flow field visualization using vector field perpendicular ...karlu20/papers/palmerius09flow.pdfflow...

Flow Field VisualizationUsing Vector Field Perpendicular Surfaces

Karljohan Lundin Palmerius∗ Matthew Cooper† Anders Ynnerman‡

Norrkoping Visualization and Interaction Studio,Linkoping University, Sweden

Abstract

This paper introduces Vector Field Perpendicular Surfaces as ameans to represent vector data with special focus on variationsacross the vectors in the field. These surfaces are a perpendicu-lar analogue to streamlines, with the vector data always being par-allel to the normals of the surface. In this way the orientation ofthe data is conveyed to the viewer, while providing a continuousrepresentation across the vectors of the field. This paper describesthe properties of such surfaces including an issue with helicity den-sity in the vector data, an approach to generating them, several stopconditions and special means to handle also fields with non-zerohelicity density.

CR Categories: I.3.5 [Computer Graphics]: ComputationalGeometry and Object Modeling; I.3.6 [Computer Graphics]:Methodology and Techniques; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism

Keywords: vector field visualization, perpendicular surfaces, sur-face representation

1 Introduction

The search for better ways to visualize complex structures and dy-namics in volumetric data is an active area of research. Examplesof applications driving this development are the computational fluiddynamics (CFD) analysis of aerodynamics and combustion. Thereexists a wide range of approaches for visualizing vector fields, seee.g. [Schroeder et al. 1998; Post et al. 2003; Laramee et al. 2004;Laramee et al. 2006]. The popular integration-based techniques —streamlines, stream-ribbons, stream-surfaces, etc. — represent thedata at discrete regions in space, but describe the orientation contin-uously along the direction of the field. A similar effect is achievedusing more recent methods, such as dye advection[Weiskopf 2004]or volume LIC[Interrante and Grosch 1997; Interrante and Grosch1998]. The distribution is discrete across the vectors in the fieldand the representation continuous along the vectors. Therefore, thestreamline and its peers can be thought of as visualizing the datawith special focus on variations along the field.

These stream-based approaches to visualizing vector data are gener-ally considered both intuitive and powerful. Sometimes, however,

∗[email protected]†[email protected]‡[email protected]

the variation across the vectors in the field, instead of along thevectors, is of more interest. This can be, for example, along thefront edge of a wing to visualize variations in wind bifurcation, inflow data to emphasize vortex structures, or in magnetic fields tovisualize how a change of position might affect the direction of anattractive force.

This paper introduces Vector Field Perpendicular Surfaces, as aperpendicular analogue to streamlines, as a complement for rep-resenting the field data with particular focus on the cross-field vari-ations. When a streamline grows from a seed point forwards andbackwards in the orientation of the vector field, this surface growsoutwards, perpendicular to the field and such streamlines. In thesame way that a streamline can be used to represent vector dataand describe the behaviour of the properties along the local vectors,the vector field perpendicular surfaces can be used to represent thesame data and describe the behaviour across the orientation of thevectors. A local surface patch representing the orientation of thedata is in the work presented here generated by starting at a seedposition and marching a surface edge outwards and perpendicularto the vector data.

The contributions of this paper are:

• The introduction of Vector Field Perpendicular Surfaces asthe perpendicular analogue to streamlines

• The description of common properties of such surfaces andvisualization examples

• Support for the surface representation of vector fields withnon-zero helicity density

• Several appropriate stop conditions and suggestions on howto implement them

• A straightforward and fast approach for generating the sur-faces

• An open source implementation

2 Related Work

Apart from dense methods, such as LIC[Cabral and Leedom 1993]that display changes in any represented direction, there have beenfew approaches presented that allow for focus on changes across thefield. The probe presented by Leeuw and van Wijk in [de Leeuwand van Wijk 1993] is capable of displaying the local change acrossand along the field for discrete probed points. This is done by cal-culating a velocity gradient tensor at the position of the probe andfrom this extract properties, such as curvature, that are mapped tothe visual appearance of the probe. It should be noted that thisprobe only visualizes the properties at one point, only an analyti-cal extraction of the changes perpendicular to the field and not thechanges themselves over larger areas.

