tessellating algebraic curves and surfaces using a … · 1 introduction with modern graphics...

TESSELLATING ALGEBRAIC CURVES AND SURFACESUSING A-PATCHES

Curtis Luk and Stephen MannUniversity of Waterloo, 200 University Ave W., Waterloo, Ontario, Canada

[email protected], [email protected]

Keywords: Algebraic surfaces, A-patches, Tesselation.

Abstract: This work approaches the problem of triangulating algebraic curves and surfaces with a subdivision-style algo-rithm using A-patches. An algebraic curve or surface is converted from the monomial basis to the Bernstein-Bezier basis over a simplex. If the coefficients are all positive or all negative, then the curve or surface doesnot pass through the domain simplex. If the scalar Bernstein coefficients are of mixed sign and have a layerseparating the positive from the negative, then the patch is in A-patch format and can be efficiently tessellated.Cases of mixed sign without a separating layer are resolved by subdividing the structure into a set of smallerpatches and repeating the algorithm.Using A-patches to generate a tessellation of the surface has the advantage of reducing the amount of subdi-vision required. And because of the A-patch properties, we are guaranteed that features within the designatedregion will not be missed.

1 INTRODUCTION

With modern graphics cards, it is possible to renderimplicit surfaces directly on the GPU without tessel-lating them. However, tessellation is still one viablemethod of rendering implicit surfaces, and fast, ac-curate methods of tessellation remain important. Oneprevalent method of triangulating implicit surfaces in-volves the use of subdivision, which is the processof dividing space into regions that can be recursivelysplit into increasingly smaller regions.

There are several advantages in using A-Patchesto construct a triangular approximation of an alge-braic surface, which are the ease in which A-Patchescan be evaluated for low degree polynomials and theease in which points on the surface can be foundwithin an area that is in A-Patch form. Most im-portantly, it can be shown that a domain that con-forms to specific A-Patch configurations has a smallnumber of single-sheeted orientations that can existwithin the patch. This removes various ambiguitiesthat could pose problems when evaluating complexsurfaces, such as self-intersections and singularities.

Some subdivision algorithms require subdomainsto be highly subdivided to generate a tessellation witha high sampling rate, which is used to generate a highquality surface approximation. The A-Patch method

can finely tessellate surfaces with less use of repeatedrecursive subdivision of the target space. More impor-tantly, our method will not misclassify multisheetedregions as having a single sheet, nor will it miss small,disconnected components within the region of inter-est.

1.1 Background

An implicit surface in 3-space is the set of all pointswhere F(x,y,z) = 0. If F is polynomial, then the sur-face is said to be algebraic. The problem of tessel-lating an implicit surface has been tackled in the past,and such schemes can be classified in one of severalcategories. Subdivision schemes such as MarchingCubes (Lorensen and Cline, 1987) attempt to find apolygonalization of an implicit surface by dividingthe space into a three dimensional grid. The grid cellsare further divided to smaller components. This classof algorithm can make the surface extremely costly toevaluate at areas of high precision.

Ray casting is another method to generate a polyg-onalization of an algebraic surface, which is done byfinding intersections between lines projected from anorigin point and the surface. Although ray casting cantessellate a surface to any arbitrary degree of precisionby casting an increasing number of rays, it must rely

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on alternate methods for detecting anomalies such asself-intersections and singularities. Also, although themethod can sample an arbitrary number of points onthe surface, ray casting cannot infer the topology ofthe surface without using a point cloud surface con-struction algorithm (Wang et al., 2005).

There exists another class of approximating alge-braic surfaces using piecewise parametric surfaces.Bajaj and Xu propose one such technique (Bajaj andXu, 1997), with parametric surface patches beinggrown around pre-computed seed singularities. Thismethod provides the advantage of generating a para-metric representation of the surface (as opposed toa piecewise linear approximation) of the surface, al-though the process involved requires several majorsteps, including computation of the singularities, di-rect triangulation, and fitting parametric patches overthe triangulation. On the other hand, Jepp, vanOverveld, Wyvill, and Wyvill (Wyvill et al., 2000)propose a method combining marching cubes withsubdivision surfaces to approximate an implicit sur-face. This method is capable of generating approxi-mations at real-time speeds, but the accuracy of thesurface in relation to the defined algebraic surface issacrificed as a result.

