August 18, Thursday 11:00, Room 570, Education Building
A Complex View of Barycentric Coordinates with Applications to Planar Shape Deformation
Lecturer: Ofir Weber
Affiliation : Courant Institute of Mathematical Sciences, New York University
Barycentric coordinates are heavily used in computer graphics
applications to generalize a set of given data values. Traditionally,
the coordinates are required to satisfy a number of key properties,
the first being that they are real. In this work we relax this
requirement, allowing the barycentric coordinates to be complex
numbers. This allows us to generate new families of barycentric
coordinates, which have some powerful advantages over traditional ones
and are especially useful for creating detail-preserving planar shape
We first use Cauchy.s theorem to construct complex barycentric
coordinates that can be used to generate holomorphic functions. Such
functions can be interpreted as conformal planar mappings as long as
the derivative of the function doesn.t vanish.
We then construct another type of complex barycentric coordinates
based on the Hilbert transform. Combined with a novel 2D shape
deformation system, we show how to generate .foldovers free. pure
conformal planar deformations. Beyond deforming a given shape into a
new one at each key frame, our method also provides the ability to
interpolate between shapes in a very natural way, such that also the
intermediate deformations are conformal.
Finally, we extend and generalize these results by investigating the
complex representation of real-valued barycentric coordinates, when
applied to planar domains. Furthermore, we show that a complex
barycentric map admits the intuitive interpretation as a
complex-weighted combination of edge-to-edge similarity
transformations, allowing the design of ``home-made'' barycentric maps
with desirable properties. Thus, using the tools of complex analysis,
we provide a methodology for analyzing existing barycentric mappings,
as well as designing new ones.
Joint work with: Craig Gotsman, Kai Hormann and Mirela Ben-Chen`
Ofir Weber received his Ph.D. from the Department of Computer Science,
Technion in 2010.
He is currently a postdoc researcher at the Courant Institute of
Mathematical Sciences, New York University.
His main research interests are theoretical and computational methods
in geometry and their applications to problems in computer graphics,
shape/image deformation, parameterization and quadrilateral remeshing.