Quality Measure of Quadrilateral Mesh

Quality Measure of Quadrilateral Meshes

Phillip LuAdvised by Dianna Xu

Haverford College, Department of Computer Science

Abstract

A method of analyzing the quality of quadrilateral meshing algo-rithm is presented. By running each quadrilateral of the mesh throughJohn Robinsons Continuum-Region-Element (CRE) method, the re-sultant numeric values correlate with the shape parameters of eachquadrilateral. We find that Robinsons Jacobian determinant, evenafter accounting for scale, is only as good as a mesh quality measureas aspect ratio as defined by the CRE method.

Keywords. quadrilateral mesh, mesh quality, mesh analysis

1 Introduction

Polygonal meshes are discrete, approximate representations of 2D or 3Dshapes constructed from simple polygons: a collection of points, edges, andpolygonal faces that represents a 2D or 3D object. Though mainly used incomputer graphics, these meshes have a variety of applications, from high-order surface modelling to data compression [3].

Since most 3D geometrical objects have two dominant directions, quadri-lateral meshes are particularly well suited for representing such objects, giventhat individual quadrilaterals have two pairs of edges that run (ideally) or-thogonal to each other [3]. Applications for which preserving the shape andform of the 3D object is of importance would therefore prefer quadrilateralmeshes over triangular meshes, since transformations of a triangular meshwould not guarantee that the overall shape is preserved. However, due to thedifficulties inherent in creating quadrilateral meshes, the majority of research

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effort has gone into triangle mesh construction. As such, though there areseveral methods of measuring the quality of a triangle mesh [2, 5], there isno widely accepted similar metric to measure the quality of a quadrilateralmesh.

Beyond simple metrics like the amount of polygons in a mesh, it is difficultto find a measurement of quality that represents the entire mesh withoutresorting to measurement of the individual polygonal elements. As such,almost all mesh quality measures examine the shape quality of each polygon.

As Eppstein states in his presentation to the Meshing Roundtable in 2001,there are numerous measures of a triangle, such as ratio of circumcricle toincircle radii, ratio of diameter to height, or perimeter squared to area [5].However, he mentions that the quality guarantees for quad/hex meshes [are]much less developed. It is easy to see that finding a metric for triangles iseasier than finding one for quadrilaterals: while triangles can always be in-scribed within a circle (the smaller the circle is, the better a triangle is),not all quadrilaterals can be inscribed within a circle (i.e. non-cyclic quadri-laterals). While diameter to height is a consistent measure of a trianglesquality, it is not for that of a quadrilaterals, as using that measure wouldreturn the same value for both squares and a rhombus (same diameter, sameheight). However, we would not want the measure for a square and rhom-bus to be the same, since the preference for quadrilateral meshes over othertypes of meshes is partially due to the quadrilateral having two dominantlocal directions, typically associated with principal curvature directions. Wewould thus prefer to have a mesh composed of mostly squares, where the twodominant local directions are, for the most part, orthogonal to each other,and thus accurately representing curvature directions, as opposed to a meshcomposed of mostly rhombuses, especially extremely tapered ones.

Measuring the quality of a quadrilateral is significantly more difficult thanthat of a triangle, since there are more geometric features in a quadrilateralthat need to be properly reflected in the metric itself (i.e. skew, aspect ratio)than in a triangle. While John Robinson was primarily concerned aboutfinite element stress testing in his Continuum-Region-Element (CRE) methodpaper, he shows that the determinant of the quadrilaterals Jacobian matrixis a numerical variable that reflects all significant features of a quadrilateral[8]. This paper will attempt to evaluate the Jacobian determinants viabilityas a measurement of quadrilateral mesh.

Throughout this paper, I will use the shorter term quad to refer toquadrilateral.