Weinkauf and Theisel describe in [Weinkauf and Theisel 2002]Normal Surfaces, surfaces that have the vector field as normals, and

3.2 Standard Patterns 3 THE BASIC PRINCIPLES

describe many properties of these. They do not, however, extractthe surfaces or use them for visualization. As another example offeatures perpendicular to the vector field, Bachthaler and Weiskopfdescribe in [Bachthaler and Weiskopf 2007] the use of an LIC thatis orthogonal to the flow. This is also combined with animation toincrease the perception of features in the visualization.

Zhang et al.[Zhang et al. 2003] presented a method they call“streamsurfaces” (not to be confused with the stream-surfacesmentioned above), where a surface is generated along the two majoreigenvectors of a tensor field, starting at a seed position. This resultsin a visual appearance similar to our method, however their methodcannot correctly handle helicity. When visualizing diffusion tensordata this might not produce noticeable artifacts, however the oc-currence of helicity must be correctly handled in the generalizationof this concept. A continuous surface that is perfectly orthogonalto the vector field can only be defined if the field has zero helicitydensity in the local region.

Sondershaus and Gumhold[Sondershaus and Gumhold 2003]present an alternative use of the surface representations presentedby Zhang et al. In it they distribute points over the surfaces insteadof rendering them directly, to avoid occlusion, and apply lightingmodels and colouring to emphasize shapes and structures. Theyalso present an algorithm for finding such surfaces, using face-based coding. Just like Zhang et al., however, they fail to acknowl-edge or handle the issue with helicity.

3 The Basic Principles

The vector field perpendicular surfaces method for vector field visu-alization represents the data visually using surfaces that are, at anypoint on the surfaces, perpendicular to the vector field. Through theorientation of the surface, they describe the structure of the field ina continuous manner across the vectors in the field. By inspectingthe surface across the field, the variations of the field can be visuallyexplored, see example in figure 7. Such surfaces can be generatedas patches from discrete seed points the same way that streamlines,for example, are distributed in a flow field.

We can define the perfect vector field perpendicular surface as acontinuous set, S , of points,~x, with the following property:

lim|~x1−~x0|→0

~V (~x0) · (~x1−~x0)∣∣∣~V (~x0)∣∣∣∣∣∣~x1−~x0

∣∣∣ = 0 (1)

where ~V is the vector field and where ~x0 and ~x1 are points on thesurface. As described in [Weinkauf and Theisel 2002], such sur-faces are undefined at critical points, never intersect or touch eachother, and also never self-intersect. This simplifies the surface con-struction since such special cases need not be considered. They arealso undefined where the flow is of zero magnitude.

3.1 Non-zero Helicity

Putting the definition of the vector field perpendicular surfaces inequation 1 in a Riemann sum it follows that∫

C~V ·d~s = 0 (2)

for any curve, C, on the surface. A way of writing curl explicitly asrotation is

~n ·~∇×~V (~p) = limr→0+

1πr2

∮Cr

~V ·d~r (3)

where Cr is a circle around ~p with radius r and~n is the normal of thatcircle. By letting Cr be a curve in the surface defined by equation 1we see that~n = ~V (~x0)

/∣∣∣~V (~x0)∣∣∣ and thus, using equations 2 and 3,

we get~V ·~∇×~V = 0 (4)

which describes a helicity free field. This means that the surface canonly be defined if the field has a zero helicity density in the regionof the surface. If the vector field is conservative, i.e. ~∇×~V = 0 inwhich case it will also be helicity free, the surface perpendicular tothe vector field will be a section of an iso-surface of the potential ofthe field, with the iso-value being the potential at the surface.

If there is helicity in the field at the location to be visualized, thena modification of the mathematical definition must be made. Oth-erwise it is impossible to generate such a visual representation. Wedefine a non-perfect vector field perpendicular surface as a piece-wise continuous, connected set, S ε , of points which has the fol-lowing property:

lim|~x1−~x0|→0

∣∣∣~V (~x0) · (~x1−~x0)∣∣∣∣∣∣~V (~x0)

∣∣∣∣∣∣~x1−~x0

∣∣∣ ≤ ε (5)

where ε is a small number indicating the maximum level of errorand where~x0 and~x1 are points in the surface. Since there are no per-fect perpendicular surfaces in fields with a non-zero helicity densitythe visual representation with vector field perpendicular surfaces ofsuch vector fields must allow for a certain orientation error. Themaximum orientation error, ϕ , for a surface at a certain positioncan be calculated from equation 5 as ϕ = arccos ε .

