm). Even though the 

adaptive algorithms offer a variety of results (because of the 

three parameters: combined ε, maxSize, and minSize), we 

present only a small range here. The tables (see Tables 1, 2, 

3) can be used to find an adequate subdivision level for the 

uniform approach that provides an image quality comparable 

to that of the adaptive algorithms. 

comb. average Ray intersections in subdivision level 

ε level 0–1 2 3 4 5 6 7 

0.005 4.74 0 0 960 78606 118007 17546 3199 

0.010 4.32 0 0 7260 143327 58230 8026 1413 

0.020 4.10 0 0 24309 149340 37668 3876 443 

0.040 3.98 0 60 38697 140984 31456 1635 170 

0.080 3.94 0 668 41568 139241 29147 707 30 

0.200 3.92 0 1163 42603 138283 28135 305 11 

0.500 3.92 0 1234 42812 138087 27739 218 0 

Table 1: Number of ray intersections per subdivision level 

for Frogship (minSize=2 Pixels, maxSize=12 Pixels). 


ε level 0–1 2 3 4 5 6 7 

0.005 4.26 0 366 43581 406663 125442 32905 3434 

0.010 3.89 0 3749 196371 335172 82529 23545 1366 

0.020 3.57 0 26663 324712 210037 62414 18109 615 

0.040 3.26 0 138635 298292 152784 53122 15835 338 

0.080 3.06 0 232187 221692 129657 48624 15207 248 

0.200 2.98 0 267544 186845 115271 47336 15055 204 

0.500 2.96 0 274966 176705 112317 47033 15039 195 


for Col-Henge (minSize=2 Pixels, maxSize=12 Pixels). 

At a given quality, for simple scenes the number of faces 

generated by the uniform approach is quite similar to the 



ε level 0–1 2 3 4 5 6 7 

0.005 4.48 0 5671 119254 262671 274859 103913 6074 

0.010 4.32 0 27545 117212 299910 245126 84493 2089 

0.020 4.24 0 43791 104466 322688 223068 76369 511 

0.040 4.19 0 48299 121455 308671 216631 73181 196 

0.080 4.16 0 49585 132244 296500 214256 71473 43 

0.200 4.13 0 69865 117587 293285 212855 70949 9 

0.500 4.12 0 74910 112192 292307 212612 70600 4 


for Eggs’n’Bunnies (minSize=2 Pixels, maxSize=20 Pixels). 

ShaoLin uni. uni. uni. uni. uni. 

lev. 3 lev. 4 lev. 5 lev. 6 lev. 7 

F’ship 83M 24M 48M 140M 495M 1635M 

C’H’ 491M 189M 690M — — — 

E’n’B 1499M 1090M — — — — 

Table 4: Memory requirements for ShaoLin (ε = 0.01) and 

the uniform approach. 

number of faces generated by an adaptive approach. Due to 

the higher complexity of an adaptive algorithm, the uniform 

approach is comparable to the adaptive algorithm in this case 

regarding computation time and image quality (see Table 1 

and Fig. 15, left). 

With growing complexity of the scene, the ShaoLin algorithm 

can show its advantages: The number of generated 

faces is significantly smaller than in the uniform approach, 

saving run time and memory. These savings exceed by far 

the extra costs for the adaptive approach ShaoLin (see Fig. 

15 and Table 3). 

High memory requirements of the uniform approach lead 

to a limited subdivision level as shown by Table 4. It was, 

for instance, impossible to render the Eggs’n’Bunnies-scene 

with the uniform approach at subdivision level 4. 

Mainly due to the bigger bounding volumes defined by the 

1-neighbourhood, the A3 algorithm is noticeably slower than 

the ShaoLin algorithm (see Fig. 15). Because of the same 

refinement criteria, A3 yields similar level maps and has the 

same memory consumption. 

10. Conclusion 

With the ShaoLin algorithm we present an adaptive ray tracing 

method for subdivision surfaces. The algorithm is fast, 

yields high quality images and consumes little memory in 

comparison to the uniform approach. Furthermore, it is easy 

to implement, does not need precomputation and requires 

only the basic set of subdivision rules. The algorithm is also 

robust in the sense that it deals with all features of subdivision 

surfaces like creases and corners. Finally, the proposed 

ray-intersection algorithm is applicable, e.g., for simple collision 

detection, for radiosity calculations, and for picking 

objects interactively.

Time [sec] 

800 

700 

600 

500 

400 

300 

200 

100 

Frogship 

ShaoLin 

A3 

Uniform, level 7 








0 

0.005 0.01 0.02 0.04 0.08 0.2 0.5 

Combined Epsilon 


Time [sec] 

2000 

1500 

1000 

500 

Col-Henge 

ShaoLin 

A3 






0 

0.005 0.01 0.02 0.04 0.08 0.2 0.5 


Time [sec] 

2500 

2000 

1500 

1000 

500 

Eggs N’ Bunnies 

ShaoLin 

A3 





0 

0.005 0.01 0.02 0.04 0.08 0.2 0.5 


Figure 15: Computation times for Frogship, Eggs’n’Bunnies and Col-Henge with resolution 800 × 800. 

Figure 16: Visual results of the ShaoLin algorithm (Eggs’n’Bunnies, GSS Europa, Col-Henge). 

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