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my
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Research 
My
research is primarily in the field of Geometric Modeling.
My interests span the following areas.
 industrial
curve and surface applications
 mathematical definitions of shape
 Bspline
and Bezier methods, and more generally, NURBS
 classical
geometry in Computer Aided Geometric Design (CAGD)
 triangulations
 Voronoi
diagrams
 scientific
visualization
Please
visit my Publications page
to see a full listing of my articles and books.
Below
I have summarized my research.





Screen Mutations
Anamorphism project in collaboration with Louisa Zahareas
Video
Louisa was awarded a shortlist award from the 2015 London International Creative Competition
weheart blog post
dezeen magazine
fastcodesign
inhabitat



Agnostic G^1 Gregory Surfaces
We discuss G1 smoothness conditions for rectangular and triangular Gregory patches. We then incorporate these G1 conditions into a surface fitting algorithm. Knowledge of the patch type is inconsequential to the formulation of the G1 conditions, hence the term agnostic G1 Gregory surfaces.
On the left is a bicubic (rectangular) Gregory patch and a quartic (triangular) Gregory patch, both given a Bezierbased formulation.
The figure below illustrates the basic steps of the method. Left: the given data is point, normal, and a connectivity structure consisting of triangular and quadrilateral faces. Middle: cubic boundary curves and tangent ribbons are constructed for each face, resulting in G0 splinelike "guess" surface. Right: geometry parameters are fed into a linear system to force G1 continuity across interior edges.
G. Farin, D.Hansford, Agnostic G1 Gregory surfaces, Graphical Models, volume
74, pp. 346350, 2012. pdf
Please note that there is an error in the typsetting in the pdf on page 348 at the end of Section 4. The right hand side of the linear system should have only three rows. The "4th" row is part of the 3rd row.



Natural
Neighbor Extrapolation Using Ghost Points
Among
locally supported scattered data schemes, natural neighbor
interpolation has some unique features that makes it interesting
for a range of applications. However, its restriction to the
convex hull of the data sites is a limitation that has not
yet been satisfyingly overcome. We use this setting to discuss
some aspects of scattered data extrapolation in general, compare
existing methods, and propose a framework for the extrapolation
of natural neighbor interpolants on the basis of dynamic ghost
points.
This paper
serves as a survey of extrapolation methods as well, and towards
this effort, we extend on the classification of extrapolation
approaches that was introduced in a technical report by Peter
Alfeld in 1983.
T. Bobach, G. Farin, D. Hansford, G. Umlauf, Natural neighbor extrapolation using ghost points, CAD, volume
41, issue 5, pp. 350365, May 2009 pdf
Research
supported by NSF grant 0306385 "Splines over Iterated
Voronoi Diagrams" and the International Graduate School DFG
grant 1131 on "Visualization of Large and Unstructured
Data Sets  Applications in Geospatial Planning, Modeling
and Engineering".



DINUS
 Double Insertion, Nonuniform, Stationary Subdivision Surfaces
Double
insertion, nonuniform, stationary subdivision surfaces (DINUS)
generalize both nonuniform bicubic spline surfaces and CatmullClark
subdivision surfaces. DINUS allows arbitrary know intervals
on the edges, allows incorporation of special features, and
provides limit point as well as limit normal rules. It is
the firs subdivision scheme that gives the user all this flexibility
and at the same time all essentail limit information, which
is important for applicaiton in modeling and adaptive rendering.
DINUS is also amenable to analysis techniques for stationary
schemes. We implemened DINUS as an Autodesk Maya Plugin to
show several modeling and rendering examples.
K.
Mueller, C. Fuenfzig, L. Reusche, D. Hansford,
G. Farin, H. Hagen, DINUS  Double insertion, nonuniform, stationary subdivision surfaces, ACM Transactions
on Graphics, volume 29, number 3, pp. 121, 2010. pdf
Research
supported by the DFG (German NSF).