By allowing a small error, the surface can describe the flow struc-ture over a small region, however as the surface grows the amountof error will increase. To ensure a bound error in the surface orien-tation we must tear the surface into smaller pieces, each of whichprovides a good representation of the flow. The size of these pieceswill be dependent on the amount of curl, that is how close to zerothe vector field helicity density is in the local region, and the amountof error allowed in the visual representation.

3.2 Standard Patterns

The surfaces react in a predictable way to form identifiable patternsfor certain structures in the vector data, see figure 1. Divergenceand convergence in the flow are rendered as cone or spherical cappatterns, depending on the radial structure of the divergence or con-vergence, see figure 1(a). The combined divergence in one directionand convergence in the other also produces a very distinct pattern,see figure 1(b).

These three patterns, convergence, divergence and combined, aresimilar to what you might get with Leeuw and van Wijks flow fieldprobe[de Leeuw and van Wijk 1993] if the data at the probed pointare similar to the surrounding points described by the larger sur-face. A rotational field, however, typically has a non-zero helic-ity and therefore requires tearing of the surfaces which produces adistinct fan-like pattern, see figure 1(c). Each patch in the tearedsurface describes the local flow direction and the combined pictureshows the rotational nature of the flow. Also, the amount of tearingindicates the magnitude of curl in the area.

Observe that these are only typical patterns and that a single vec-tor field perpendicular surface can show several of these patternsdescribing the variations across the vector field.

c©ACM 2009. This is the authors version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution.The definitive version was published in Proceedings of Spring Conference on Computer Graphics, 2009

2

4 FAST SURFACE CONSTRUCTION

(a) Divergence or convergence in the field ren-ders surfaces in the shape of a cone or sphericalcap.

(b) A combined convergence and divergencerenders surfaces in the shape of saddle sur-faces.

(c) A vortex forces the surface to tear apart andproduces a fan-like pattern.

Figure 1: The vector field perpendicular surfaces provide certain general patterns in response to typical features in the vector data. Each suchpattern can be easily identified and associated with the corresponding structure.

Figure 2: Vector perpendicular surfaces visualization used to visu-alize vortices in CFD data.

3.3 Vector Property Extraction

The vector perpendicular surfaces can also be used to visualizeother properties of 2D manifold nature extracted from scalar, vec-tor or tensor data. Zhang et al. have already shown in [Zhang et al.2003] that creating a surface perpendicular to the minor eigenvectorof a diffusion tensor field can provide an intuitive visualization ofthe planar class tensors in the data. Other properties represented byvectors can be extracted and visualized in a similar manner. For ex-ample, the gradient of scalar data can be used to control the surfaceorientation, which results in iso-surface patches around the selectedseed points.

In flow data an example of a property of interest is the location andshapes of helices. This property can be estimated by calculatingthe cross product between the flow and the vorticity, as shown byLawrence et al. in [Lawrence et al. 2000],

~Vh(~x) =(~∇×~V (~x)

)×~V (~x) (6)

where ~V is the vector data. Using this property, ~Vh, to define thesurface orientation, we get the examples shown in figure 2.

4 Fast Surface Construction

Vector field perpendicular surfaces are generated, just like stream-lines, from seed points distributed in the data. Just like with stream-

(a) For each non start/end point in the edge segmentnew points are added to a new edge segment.

(b) If a stop criterion is fulfilled at a point, this pointwill become the end point of the currently generatedsegment and the start point of a new segment.

Figure 3: The surface is generated by the outward propagation of anedge. For each edge segment on ring R (SR, green-white-red points)zero or more new edge segments are generated for ring R+1 (SR+1,green-blue-red).

lines these seed points may be pre-defined or automatically or in-teractively distributed. From each such seed the surface is com-puted by first generating an initial seed surface. The sequence ofrim points of this surface defines an edge, E. The surface is theniteratively grown by producing new points defining a new edge out-side the previous one, see figure 3(a). We call each such iterationa ring, starting with ring zero being the initial, closed edge aroundthe seed point.

Since points are iteratively added in a structured manner it isstraightforward to define triangles that connect these points. Thetriangles then become well structured and it is simple to store thesurface in the form of triangle strips, a structure that is more mem-ory efficient and faster to render than unordered triangles.