1.2 Previous Work

The concept of the trivariate Bernstein-Bezier basisrepresentation within a tetrahedral volume is not new.Sederberg (Sederberg, 1985) defines such a structure,which he calls an algebraic surface patch, as a toolfor modeling free form algebraic surfaces due to itsability to define a wide variety of surfaces with alow degree compared to parametric surfaces and sincethey inherently define half-spaces, which is useful formodeling solid geometry. Bajaj (Bajaj and Ihm, 1992)proposed using A-patches for fitting surface data andlater (Bajaj et al., 1995) proposed a method of build-ing models using piecewise A-patches. Specifically,a set of A-patches is used to construct models thatare C1 continuous and match the topology of the orig-inal model specification. A-patches were also pro-posed for constructing algebraic surfaces (Bajaj andXu, 1997), although unlike the method presented hereit uses the patches as spline surfaces that approximatethe algebraic surface as opposed to using A-patchproperties to compute points on the surface.

The use of subdivision algorithms for tessellatingalgebraic surfaces is not a recent phenomenon, withBloomenthal (Bloomenthal, 1988) proposing a tes-sellation method that relies on repeated subdivisionof a domain space and surface-line intersection eval-uation. Further, more recent work on subdivision-

style polygonalization algorithms have been pursuedby Belyaev and Ohtake (Ohtake and Belyaev, 2002)on implicit surfaces with sharp features. Unlike pre-vious subdivision methods, our A-patch scheme relieson the tetrahedron as opposed to the cube as a unitspace. This is due to the geometric requirements dic-tated by the structure of the A-patch, although subdi-viding a tetrahedron is a more difficult task than sub-dividing a cube.

This paper will investigate both the theoretical andimplementation aspects of the basic A-patch surfaceapproximation method. We begin with a review ofA-patches, after which we present an algorithm forfinding a piecewise linear approximation to a 2D al-gebraic curve. We then generalize the algorithm totessellate algebraic surfaces in 3D.

2 A-patch BASICS

An algebraic surface patch (also known as an A-patch) is essentially a piece of an algebraic surfacethat is constrained within an arbitrary tetrahedron.The contour of the surface contained within the tetra-hedron is determined by the weight of its controlpoints. In A-patch form, the algebraic surface is rep-resented in the Bernstein-Bezier basis. Normally, al-gebraics are represented as a vector of scalars that cor-responds to the monomial basis

Mn(P) = xaybzc,

where 0 ≤ a,b,c and a + b + c = n. The Bernstein-Bezier basis in three-space is

Bn~i (P) =

(n~i

)pi0

0 pi11 pi2

2 pi33 ,

where ~i = (i0, i1, i2, i3) with 0 ≤ i0, i1, i2, i3 and i0 +i1 + i2 + i3 = n and where p0, p1, p2, p3 are theBarycentric coordinates of P relative to a domaintetrahedron T = 4ABCD. The A-patch uses theBernstein-Bezier basis to weigh scalar values in thefollowing relationship:

F(P) = ∑~i

CiBn~i (P).

The scalar coefficients Ci (also called control points)have no location in the space we are interested in, al-though one can imagine them to be distributed evenlyacross the triangular domain as Bernstein/Bezier con-trol points of the patch as seen in Figure 1; this visual-ization will be particularly helpful when talking aboutthe separating layer of control points in the later sec-tions of this paper. For a Bernstein/Bezier represen-tation to be an A-patch, its coefficients must satisfy

TESSELLATING ALGEBRAIC CURVES AND SURFACES USING A-PATCHES

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C0020C0110C0200

C1010

C0011

C011C020

C110

C002

C200

C101 C1100

C1001

C2000

C0002

Figure 1: Visualizing the Bezier coefficients in the domain.

certain conditions on their signs, as we will discuss inSections 4 and 5.