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2 Related Works

As mentioned before, this paper will be an evaluation of the validity of Robin-sons CRE method [8] in mesh quality measurement. Thus, this paper willdraw heavily from the terminology and methods of Robinsons paper andbook (i.e. aspect ratio, skew, and taper are all defined within his works,and will be integrated within my analysis). Robinson was concerned abouthaving a general method of finite element stress testing: a method of analyz-ing the forces on and within physical structures (planes, bridges, dams, etc.)by breaking them down into many small discrete structural pieces (finiteelement) [9]. Since these individual finite elements are made up of sim-ple polygons (polygons with non-intersecting perimeters), there is significantwork done on analyzing the effects of pressure, torque, and bending on mate-rials modeled by simple polygons. As such, the CRE method, while originallydeveloped as an element testing procedure, can be used to measure a quadsaspect ratio, skew, and taper, amongst other measurements of shape.

The first quad mesh algorithm I will be testing my quality measurementon will be Atalay, Ramaswami, and Xus quad-tree algorithm [1] for gener-ating a quad mesh. I will be using the mesh quality measures section oftheir paper and comparing my results with theirs. The algorithm itself willbe discussed in section 2.1

While the Bommes et al. survey of quad-mesh algorithms [3] containsmultiple quad-mesh algorithms, few would mesh over a 2D point cloud, fewerstill have publicly available executables. Thus, I will be using JonathanRichard Shewchuks publicly available triangle mesh generator [10] to createan initial triangle mesh, from which I will convert to a quadrilateral mesh.A discussion of this process will be in section 2.2.

2.1 Atalay, Ramaswami, and Xus Quadtree Mesh Al-gorithm

Atalay, Ramaswami, and Xus algorithm for generating quad meshes relies onusing quadtrees effectively [1]. A quadtree, effectively a tree data structuresuch that each node has either no children (leaf), or exactly 4 children. Atalayet al.s algorithm first takes the point set, and creates a quadtree, each nodesplitting whenever it has more than two elements in it. This effectivelymeans that each node will contain at most one point. Further, Atalay et al.salgorithm ensures that no two neighbouring structure are two or more levels

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Figure 1: A visualization of a quadtree with associated point set. [6]

apart, and uses this property to apply quad templates for the deepest levelof the subdivision. After that, the algorithm can apply general templatesfor stitching together nodes of arbitrary levels. The existence of generaltemplates allows geometric analysis of the worst-case configuration, whichyields the minimum angle of 18.43 and a maximum angle of 171.86.

Atalay et al.s algorithm takes 2D point set information and outputs aGeomview Object File Format (.OFF) file, that consists of the meshs verticesand faces.

2.2 Shewchuks Triangle

As previously mentioned, I will be running the analysis program over theoutputs of Shewchuks Triangle program. Triangle can take in point sets,

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and output a Delaunay triangulation (a triangulation such that no point isinside the circumcircle of any triangle in the triangulation). A Delaunay tri-angulation will maximize the minimum angle of the resultant triangle mesh,given that no additional points can be added [4]. Every point set is alsoguaranteed to have a Delaunay triangulation.

An advantage to using Shewchuks Triangle program is that I am able tospecify certain quality guarantees. I will thus be using Triangle to output twosets of data: one with no quality guarantee, and one with a quality guaranteethat the minimum angle will be at least 20 degrees.

This by itself should not be of any concern. After all, my goal here is notto analyze the quad meshing algorithms in-depth, but instead to use Robin-sons metrics on the meshes and comparing the result. However, ShewchuksTriangle only outputs triangle meshes, and so we need a way to convert thetriangle mesh into a quad mesh. While there exists several methods of tri-to-quad mesh algorithms, few of them offer an implementation of them online.Instead, we will do a crude tri-to-quad algorithm that is simple to implement,though offers nothing in terms of quality guarantees.

For each triangle in the triangle mesh, we add four additional points atthe centroid, and the three midpoints at their respective edges. We thenadd three edges, from centroid to the three midpoints. We now have threequadrilaterals in place of the triangle. We do this conversion for all thetriangles in the mesh, and we will end up with a quad mesh. See Figure 2for example.

While this method is crude and offers no specific guarantee, this algorithmwill suffice for providing a mesh generator to compare to. Generally, weexpect a tri-to-quad mesh algorithm to have worse metrics than direct quadmesh generation algorithms, especially tri-to-quad mesh algorithms that donot apply mesh smoothing or mesh simplification afterwards. This gives us avery general hypothesis to test: our metric for mesh quality should be betteron Atalay et. als algorithm than on Shewchuks algorithm after tri-to-quadmesh conversion.