A variety of stop criteria are defined for the surface, for examplelimiting the length of growth and thereby the size of the surface,see section 5 for details. When a stop criterion is fulfilled for acertain point at the edge, the surface should be terminated at thatpoint, but may continue the propagation on both sides of the point,see figure 3(b). To handle this the edge is divided into segments, S.Thus, conceptually the edge at ring R, ER, consists of one or severaledge segments, ER =

SR

0 , . . .

. Each segment, in turn, consists ofat least three points, SR

i = ~p0, ~p1, ~p2, . . ., a start point, one ormore middle points and an end point. Why each segment consistsof at least three points will be apparent from the approach chosento define new surface points.


3

4.2 Edge Propagation 4 FAST SURFACE CONSTRUCTION

Figure 4: From the seed point (centre) six points are distributed onestep outwards, 60 apart. These points form an initial seven pointsedge segment (E0 = S0) with common start and end points, greenrespectively red.

The algorithm is also summarized in the form of pseudo code at theend of this section.

4.1 Surface Initialization

The first edge for the surface is generated around a seed point, ~pseed,as an initialization before the edge propagation can commence. Tofind the position for the first point of this edge, ~p0, a unit lengthvector, q, being orthogonal to the local vector data, ~V (~pseed), isfirst defined. The choice of q will affect the visual appearance ofthe generated surface, however it is not known at this point whichdirection will be beneficial. One step is then performed by integra-tion along the vector function ~Q(~x, q) over position, ~x, starting atthe point ~pseed and with length δstep. The end point of this integra-tion is then used as ~p0. The function is the projection of the vectorq onto the plane defined by the vector field at the point~x,

~Q(~x, q) = q− ~V (~x)(~V (~x) · q

)/∣∣∣~V (~x)∣∣∣2 (7)

and it is the integration along this function that produces the curvedsurface.

To define the rest of the points in the initial surface the directionvector, q, is rotated around the vector at the seed position, ~V (~pseed),and the same procedure performed again and again until the circleis closed. The rotation described here is clockwise, however thedirection of rotation should have a minimal impact on the visualappearance.

The initial points are distributed 60 degrees apart in order for thetriangles in the generated surface mesh to, as far as possible, ap-proximate equilateral triangles. This means that the initial surfacerim is of six points forming a hexagon, see figure 4. Since at thispoint the rim is closed, the start and end points will be the same. Wenow have the first edge consisting of a single segment, E0 =

S0

where S0 = ~p0, ~p1, . . . , ~p5, ~p0.

4.2 Edge Propagation

For each segment in the edge of a ring, R, new segment(s) are gen-erated defining a new edge of the subsequent ring, R + 1. Thus,the edge propagates through the data and the surface grows ringby ring. To generate new points for a segment in the subsequentring the points in each segment in the current ring are processed inturn, see figure 3(a). When a point is processed the ‘triplet’ madeup of the preceding, current and subsequent points is used. Sincea segment contains start, end and middle points, we are guaranteed

~p j

αβ

qfqr

~p j+1~p j−1

Figure 5: New points generated for each triplet,~p j−1, ~p j, ~p j+1

:

β is chosen to be close to 60 which, together with the angle α ,determines the number of new points to generate to evenly fill theavailable space.

to have at least one triplet to process. To determine the numberof points to generate for the subsequent ring we first estimate theavailable space in front of the triplet of points, ~p j−1, ~p j and ~p j+1,

qr =~p j−1−~p j∣∣~p j−1−~p j

∣∣ (8)

qf =~p j+1−~p j∣∣~p j+1−~p j

∣∣ (9)

α =

arccos qr · qf if qr ·(

qf×~V (~p j))≥ 0

2π− arccos qr · qf if qr ·(

qf×~V (~p j))

< 0(10)

Points are added by filling the available space in front of eachtriplet. The point separation should, as before, be as close to 60degrees as possible, see figure 5, so the number of points to gener-ate, Nnew, and the point separation, β , are determined by

Nnew =[

2α

π

](11)

β =α

Nnew +1(12)

where the angle bracket signifies rounding to the closest integervalue. The first point position is then determined by rotating qr byβ around the local vector and performing one step, as described insection 4.1, in that direction. The positions of any further points aredetermined the same way by again rotating the vector qr by β andthen performing one step from the centre point, ~p j, of the triplet.