The two dimensional representation of a polyno-mial curve in monomial form is similar to its threedimensional surface counterpart. The only differenceis that there is one less variable. This means that themonomial basis becomes

Mn(P) = xayb,

where 0≤ a,b and a+b = n. The bivariate Bernstein-Bezier basis is

Bn~i (P) =

(n~i

)pi0

0 pi11 pi2

2 ,

where~i = (i0, i1, i2) with 0≤ i0, i1, i2 and i0 + i1 + i2 =n and where p0, p1, and p2 are the Barycentric coor-dinates of P relative to a domain triangle T =4ABC.

Standard change of basis algorithms can be usedto convert between the monomial and Bernstein repre-sentations. Evaluating the algebraic function in Bern-stein form at a point in the region can be accom-plished by performing a de Casteljau evaluation of theA-patch. de Casteljau’s algorithm also subdivides acurve into two sets of control points, each of whichis the representation for the curve over a subdomaindetermined by the evaluation point. de Casteljau’s al-gorithm can also subdivide higher order Bezier func-tions, although these methods become more complexas the dimension increases. For details on Bernstein-Bezier representations, change of bases, and subdivi-sion algorithms, see (Farin, 2002).

2.1 Conditional A-patches andApplication

A-patches have six characteristics that are definedby Sederberg in his paper regarding algebraic sur-face patches (Sederberg, 1985). These come intoplay when we devise methods to generate tessella-tions from A-patches. We list the two properties thatare most important for our purposes:

Point Interpolation Property. F(P) at any of thetetrahedral vertices corresponds to the weight of thecontrol point at that vertex.

Line Interpolation Property. If the weights of allthe control points along an edge are zero, then theedge interpolates the surface F(P) = 0.

More importantly, Sederberg (Sederberg, 1985)(Sederberg and Anderson, 1985) and Guo (Guo,1995) showed that if monotonicity conditions holdon edge lines of a tetrahedron then all parallel lineswill intersect the algebraic surface at most once. Ba-jaj (Bajaj et al., 1995) elaborates on this by formaliz-ing the concept of the three-sided and four-sided al-gebraic patch, both of which are shown to be single-sheeted and non-singular. In addition, the algebraicpatches Bajaj propose do not require the monotonicconditions outlined by Sederberg, and instead imposerestrictions on the signs of the coefficients.

We restate Bajaj’s definition of three and four-sided A-patches:

Definition: Three-sided Patch. Let the surfacepatch SF be smooth on the boundary of the tetra-hedron T = 4e0e1e2e3. If any open line segment(e j,α∗) with α∗ ∈ S j = {(α1,α2,α3,α4)T : α j =0,αi > 0,∑i6= j αi = 1} intersects SF at most once(counting multiplicities), then we call SF a three-sidedj-patch.

Definition: Four-sided Patch. Let the surfacepatch SF be smooth on the boundary of the tetrahe-dron T =4e0e1e2e3). Let (i, j,k, l) be a permutationof (1,2,3,4). If any open line segment (α∗,β∗) withα∗ ∈ (eie j) and β∗ ∈ (ekel) intersects SF at most once(counting multiplicities), then we call SF a four-sidedi j-kl-patch.

Using the above information we can generateapproximations of surfaces and curves with threeand four-sided A-patches and their two dimensionalequivalent. To find this using just the Bernstein co-efficients and the domain tetrahedron, we must find aset of edge points whose weights have opposite signs.From there we can find the point between the pairswhere F(P) = 0. Sections 4 and 5 will elaborate onthe basic outline provided here.

3 THEOREMS

From the basic properties of A-patches we can de-rive a number of theorems that will be relevant to theA-patch tessellation algorithm. Although Bajaj statesconditions for a single-sheeted surface, we must alsodetermine regions that do not contain the surface. Thefollowing theorems justify the method to find theseregions by finite subdivision of the region and inves-tigation of the coefficients of the empty regions.

Theorem 1: Given a d dimensional simplex T =4V0V1 . . .Vd and a dimension d, degree n polynomial

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F with Bernstein coefficients C~i over T , if C~i > 0 forall~i then F(p) > 0 for all p ∈ T .