3 Method

The CRE method was originally developed as a procedure for understandinghow forces work on and within 3-dimensional, complex objects, representedby and broken down into numerous discrete, elementary pieces. While the

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Figure 2: Making a quad mesh from a triangle mesh.

(a) Shewchuks Triangles Delaunay triangulation of 50 random points.

(b) Quad mesh by running the tri-to-quad mesh algorithm over the triangulation.

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method was designed to work on any shape of element with any number ofnodes, the elementary shape he derives all the equations in the paper fromis the quad [8]. Since bad shapes can affect the results of finite elementanalysis, Robinson developed the CRE method to find the shape param-eters that differentiate one shape to another, so as to be able to identifybad shapes prior to conducting finite element analysis via an automatedanalysis system.

In his paper, Robinson shows that any quad shape can be constructedgiven four parameters: aspect ratio (AR), skew (), and the respective taperin two principle directions (Tx, Ty) [8]. He further shows that given the coor-dinates of the four points on the quad, one can plug in equations to find allfour shape parameters, the proof of which can be found in Robinsons paper.I will explain these equations in detail in the sections 3.1 and 3.2 below.

The method and structure for comparing the different algorithms will bediscussed in section 3.3.

3.1 Local Cartesian Coordinates

Each of our quads exist as four points in some global Cartesian coordinatesystem. To ease calculations and to make sure the quad is in the rightorientation (see Figure 3), we need to convert to a local Cartesian coordinatesystem [9]. To do this, the four vectors from the centroid going through

the four bisectors of the quad edges is used (i.e.V 05 being the vector from

the centroid to the bisector of the segment from vertex 1 to vertex 2. See

Figure 4). The correspondingV 06 is used as the vector to define the local x

vector (x ). To find the corresponding y vector, we cross V 06 with V 07 toget the normal vector to the plane of the quad (corresponding to the localz vector, z ). With x and z defined, we can then find the y by crossingz with x . x and y will be referred to as x and y in the rest of the paper,x-axis being the axis that is in the direction of x .

With the local coordinate system defined, we can now convert the fourpoints into the new coordinate system by projecting the old Cartesian coor-dinates onto the new vectors, like so:

xi =V 0i xyi =V 0i y

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where i from 1 to 4 indicates the four respective vertices of the quad, andV 0i is the vector from the centroid to that particular vertex. Thus, we nowhave all four vertices in the local coordinate system.

The interpolation function from the old system to the new system canbe also expressed in the following equation, where and are curvilinearcoordinates with limits 1;

x = e1 + e2 + e3 + e4y = f1 + f2 + f3 + f4

The curvilinear coordinates (, ) of the four points are (1,1), (1,1),(1,1), (1,1) for points 1, 2, 3, and 4 respectively (Figure 3). The e and fcoefficients are given in the following equations:

Figure 3: Points oriented counter-clockwise starting from bottom left.Point 0 indicates centroid. Bisectors shown.

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Figure 4: Midpoints defined in counter-clockwise fashion

e1 = 14(x1 + x2 + x3 + x4) f1 = 1

4(y1 + y2 + y3 + y4)

e2 = 14(x1 + x2 + x3 x4) f2 = 1

4(y1 + y2 + y3 y4)

e3 = 14(x1 x2 + x3 + x4) f3 = 1

4(y1 y2 + y3 + y4)

e4 = 14(x1 x2 + x3 x4) f4 = 1

4(y1 y2 + y3 y4)

We can now produce the shape parameters by using the coefficients pro-duced above.

3.2 Shape Parameters

The four coefficients of e and f each are closely related to the features of thequad. While I will be explaining how each coefficients affect the features,Figure 5 provides a quick glance at what the coefficients do.

e1 and f1 may be the most straight-forward coefficients to explain. Sincee1 is the sum of the four x coordinates divided by four, this is the averagevalue of x. Likewise, f1 represents the average value of y. The coordinates(e1, f1) thus points to the centroid of the quad.