With this fundamental approach the last point generated from onetriplet will almost coincide with the first point of the subsequenttriplet, see figure 5. The mean of these two points is used as thepoint connecting the two triplets. This approach allows us to use aflexible approach to the estimate of the surface growth length (sec-tion 5.1). Observe that one reason that these two points do not al-ways coincide is that in fields with non-zero helicity the integrationover two different paths towards the same position will not inter-sect, as described in section 3.1. Even though these two paths areboth orthogonal to the vector field the surface connecting the twopaths will not be, which introduces a small error in the surface ori-entation compared to a perfectly perpendicular surface.

Curvature in the data can cause the base of the triangles in a regionto become shorter and shorter for each ring, see figure 6. To avoidthe case where this may cause a triangle to become obtuse (havinga point outside the base) the following condition is added:

qf ·(~pnew−~p j

)<∣∣~p j+1−~p j

∣∣ (13)


4

5.1 Length Limit 5 STOP CRITERIA

Only if this condition is true will the point fall within the base of alltriangles associated with the current point, ~p j. By not using pointsfor which it is false the points distribution can be kept structured,see figure 6.

When all segments for an edge have been processed and an edge forthe subsequent ring has been generated, it is then necessary to closethe ring of that new edge. Otherwise, the resulting ring will have anotch, a cleft in the edge, where the first and last segment share theirstart and end point, see figure 3(b). If this shared point fulfils a stopcondition, the ring is not closed and a flag is set so that it will notbe closed in subsequent rings either. If the point is a valid candidatefor propagation, however, new points are first added to fill the cleftif necessary, as described above. The last point added is then usedas the new start and end point for the first and last segment. Afterthis the edge is fully connected at the start/end point, similar to theinitial hexagon surface in figure 4.

4.3 Summary

The steps described above can be summaries in pseudo code in thefollowing way. The same notation is used here as is used above.

1 initialize E0 and set R = 02 while ER is not empty3 for each Si in ER4 add ~p0 in SR

i to new segment SR+1k

5 for each triplet ~p j−1, ~p j, ~p j+1 in Si6 if stop criterion met for ~p j7 add ~p j to SR+1

k8 if size(SR+1

k ) > 29 add SR+1

k to ER+110 increment k11 add ~p j to SR+1

k12 continue from line 513 estimate angle α between qf and qr14 Nadd =

[ 2α

π

]and β = α

Nadd+115 do Nadd times16 rotate qr by β

17 ~pnew = ~p j +∫

~Q(~x, qr)18 if first of the Nadd points and len(SR+1

k ) > 119 merge ~pnew with last added point20 else if qf ·

(~pnew−~p j

)<∣∣~p j+1−~p j

∣∣21 add ~pnew to SR+1

k22 add ~p j+1 in SR

i to new segment SR+1k

23 add SR+1k to ER+1

24 if is closed25 if stop criterion met for ~p026 is closed = false27 else28 join SR+1

0 and SR+1k

29 increment R

5 Stop Criteria

The vector field perpendicular surfaces are generated by starting ata seed and propagating outwards. The propagation of the surfacestops at a point on the rim when a stop criterion is fulfilled at thatpoint. An obvious such stop criterion is the lack of data: if thelocal vector is shorter than some threshold value, the uncertainty isdeemed too high and the surface is terminated at that point.

Figure 6: The surface is seeded at the blue sphere and propagatesover the convex field. The handling of curvature can be seen in thebottom left and right region of the surface.

This section describes additional stop criteria and how they are esti-mated at run-time. Observe that the stop criteria introduce parame-ters that, just like the step length, can be adjusted to fit the data andthe purpose and properties of the visualization.

Terminating at a point leaves a cleft of approximately 60 degreesin the edge, see figure 3(b). If the stop criterion is not fulfilled forthe points around that cleft, the surface may start to grow inwardsthrough the cleft, resulting in a self intersecting surface growingsimultaneously in all directions around the point of surface termi-nation. This can be avoided by simply not expanding the pointsnext to a terminated point. By doing this not directly after the ter-mination, but after one or a few rings, the level of sparseness ofthe resulting surface can be somewhat controlled. This measure hasbeen taken in the implementation used to make the example images.

5.1 Length Limit

The length limit is a useful stop criterion to avoid creating a surfacethat grows to infinity. Therefore each point, ~p j, is associated withan accumulated length, L j. Each time a new point is generated itsaccumulated length is estimated as the accumulated length of thepoint being expanded (described in section 4.2) plus the length ofthe step. Since the steps made follow a triangle pattern and aregenerally not performed in the direction of the surface expansion(see figure 5) the length increment is subject to a modulation:

L j = Li +δstep cos(

β − α

2

)(14)

where Li and L j are the lengths associated with the old and newpoint, respectively. When two points are merged the edge growthdirection of these two points will be different and thus the estimatedlength will differ even though the points almost coincide. The meanlength of the two points is then used as an estimate for the newpoint.