Proof: This follows immediately from the Bern-stein basis functions forming a convex combinationover T .

Theorem 2: Given a dimension d, degree n alge-braic function F whose value is strictly greater thanε > 0 on the interior of a simplex T and where theBernstein representation of F over T has coefficientsof mixed sign. Then over any aligned subsimplex Tsof T with edge lengths no greater than 1

2d−N(2d+2)n/εe ofthe size of those of T , the Bernstein coefficients of Fover Ts are all non-negative, where N < 0 is the mostnegative Bernstein coefficient of F represented overT .

Proof Sketch: The proof proceeds by consideringan arbitrary, aligned subsimplex whose edges are halfthe length of the initial simplex. We can show that themost negative coefficient of the subsimplex will be atleast a fixed amount larger than the most negative co-efficient of the initial simplex, where the fixed amountis dependent on ε, d, and n (i.e., it is independent ofthe Bernstein coefficients). From here, we just findthe number of “halvings” needed to ensure that all theBernstein coefficients are positive. See (Luk, 2008)for details.

Lemma 3: For any set of simplices covering Twhere each simplex can be embedded in an alignedsubsimplex of T of size 1

2d−N(2d+2)n/εe then the Bern-stein coefficients of F over each subsimplex will benon-negative.

Proof: The lemma follows immediately from theTheorem 2.

Comments: The theorems and lemmas also holdfor F(p) < 0, F(p) < ε < 0, etc. Also note thatLemma 3 is a very loose bound on the number of sub-divisions needed to have all the coefficients be of onesign when working in a region where F(p) > 0; ingeneral, we expect (and observed) much faster con-vergence than this. While Theorem 1 is a well knownsimple observation, Theorem 2 and Lemma 3 are newand are a specialized generalization of the propertythat the Bezier curve control polygon converges to thecurve under repeated subdivision.

4 APPROXIMATING ALGEBRAICCURVES

We begin by presenting a method for finding a piece-wise linear approximation to a 2D algebraic curve.We assume that the user has selected a region of in-terest, and that this region has been triangulated. We

Figure 2: A possible separating layer in an ideal two-pointed A-patch. The white have negative sign; the blackhave positive sign; the magenta is the separating layer, andthe coefficients may have either sign.

then find the Bernstein representation for the alge-braic function over each triangle.

After converting to Bernstein-Bezier format, it isnecessary to determine whether each triangle containsthe curve. This can be determined by examining thesign of the Bernstein coefficients. If all the coeffi-cients of a triangle are non-zero and of one sign, thenthe curve does not pass through the triangle (Theorem1), and the triangle need no longer be considered.

Triangles with Bernstein-Bezier coefficients ofmixed sign can be categorized into two differentclasses. The first is what we will refer to as a two-pointed A-patch. For a Bernstein representation tobe a two-pointed A-patch with domain triangle T =4ABC, there must exist a vertex control point (as-sume it is A for discussion) such that F(A) does nothave the same sign as F(B) and F(C). In addition,there must exist a layer of mixed-sign control pointsthat separates a set of positive control points from aset of negative control points, illustrated in Figure 2.See (Bajaj and Xu, 1997) for a more formal definitionof an A-patch.

If the Bernstein representation is a two-pointedA-patch, then the curve can be approximated withinthe domain of the patch as described in the next sec-tion. Otherwise, we will perform a 4-to-1 subdivisionof the triangle. The control points of the four newpatches are computed and the search for a separatinglayer is performed on each subtriangle.

In our implementation, we used a 4-to-1 subdi-vision (Bohm, 1983); however, a 2-to-1 subdivisionwould produced similar results and would be easierto implement (Peters, 1994).

4.1 Approximating the Curve

Let 4ABC be the domain for a two-pointed A-patchF with a separating layer. In this situation, two ofthe corner Bezier coefficients will have one sign (e.g.,


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Figure 3: Curve properties that are not present in an A-patchwith a separating layer.