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e2 and f3 both represent a similar concept: the half-length of the edges ofthe original rectangle. We can see this by construing e2 as actually being14((x2 x1) + (x3 x4)) and f3 as being 14((y3 y2) + (y4 y1)). Keeping inmind the orientation of the points (Figure 3), (x2x1) and (x3x4) representthe edges of the original rectangle parallel to the x-axis, while (y3 y2) and

Figure 5: Physical meaning of the e and f coefficients [8]

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(y4y1) represent the edges of the original rectangle parallel to the y-axis. Byhalving the respective sums, we find the average length of the line segmentsparallel to their respective axis. By halving the result again, we find thehalf-length of the line segments.

e3 and f2 both represent a similar concept: the length between the bi-sector of a segment on the original rectangle to its respective point on askewed parallelogram (see Figure 5c). Again, we can see that e3 is actually12(12(x3 x2) + 12(x4 x1)). Since (x3 x2) and (x4 x1) represent twice thedistance between the vertex to the respective point on the original rect-angle in the x-axis, 12(x3 x2) and 12(x4 x1) represent just the distance.12(12(x3 x2) + 12(x4 x1)) thus represents the average distance of the two.The logic follows for f2 on the y-axis.

e4 and f4 both represent the distance from a point on the tapered quadri-lateral (resembling a trapezoid) to its respective point on the original rect-angle. The logic is similar to that of the other coefficients. Taper causespoints 1 and 3 to move in one direction, and points 2 and 4 to move in theother (as seen when comparing the movement of the vertices on a rectangleas it gradually becomes a trapezoid). Thus, to measure the average distanceof movement on the x-axis, we add x1 and x3 while subtracting x2 and x4 tofind the total distance of taper. Dividing by 4 yields the average distanceof taper.

The shape parameters are given in Robinsons paper as the following [8]:

aspect ratio = e2f3

orf3e2

(largest)

skew = e3f3

taper in the x-direction(Tx) = f4f3

taper in the y-direction(Ty) = e4e2

We can easily see how e2 and f3 are related to the aspect ratio. e2represents the half-width, f3 represents the half-length; by dividing one overthe other and taking the bigger of the two, we get the aspect ratio (widthover length, or length over width, whichever is bigger).

Regarding skew, since e3 represents the absolute distance of skew in thex direction, we need a way of scaling so that if the shape is identical, our

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metric for skew does not increase as the shape becomes larger. f3 is chosenfor this, as it is a scale of the associated parallelogram in the y direction.

Thus, skew = e3f3

Regarding the two tapers, e4 and f4 represent the emphabsolute distanceof taper in the x and y direction, respectively. Similar to what we did withskew, we need to ground the distance to the size of the quadrilateral. Thus,taper in the x direction would use the half-width e2, and taper in the y

direction would use the half-length f3. Thus, taperx =f4f3

, and tapery =e4f2

From Robinsons paper [8]: the Jacobian matrix for a flat (projected)quadrilateral is given by:

[J] =

x

y

x

y

= [(e2 + e4) (f2 + f4)(e3 + e4) (f3 + f4)]

The determinant of [J] can written as det[J] = e2f3+e2f4+(e4f3e3f4).Which, in turn, can be refactored using the shape metrics in the following:

det[J] = f 23 (AR)(1 + Tx + (Ty Skew

ARTx))

As we can see, all four shape parameters are present within the deter-minant, in addition to the half-length of the basic rectangle. For analysis,we will be taking the determinant at the third vertex of the quadrilateral(top-right), as that vertex has the curvilinear coordinate where = = 1.

Due to the presence of the half-length, the size of the quadrilateral willaffect its Jacobian determinant. This, plus the possibility of having negativetapers which can actually reduce the determinant to below the determinantof a square, have led me to suggest a new metric, Modified determinant, forquadrilateral quality, by removing the mentioned metrics from the equation:

ModDet[J] = (AR)(1 + Tx + (Ty SkewAR

Tx))

Again, we will use only the third vertex of the quadrilateral for analysis.