When the accumulated length, L j, exceeds a preset length thresh-old, CL, the surface should be terminated at the associated point. Toget a smooth edge of the final rim, however, the step length is ad-justed at the last step so that the last rim points end up at the exact


5

6.1 Examples 6 RESULTS

length specified:

δ′step = δstep

CL−L j

cos(β − α

2) (15)

In special cases, where a triplet forms a very sharp angle, the co-sine term (equation 14) may cause a length decrement instead of anincrement. To avoid this a lower limit may be set on that term.

5.2 Maximum Winding Angle

A vector field perpendicular surface can in a high curvature fielddescribe a closed loop which can be visually confusing. The wind-ing angle, ω j, associated with point ~p j is used to add control overthis property of the surface. It is estimated by comparing the di-rection of propagation to and from the point ~p j. For each point theaccumulated winding, Ω j, is tracked so that the surface can be ter-minated at the point if Ω j > CΩ, where CΩ is a preset accumulatedwinding limit.

Too large a change in vector orientation in a field may introduce toolarge an error in the integration, or may simply cause strange sur-faces. A winding limit, Cω , is also introduced that makes it possibleto terminate the surface in regions containing too large a change.

5.3 Maximum Orientation Error

As is described in section 3.1, only vector fields with zero helicitydensity will have surface manifolds that are at all points perpendic-ular to the field. Fields with regions of non-zero helicity will needa certain allowed orientation error for a continuous surface to begenerated. In certain cases the error might become too large for anappropriate representation of the data, so an enforced limit, Cϕ , ofthe orientation error, ϕ , must be introduced.

We estimate the orientation error associated with a point, ~p j, as thedeviation from a right angle of the angle between the normalizedlocal vector and the direction to the subsequent point in the segment

ϕ =

∣∣∣∣∣∣arcsin

~V (~p j) ·(~p j−~p j−1

)∣∣∣~V (~p j)∣∣∣ ∣∣∣~p j−~p j−1

∣∣∣∣∣∣∣∣∣ (16)

When this value exceeds Cϕ the surface is terminated at that point.Since the current edge segment becomes divided into two segmentsaround this point, the surface will continue to propagate on bothsides. This will produce an effect similar to tearing the surface tomake it fit the orientation of the vector field. This effect is clearlyvisible in figure 7.

6 Results

The implementation of the presented work has been used to producevisualization examples for demonstration purposes and to evaluatethe algorithm performance.

The vector field perpendicular surfaces approach to vector visual-ization has been implemented in C++ using H3D API, an X3D-based scene graph system for implementing multi modal interac-tion, and the Volume Haptics Toolkit (VHTK) which is a H3D APItoolkit for volume haptics and visualization. This reference imple-mentation is freely available in the source distribution of the GNUGPL release of VHTK.

The surface generating scene graph node extends the IndexedTrian-gleSet type, which handles point coordinates, point colours, and anindex array that specifies the point structure for building triangles.The surface colour is controlled through an RGB transfer functionfrom the magnitude of the vector field. It is, however, possible toreplace this with any scalar property, or even another scalar field.

The integration of the function in equation 7 is in this implemen-tation estimated using fourth order Runge-Kutta. Each step out-wards in the algorithm is performed using one single integrationstep. Thus, the step length, δstep, is here both describing the trian-gle size in the final mesh and the step length in the Runge-Kuttaintegrator.

6.1 Examples

The first data set used for the demonstration comes from a CFDsimulation of the air flow around an experimental unmanned aerialvehicle (UAV) called SHARC. The vector field used to representthe air flow is 32 bit floating point vectors and has a resolution of1283 voxels. An example of the visualization of this data set usingvector field perpendicular surfaces is shown in figure 7.

Experience with the vector field perpendicular surfaces in real datasets indicates that it is possible to learn how to recognize these pat-terns and associate them with structures in the vector data for im-proved understanding of, for example, complex flow. The air flowdata is almost free from helicity in some regions in which the sur-face plainly forms after the behaviour of the flow. This can be,for example, the convergence in the flow indicated by the surfacesin the upper circle in the figure. In other regions, especially nearvortices, the vector field has helicity and the surface is torn to fitthe data, as shown in the lower circle. A maximum orientation er-ror of 5 degrees has been allowed, which is close to the perceptuallimit[Knill and Kersten 2003].