F(A) > 0 and F(B) > 0), and the third coefficient(e.g., F(C) < 0) will have the opposite sign. Be-cause of the separating property, the algebraic func-tion’s value at any point on the edge AB will have thesame sign as F(A) and F(B). We will sample AB fora sequence of points, P1,P2, . . . ,Ps, and for each pointPi, we consider the line between it and C.

Again, because of the separating layer, there isone zero of F along the line segment PiC. A nu-merical search can quickly find this zero. Our piece-wise linear approximation to the curve in this regionis made by connecting this sequence points at whichF is zero. The single-sheeted properties of the patchwill guarantee that by interpolating from one vertex tothe other we do not need to consider certain features,such as those seen in Figure 3.

Figure 4 shows some examples of implicit curvesevaluated in this manner. Lemma 3 ensures that wewill eventually stop subdividing in regions that do notcontain the curve. However, if there is a singularityin the curve, an A-patch over any triangular regioncontaining this singularity will not have a separatinglayer. As the algorithm refines in this region, the sin-gularity will become more and more localized, but ifthe precise location of the singularity is desired, an-other method must be used to find it.

Figure 5 shows the subdivision of space of a curvethat has a singularity. Notice the increasing depth ofsubdivision near the singularity. Also note that sub-division stops fairly quickly in regions not containingthe curve, and that away from the singularity, the tri-angles were in A-patch format at a fairly coarse level,even if the curve just clipped a triangle or if the curvepassed through the triangle at unusual angles.

5 TESSELLATING ALGEBRAICSURFACES

We now consider algebraic surfaces. To improvecomputational efficiency of the surface constructionalgorithm it is essential to have a good subdivisionscheme. Unfortunately, unlike triangles the regulartetrahedron cannot be constructed from a small num-ber of regular tetrahedra, which makes both subdivi-

sion and tetrahedron layout non-trivial problems. Wechose to abandon the use of regular tetrahedra andworked with axis-aligned grids of cubes, with eachcube containing a set of tetrahedra instead. The useof axis-aligned cubes has advantages in making thestructure easier to understand and implement, as wellas accommodating a reasonable subdivision scheme.

We considered two schemes for A-patch layoutand subdivision based on the axis-aligned cube. Thefirst method is defined by dividing the cube into fiveseparate tetrahedron, shown in Figure 7, left. This ar-rangement results in a cube that is created with a mini-mum number of A-patches. Unfortunately, in this lay-out adjacent cubes must be placed in a specific waysuch that the faces of each A-patch are shared withits neighbour otherwise tearing artifacts will appearwhen the final result is rendered.

The second method divides the cube into twelvetetrahedron, outlined in Figure 7, right. This configu-ration has a couple of advantages, the first being thatthe faces of every patch along the face of a cube matchperfectly with the faces of an adjacent cube’s patches,provided that all the cubes follow the same A-patchlayout. A second advantage is that the tetrahedronwithin the cube are of the same size and shape. Fi-nally unlike the previous method there is no require-ment for the cubes to be arranged in an octet to guar-antee that the patch faces match up, which simplifiesthe implementation. Unfortunately this method alsoyields over twice the number of patches than the 5-patch cube over the same domain, which slows downthe algorithm by a corresponding amount. Due to thedecrease in the number of A-patch computations wedecided to use the first method in the final implemen-tation.

5.1 Criteria for Subdivision andEvaluation

As with the evaluation of two dimensional A-patcheswe must determine whether or not the function lieswithin the domain. Again, by analyzing the patternof the control points in the patch we can infer variouscharacteristics about it.

A-patches with mixed-sign control points fall un-der one of three categories, two of which are suitablefor evaluation. The two ideal patches are outlinedby Bajaj (Bajaj and Xu, 1997), which he refers toas three-sided and four-sided A-patches. Both threeand four-sided A-patches contain a separating layerof control points similar to that found in the two-dimensional case, although in this instance the con-trol points extend in an extra dimension. Three-sidedA-patches contain one vertex control point that has


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Figure 4: Curve approximations generated by the implementation. From upper left clockwise: x4 +3x2y2 +2y4−2x2 +3y2 +0.2 = 0, x4 + 4x2y2 + 4x2y−3x2 + 3y2−4y = 0, x4−2x2 + 2y2 = 0, and 2x3−2x4− y2 = 0. The right two examples showfunctions that have areas that cannot be approximated exactly at the point of the singularity.