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Since the aspect ratio, skew, both tapers, and the Jacobian determinantare all calculable from only the four points of a quad, these calculations canbe implemented in a relatively straight-forward way. Since quad meshes aremade up of quad elements by definition, I can run the CRE method over allthe quads in the mesh and find out the average of all four respective shapeelements and the Jacobian determinant.

Running the mesh generation algorithm over a set of different inputswould yield many quad meshes derived from an algorithm. By using theaverage value of the shape elements from all the quads from all the meshesgenerated by a single algorithm, we can compare that average to that of otheralgorithms.

3.3 Comparison Method

I have written a program that takes in a quad mesh in Geomview Object FileFormat (.OFF), and outputs the average and worst metric of quads in themesh using the method discussed previously. Using this program, for eachmesh generation algorithm I examine, I will run and record the metrics overseveral selected point sets. This includes 6 sets of points placed randomly invarious quantities (10, 50, 100, 200, 500, 1000) and a set of 303 points thatis a polygonal representation of Lake Superior. The 6 sets of random points,once generated, is tested on all the algorithms: all the meshing algorithmsare tested on the same data set.

The metrics that I will specifically compare with are the following:

Total quadsAverage/worst ARAverage/worst Skew valueAverage/worst Taper in the x-axisAverage/worst Taper in the y-axisAverage/worst Jacobian determinantAverage/worst Modified Jacobian determinant

4 Comparisons and Results

The tables below show the average (Table 1) and worst (Table 2) metrics ofthe resultant meshes using the specified algorithm and the specified input.

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Table 1: Average metrics

10 pts 50 pts 100 pts 200 pts 500 pts 1000 ptsLake Superior

(303 pts)Total quads 801 3768 7613 15074 37219 75057 24444Avg. AR 1.182521 1.207838 1.191341 1.200362 1.202959 1.197838 1.198281Avg. skew value 0.187232 0.176465 0.176154 0.176752 0.175277 0.176673 0.185968Avg. taperx 0.15145 0.156579 0.151035 0.150312 0.149911 0.149348 0.155988Avg. tapery 0.173889 0.16305 0.163772 0.159986 0.158285 0.161002 0.172206Avg. Jacobian det. 119.4661 31.66326 16.43845 8.39404 3.417593 1.698783 0.013589Avg. mod. det. 1.609149 1.631948 1.601269 1.608572 1.610947 1.605722 1.629233

Table 1.a Atalay et al. quadtree meshing average metrics



Table 1.b Shewchuk Triangle tri-to-quad meshing average metrics



Table 1.c Shewchuk Triangle with minimum 20 angles, tri-to-quad meshing aver-age metrics

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Table 2: Worst metrics


(303 pts)Worst AR 3.000000 3.000004 3.000003 3.000007 3.000027 3.000055 3.00073Worst skew value 1.145833 1.333333 1.333333 1.333334 1.875 1.875002 1.875004Worst taperx 0.6 0.6 0.6 0.6 0.600002 0.600002 0.600029Worst tapery 0.730798 0.730798 0.730798 0.730799 0.7308 0.730805 0.730854Worst Jacobian det. 4783.264 2990.188 3140.657 3153.046 3157.515 3160.102 17.503433Worst mod. det. 4.89 4.890005 4.890003 4.890004 4.89001 4.890047 4.890585

Table 2.a Atalay et al. quadtree meshing worst metrics


(303 pts)Worst. AR 42.79535 281.9245 466.2196 559.2251 1777.939 1174.521 530.1884Worst. skew value 28.77125 127.9088 294.6275 329.0671 1254.819 624.1278 351.1376Worst. taperx 0.25 0.25 0.25 0.25 0.25 0.25 0.25Worst. tapery 1.823832 1.044526 7.098919 4.005084 9.556236 3.096078 3.855682Worst. Jacobian det. 2448.629 3334.426 4191.789 1702.153 107129.7 964.6050 8.472251Worst. mod. det. 64.19302 422.8869 699.3325 838.8374 2666.950 1761.781 795.2556