For a second example the blunt fin CFD data set[Hung and Buning1984] has been resampled to a regular grid of 2563 voxels with 32bit floating point vectors in order to fit the currently used frame-work. The visualization of this data set, see figure 8, shows howthe cross sections represented by the vector field perpendicular sur-faces visualize the change in flow direction at different levels fromthe surface of the fin. Stream-tubes rendering has been added toprovide a comparison with classical vector visualization. As can beseen the surfaces display, through their orientation, how the flowdirection varies across the flow, while the stream-tubes show howthe flow turns along the flow.

6.2 Performance

The time it takes to generate a vector field perpendicular surface isclearly dependent on the step length, δstep, and the allowed lengthof propagation, CL, since these have direct impact on the numberof points that need to be generated. Also, the amount of tearingin the surface and the maximum winding allowed have an impact.The performance of the algorithm, when generating the image infigure 7, is presented in figure 9 as a graph displaying the time ittakes to generate each surface when using different length limits.The step length is 0.005 m and the surface length limit for the ex-ample images is 0.1 m. Observe that since the number of points togenerate on a surface relates to the square of the surface radius, thedelay is close to linear in the log-log scale.

Because the surface is a generated 2D structure it will always beslower than generating a streamline, which is only a 1D structure.


6

7 CONCLUSIONS

Figure 7: This visualization of the SHARC CFD simulation shows both the convergence pattern (upper circle) and the vortex pattern (lowercircle), even though the streamlines used to seed the vector field perpendicular surfaces do not directly cross the centre of the vortex. Theorientation of the surface at any position shows the orientation of the flow, both around the vortices and at other locations in the flow.

Figure 8: This visualization of the Blunt Fin CFD data set showshow the change in flow direction at different levels displays in atwisted surface and that the curl causes the surface to tear. Thestream-tubes rendering provides a comparison with classical vectorvisualization.

0.001

0.01

0.1

1

10

0.01 0.1

T (s)

Surface length (m)

×××××××××××××××××××××××××××

×××××××××××××××××××××××××××××××××××

×××××××××××××××××××××××××××××××××××××××××

××××××××××××××××××××××××××××××××××××××××××××

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Figure 9: The time required to generate a single vector field perpen-dicular surface compared to the length limit of the surface, when thestep length is set to 0.005 m.

Even generating stream-tubes or stream-surfaces should have a timecomplexity of O(L), where L is the length limit. For the vector fieldperpendicular surfaces the length limit specifies the radius, so gen-erating such a surface will typically be done with a time complexityof O(L2).

7 Conclusions

While there exist a wide range of methods for vector field visu-alization, few of them produces a generally applicable representa-tion of the data with special focus on the changes across the ori-entation of the vectors in the field. The vector field perpendicularsurfaces introduced in this paper complement this array of meth-ods by rendering a surface across the vectors, with surface orienta-tion perpendicular to the field, a perpendicular analogue to stream-lines. The surface is generated in a structured manner, resulting in astraightforward triangulation, little need for caching and no searchfor neighbouring points. We have also shown that the implemen-


7

REFERENCES REFERENCES

tation provides stable rendering and is fast enough to be used forinteractive visual data exploration. The issue with errors in surfaceorientation inherent for regions with helicity in the vector field hasbeen partly solved using a stop criterion that results in a clear tearin the surface.

The examples and identified standard patterns show promising re-sults and potential use for certain situations, especially where thesearch for special properties is most effectively performed orthog-onal to the vectors of the data. The visual representation of thedata is, perhaps, not as intuitive as streamlines, but is straightfor-ward and provides a direct connection between the properties in thefield and the visual appearance. It should be noted, however, that tobe able to generate a perpendicular surface representation of vectordata with non-zero helicity density requires some orientation error.The amount of error that can be allowed without introducing a nega-tive impact on the interpretation of the data is suspected to be muchhigher than the perceptual limit and is a topic for future research.

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WEISKOPF, D. 2004. Dye advection without the blur: A level-setapproach for texture-basedvisualization of unsteady flow. Com-puter Graphics Forum (Proc. Eurographics) 23, 3, 479–488.

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