Figure 5: Subdivision near a singularity.

V0

V1 V2

v1 v2 = 12(d+1)

V0 + ...

v0 = d+22d+2

V0 + ...

C = 1d+1

V0 + ... + 1d+1

Vd

Figure 6: Variations in curve detail for A-patches containingthe input function.

a different sign than the other three while four-sidedpatches contain two pairs of vertex control points thathave different signs from each other. Figure 8 showsan example of separating layers for three and four-sided A-patches. Note that three-sided A-patchescontain triangular-shaped surfaces while four-sidedA-patches contain quadrilateral-shaped surfaces.

If all the tetrahedron within a cube have entirelypositive or negative control points and/or passes thethree or four-sided A-patch criteria then it is possibleto approximate the surface that passes through the cu-bic area using the method outlined in the next section.Otherwise we subdivide the cube into eight separatecubes. New Bernstein representations are computedfor the tetrahedrons in the subdivided cubes, and the

Figure 7: A diagram of an axis-aligned cube containing fiveand twelve A-patches.

Figure 8: The separating layer of control points in a threeand four-sided A-patch. The white have negative sign; theblack have positive sign; the magenta is the separating layer,and the coefficients may have either sign.

process is repeated until all the patches are in the de-sired configuration. This may require multiple levelsof subdivision.

5.2 Tessellating the Surface

Once we have determined which tetrahedron are A-patches, we can generate a piecewise linear approxi-mation of the surface. We will use the same methodto determine a zero-set of the implicit function withinthe domain of the patch as before, although themethod for evaluating the surface differs between athree and four-sided patch. The main difference be-tween the two situations lies in the number of pointsto evaluate and the method in which pivoting pointsare selected for finding a particular member of thezero-set.

In the case of a three-sided patch, we know thatF has one sign at three of the corners of the domain


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A

C C

A

B

B

D

D

Figure 9: Tessellation of an A-patch: White points forma triangular grid on a face/edge are of one sign; the blackpoint(s) has/have opposite sign. Red points are point on thesurface. Each red point is calculated by connecting a whitepoint to the black point and numerically searching for thezero. The green line illustrates this for one point, with thezero on the line being drawn in green rather than red.

tetrahedron T = 4ABCD, and the opposite sign atthe fourth corner. Assume that F has the same signat A,B,C. Since we have a separating arrangement ofcontrol points, F has the same sign at all point on theinterior of 4ABC. We setup a triangular grid acrossthis face, and for each point P on the grid, we con-struct a line to D. The separating arrangement of co-efficients ensure that F(P) and F(D) have oppositesigns, and further that there is exactly one zero on theline segment PD. Again, we find this zero by a nu-merical search method.

Having found a zero of F corresponding to eachpoint on the triangular grid on triangle 4ABC, weconnect these zero points in the triangular arrange-ment suggested by the triangular grid. This forms apiecewise linear approximation to F over T .

In the four-sided patch case, things differ in thattwo corners of T = 4ABCD have one sign (e.g.,F(A) > 0, F(B) > 0), while the other two cornershave the opposite sign (e.g., F(C) < 0, F(d) < 0).In this case, we uniformly sample both line segmentsAB and CD, and form all connections between sam-ple points on one edge to the sample points on theother edge. Along each line segment, there will beexactly one zero, which can be found with numeri-cal search. The result will be a rectilinear grid thatapproximates F over T . The tessellation algorithmfor both three and four-sided patches is illustrated inFigure 9. Note that unlike the other figures, in this fig-ure the points are not representative of A-patch con-trol points/coefficients, but instead are actual points inspace.

Figure 10 show several surfaces that were approx-imated using the implementation.