Table 2.b Shewchuk Triangle tri-to-quad meshing worst metrics


(303 pts)Worst. AR 3.34753 4.425987 4.043448 4.180376 4.376373 4.558736 4.615184Worst. skew value 1.656701 1.820182 1.757625 1.744029 1.812709 1.899074 1.925705Worst. taperx 0.25 0.25 0.25 0.25 0.25 0.25 0.25Worst. tapery 0.018427 0.066429 0.072425 0.079237 0.077191 0.086745 0.087063Worst. Jacobian det. 276.9116 661.7325 200.6794 136.0718 47.29270 32.69729 0.273995Worst. mod. det. 5.021296 6.63898 6.065173 6.270564 6.56456 6.838106 6.922818

Table 2.c Shewchuk Triangle with minimum 20 angles, tri-to-quad meshing worstmetrics

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Figure 6: Jacobian determinant against size of input set

An initial glance yields some interesting observations. Despite the glaringdisadvantage of the tri-to-quad algorithm, the quad meshes created by ithave much fewer faces than those created by Atalay et al.s direct quad meshgeneration algorithm. While running Triangle with a bounded minimumangle gives us quad meshes with more faces, they still have significantlyfewer faces than those created by Atalay et al.s algorithm.

The change in average metric over quantity of points does not seem to besignificant; that is, more random points in a set does not have a significantimpact on the average metrics, except for the Jacobian determinant. How-ever, knowing the way the Jacobian determinant is calculated (the Jacobiandeterminant has the coefficient of aspect ratio times the square of the mid-length), this can be attributed to the smaller elements in quad meshes withmore faces. This is supported by the fact that the Lake Superior input datayields a much smaller Jacobian determinant, no matter the algorithm. SeeFigure 6.

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Figure 7: Modified Jacobian determinant against size of input set

I initially thought there would be a significant difference between ran-domly generated point set input and a polygonal point set, but plottingmodified Jacobian determinant against input size (Figure ??) yields almostno difference between randomly generated and polygonal point set input, atleast for Shewchuks Triangle algorithm with angle guarantee and Atalay etal.s quadtree algorithm.

Comparing the average aspect ratio of non-bounded and bounded min-imum angles on Shewchuks Triangle shows a dramatic improvement fromthe 3.5 to 5.5 range to a more manageable and consistent 1.6 to 1.7 range(Figure 8). Meanwhile, Atalay et al.s quadtree algorithm generates aspectratio in the range between 1.18 to 1.2.

Curiously, the average taper in the x direction is consistently 0.25, boundedminimum angle or not. Perhaps this is an artifact from the tri-to-quad algo-rithm. After examining the individual analysis, all the quads seem to have a

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Figure 8: Aspect ratio against size of input set

taper of -0.25 under the algorithm.Looking at the worst metrics, taper in the x-direction exhibits a very

strange characteristic. While we know that the Shewchuk tri-to-quad meshcontain only quads of taper -0.25, and so the identical worst taperx is nosurprise, taper for Atalay et al.s algorithm generate meshes with extremelysimilar tapers. I can not offer any thoughts as to why this would be the case,given that six of the input point sets are essentially randomly generated, andso there would be no guarantee that a quad with a given taper would exist.This may be worth further investigation.

It is interesting to see that the worst modified Jacobian determinant forAtalay et al.s quad meshes are also very similar to each other. This can per-haps to attributed to the angle bounds that their algorithm guarantees. Thesame effect is not seen in Shewchuks Triangle with bounded angles, possiblybecause while the original triangle mesh has angle guarantees, the tri-to-quadconversion may introduce extreme angles. This can be seen in Shewchuks

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Triangle without bounded angles: the worst modified determinant increasesdramatically for bigger input sets.

Figure 9: Modified Jacobian determinant against aspect ratio

5 Evaluation

The metrics do provide the expected results: Shewchuks Triangle withoutbounded angles gives the worst meshes, followed by Shewchuks Trianglewith bounded angles, and followed by the quadtree direct meshing algorithm.While the modified determinant is indicative of four shape parameters (aspectratio, skew, taperx and tapery), Robinsons method of calculating aspect ratioseems to be sufficient for general mesh quality assessment. As we can seein Figure 9, modified Jacobian determinant correlates extremely highly withaspect ratio. This would indicate that for the input set we have analyzed,modified Jacobian determinant is as good as a mesh quality measure asRobinsons aspect ratio.

