Farthest Point Seeding for Efcient Placement of Streamlines
Abdelkrim Mebarki∗
INRIA Sophia-Antipolis
Pierre Alliez†
INRIA Sophia-Antipolis
Olivier Devillers‡
INRIA Sophia-Antipolis
ABSTRACT
We propose a novel algorithm for placement of streamlines from
two-dimensional steady vector or direction elds. Our method con-
sists of placing one streamline at a time by numerical integration
starting at the furthest away from all previously placed streamlines.
Such a farthest point seeding strategy leads to high quality place-
ments by favoring long streamlines, while retaining uniformity with
the increasing density. Our greedy approach generates placements
of comparable quality with respect to the optimization approach
from Turk and Banks, while being 200 times faster. Simplicity,
robustness as well as efciency is achieved through the use of a
Delaunay triangulation to model the streamlines, address proxim-
ity queries and determine the biggest voids by exploiting the empty
circle property. Our method handles variable density and extends
to multiresolution.
Keywords: Streamline placement, farthest point seeding, Delau-
nay triangulation, variable density, multiresolution.
1 INTRODUCTION
Vector and direction elds are commonly used for modeling physi-
cal phenomena, where a direction and magnitude, or a vector is as-
signed to each point inside a domain. A typical example of a vector
eld is given by the direction, orientation and velocity of a steady
wind. Examples of direction elds include the principal curvature
directions of a smooth surface. In this paper a ow eld refers to
either a direction or to a vector eld.
Visual depiction of ow elds is mo-
tivated by the analysis and explo-
ration of results in scientic comput-
ing. One popular method consists of
choosing a set of samples through-
out the eld, and depicting associ-
ated arrow icons to present a view of
the direction, orientation and magni-
tude in a single picture (see inset).
The most delicate task of this tech-
nique lies into the choice of sample positions so as to best balance
between sparse sampling for clear depiction at the risk of missing
ne details, versus ne sampling at the risk of a cluttered visualiza-
tion. For high range vector elds where the magnitude cannot be
used to scale the arrow icons, the iconic representation is frequently
composed of unit vectors instead, combined with color coding to
depict magnitude. Other techniques have been developed to obtain
a denser depiction by imaging ow elds, where the images are ob-
tained by advection of a random noise image [2, 3, 4]. Conversely,
some techniques propose to extract only salient features of the ow
∗Abdelkrim.Mebarki@sophia.inria.fr
†Pierre.Alliez@sophia.inria.fr
‡Olivier.Devillers@sophia.inria.fr
Figure 1: Placement of streamlines with a uniform density (with a
tapering effect [1]). The black dots depict the seed points used for
numerical integration. Processing time: 160 ms on a 2GHz Pentium
IV.
and depict them as geometric icons to facilitate comprehensive vi-
sualization [5].
One of the most popular methods in ow visualization consists
of placing a set of streamlines which are always tangential to the
ow in order to emphasize the global eld coherency (see Fig. 1).
Beside, high quality placement of streamlines has recently proven
relevant for other applications such as non-photorealistic render-
ing [6, 7] or curve-based surface remeshing [8, 9]. We next give
a few denitions and an overview of existing techniques for high
quality placement of streamlines.
Definitions A streamline is a curve everywhere tangent to the
eld. A streamline can be considered as the path traced by an imag-
inary massless particle dropped into a steady uid ow described by
the eld. In practice, a streamline is often represented as a polyline
iteratively elongated by bidirectional numerical integration started
from a seed point, until it comes close to another streamline, hits
the domain boundary, reaches a critical point or generates a closed
path. A valid placement of streamlines is obtained by saturating
the domain with a set of tangential streamlines in accordance with
a specied density. A high quality streamline placement for visual-
ization has no formal denition, although it is admittedly related to
the uniformity of streamlines as well as to the spacing with respect
to the desired density. Moreover, long streamlines are preferred
to short ones, the goal being to better emphasize the global eld
temporal coherency. At the intuitive level, and as already pointed
in [10], most streamline terminations not coincident with the ow
eld singularities are perceived as articial singularities and thus
are potential distracters for an observer.