Note that as described, it is possible that the subdi-vision of adjacent regions will be at different depths,which can lead to a non-watertight boundary. This in

Figure 10: Planar and quadratic surface tessellations gen-erated by the implementation. Clockwise from upper left:x2 +y2 +z2−0.8 = 0, x+y+z+0.2 = 0, x2 +y2−z−0.8 =0, and x2−y2−z−0.8 = 0. Magenta shows the tessellation.

Figure 11: A comparison between a standard and stitchedtessellation.

turn can lead to pixel drop-out. We implemented asimple stitching scheme to address this problem (Fig-ure 11) (Luk, 2008).

6 CONCLUSIONS

The A-patch method for approximating implicitcurves and surfaces is a sound one, both in theory andin practice. The theorems outlined in the Section 3demonstrate the convergence of the algorithm in do-mains that do not contain the surface. But althoughthe theory guarantees that we can eventually decidethe surface does not exist in regions not containingthe surface, so far we have no guarantees that the al-gorithm will find A-patches for portions of the spacethat do not contain singularities. However, our testcases show the effectiveness of the method in find-ing a surface approximation in domains that containit, strongly suggesting that the method will convergeto A-patches for portions of the surface not near sin-


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Figure 12: Approximating surface singularities to an arbi-trary precision using A-patches. The orange box identitiesan area with a high degree of subdivision.

gularities.The big advantage of our algorithm is that for al-

gebraics, it avoids certain problems in tessellating im-plicit surfaces. In particular, over the region of in-terest, our algorithm is guaranteed not to misclassifymultiple sheets as a single sheet, nor will it miss smalldisconnected features. Simplices containing such fea-tures will not be in A-patch format, and the algorithmwill subdivide the region until the multiple sheets areseparated, the small disconnect components found, orthe maximal subdivision depth is reached (in whichcase these regions are flagged as special).

Like a vast majority of the algebraic approxima-tion methods ours cannot properly identify the singu-larities of the algebraic surface unless it is coincidentwith a vertex control point of an A-patch. However,even in the worst case the A-patch method is capableof approximating a singularity point to an arbitrarydistance from the point of singularity via subdivision,which gives us an unambiguous surface contour towork on to complete the surface approximation ac-curately. Figure 12 shows how repeated subdivisionof an algebraic surface with a singularity can gener-ate an infinitely close approximation of the singularitywithout ever finding it.

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Bajaj, C. L. and Ihm, I. (1992). Algebraic surface de-sign with hermite interpolation. ACM Transactionson Graphics, Volume 11, Issue 1, pages 61–91.

Bajaj, C. L. and Xu, G. (1997). Spline approximations ofreal algebraic surfaces. Journal of Symbolic Compu-tation, Volume 23, Issue 2-3, pages 315–333.

Bloomenthal, J. (1988). Polygonization of implicit surfaces.Computer Aided Geometric Design, 5(4):341–355.

Bohm, W. (1983). Subdividing multivariate splines.Computer-Aided Design, 15(6):345–352.

Farin, G. (2002). Curves and Surfaces for CAGD. Morgan-Kauffmann.

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Lorensen, W. E. and Cline, H. E. (1987). Marching cubes:A high resolution 3d surface construction algorithm.Computer Graphics, Volume 21, Number 4, July 1987,pages 163–169.

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Peters, J. (1994). Evaluation and approximate evaluation ofmultivariate bernstein form on a regularly partitionedsimplex. ACM Transactions on Mathematical Soft-ware, 20(4):460–480.

Sederberg, T. W. (1985). Piecewise algebraic surfacepatches. Computer Aided Geometric Design, Volume2, Issues 1-3, pages 53–59.

Sederberg, T. W. and Anderson, D. (1985). Steiner surfacepatches. Computer Graphics and Applications, IEEE,Volume 5, Issue 5, pages 23–36.

Wang, J., Oliveira, M. M., and Kaufman, A. E. (2005).Reconstructing manifold and non-manifold surfacesfrom point clouds. 16th IEEE Visualization 2005 (VIS2005), page 53.

Wyvill, B., Jepp, P., van Overveld, K., and Wyvill, G.(2000). Subdivision surfaces for fast approximate im-plicit polygonization.


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