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For individual quad analysis, the modified Jacobian determinant will bemore effective in determining its quality. The determinant is affected by allfour shape parameters, and so is the ideal candidate should algorithms needto maximize for a certain quality metric.

However, there are limitations to the Jacobian determinant. As Robinsonstates, a quadrilateral has two sets of shape parameters depending on whichof the oblique axes is taken as the local x-axis, although the skew parameter isthe same in each case [8]. This means that the same quad, given a differentordering of points, could give one of two metrics. See Figure 10.

6 Conclusion

This paper examines Robinsons CRE method in-depth, and provided animplementation of the analysis to be run on quad meshes in .OFF format.The modified Jacobian determinant, given in this paper, is a valid measureof the quality of a quad mesh, although it should be noted that for meshanalysis, Robinsons method of measuring aspect ratio works similarly asan indicator of quality. Further research could be conducted by analyzingother quad mesh generation programs. Since the source code for the analysisprogram is given below, the only impediment to analyzing other quad meshalgorithms would be the acquisition of executable implementation of thealgorithms, and the necessary file type conversion, as .OFF is one of manymesh formats available.

The code for the analysis program and input data sets is available athttps://github.com/plu97/analyzeOffFile.

References

[1] Atalay F. B., Ramaswami S. and Xu D. (2012) Quadrilateral Mesheswith Provable Angle Bounds, Engineering with Computers 28, Issue 1,pp 31-56

[2] Bern M., Eppstein D., Gilbert J. (1994) Provably Good Mesh GenerationJournal of Computer and System Sciences 48, Issue 1, pp 384-409

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[3] Bommes D., Levy B., Pietroni N., Puppo E., Silva C., Tarini M., ZorinD. (2013) Quad-mesh Generation and Processing: A Survey, ComputerGraphics Forum 32, Issue 6, pp 51-76

[4] Devadoss S. L. and ORourke J. (2011) Discrete and Computational Ge-ometry

[5] Eppstein D. (2001) Global Optimization of Mesh Quality, from http://www.ics.uci.edu/~eppstein/pubs/Epp-IMR-01.pdf [Accessed Dec.19 2014]

[6] Eppstein D., Goodrich M. T., Sun J. Z. (2005) The Skip Quadtree:A Simple Dynamic Data Structure for Multidimensional Data, fromhttp://www.ics.uci.edu/~eppstein/pubs/EppGooSun-SoCG-05.pdf

[Accessed Apr. 19 2015]

[7] Remacle, J.-F., Lambrechts, J., Seny, B., Marchandise, E., Johnen, A.and Geuzainet, C. (2012) Blossom-Quad: A non-uniform quadrilateralmesh generator using a minimum-cost perfect-matching algorithm. In-ternational Journal for Numerical Methods in Engineering 89, Issue 9,pp 1102-1119

[8] Robinson J. (1987) CRE Method of Element Testing and the JacobianShape Parameters, Engineering Computations 4, Issue 2, pp 113-118

[9] Robinson J. (1988) Understanding Finite Element Stress Analysis

[10] Shewchuk J. R. (1996) Triangle: Engineering a 2D Quality Mesh Gen-erator and Delaunay Triangulator, Applied Computational Geometry To-wards Geometric Engineering, pp 203-222

[11] Tarini M., Pietroni N., Cignoni P., Panozzo D. and Puppo E. (2010)Practical Quad Mesh Simplification, Computer Graphics Forum 29, Num-ber 2, pp 407-418

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Figure 10: Jacobian Determinant measurement

(a) A quad, with aspect ratio = 1.2, mod. Jacobian determinant = 1.6733

(b) The same quad, rotated 90 to the right. Aspect ratio = 1.5, mod. Jacobiandeterminant = 2.0

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Quality Measure of Quadrilateral Mesh

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