PNG1
Triangles for Tangent Plane Continuous Surfaces on the GPU
Improving
the visual appearance of course triangle meshes is usually
done with graphics hardware with perpixel shading techniques.
Improving the appearance at silhouettes is inherently hard,
as shading has only a small influence there and geometry must
be corrected. With the new geometry shader stage released
with DirectX 10, the functionality to generate new primatives
from an input primative is available. Also, the shader can
access a restricted primative neightborhood. In this paper,
we present a curved surface patch that can deal with the restricted
data available in the geometry shader. A surface patch is
defined over a triangle with its vertex normals the three
edge neighbor triangles. Compared to PN triangles, which define
a curved patch using just the triangle with its vertex normals,
our surface patch is G1 continuous with its three neighboring
patches. The patch is obtained by blending two cubic Bezier
patches for each triangle edge. In this way, our surface is
especially suitable for efficient, highquality tessellation
on the GPU.
Face figures
(left): The brown face is our PNG1 method and the right image
is the PN patches of Vlachos et al. Notice how the the nose
and mouth shapes differ. Bottom: Our G1 patches are clearly
seen on the left in comparison with the C0 PN patches on the
right.
K.
Mueller, C. Fuenfzig, G. Farin, and D. Hansford, PNG1 Patches
for Tangent Continuous Surfaces on the GPU, Graphics Interface,
pp. 119226, 2008. pdf
Research
supported by the DFG (German NSF).





Discrete
Harmonic Functions from Local Coordinates
In this
work we focus on approximations of continuous harmonic functions
by discrete harmonic functions based on the discrete Laplacian
in a triangulation of a point set. We show how the choice
of edge weights based on generalized barycentric coordinates
influences the approximation quality of discrete harmonic
functions. Furthermore, we consider a varying point set to
demonstrate that generalized barycentric coordinates based
on natural neighbors admit discrete harmonic functions that
continuously depend on the point set.
Figure.
Middle: given data and a Delaunay triangulation. Top: given
function values for z=x^2  y^2. Bottom: approximate solution
to Laplace equation using Sibson coordinates.
T.
Bobach, G. Farin, D. Hansford and G. Umlauf, Discrete Harmonic
Functions from Local Coordinates, Accepted to Mathematics
of Surfaces XII, Sheffield, UK, September 2007.pdf
This
work was supported by the international graduate school DFG
grant 1131 on ``Visualization of Large and Unstructured Data
Sets  Applications in Geospatial Planning, Modeling and Engineering''.
Farin and Hansford were supported by an NSF grant 0306385
``Splines over Iterated Voronoi Diagrams''.





Surface
Interrogation Methods for Haptic Rendering of Virtual Objects
The process
which enables virtual objects to mimic their real world counterparts
is known as realistic rendering in haptics. Realistic sensations
could relate to any spatial feature like shape or texture.
We have proposed a system here that aims at utilizing the
shape information of a surface effectively to aid in object
recognition through a haptic interface. This paper describes
some surface interrogation techniques namely isophotes, contours
and Gaussian curvature to assist in haptic rendering by drawing
the user's attention to certain features on a surface that
cannot be perceived by realistic means. The effectiveness
of these tools, based on their behavior in an external environment,
has also been compared. The main goal of this paper is to
demonstrate that perception of virtual surfaces can be enhanced
by providing haptic feedbacks parameterized according to geometric
features identified by surface interrogation.
Anusha
Sridaran, Dianne Hansford, Kanav Kahol, Sethuraman Panchanathan,
Surface Interrogation Methods for Haptic Rendering of Virtual
Objects, World Haptics Conference, pp. 237242, Second
Joint EuroHaptics Conference and Symposium on Haptic Interfaces
for Virtual Environment and Teleoperator Systems (WHC'07),
2007. pdf
This
work was supported by National Science Foundation Grant 0554698,
Incorporation of a psychological basis in the design of
haptic user interfaces, to ASU.





Anamorphic
3D Geometry
An anamorphic
image appears distorted from all but a few viewpoints. The
curious effects of anamorphs as they are known today, were
first understood and explored by Leonardo Da Vinci who included
anamorphic drawings of a child's head in his Codex Atlanticus
(ca 1485). The appearance of anamorphs as a consciously applied
technique in the history of art is nearly simultaneous with
the restoration of the study of perspective in the Renaissance
period (early fifteenth century) by artists and architects
such as F. Brunelleschi and L. Alberti.
Here we
describe a simple method for achieving anamorphs of 3D objects
by utilizing a simple projective map (collineation), wellknown
in the computer graphics literature that takes a frustum to
an `orthographic box'. The method presented here is equivalent
to the methods employed by Niceron (ca 1638) and his contemporaries.
The novelty of this work is the creation of anamorphic 3D
digital models, and the realization that a commonly known
map can be used to create anamorphs for 3D digital models.
Additionally, we present an analytic tool for artists and
architects.
Top figure
illustrates the original data set on the top row and the anamorphic
data set on the bottom row. In the left most column we have
the viewpoint where the data sets look identical. Bottom figure
is another view of the anamorph of George Washington. Did
he tell a lie?
D.
Hansford and D. Collins, Anamorphic 3D Geometry, accepted
to Computing, Special Issue on Geometric Modeling, Vol. 79,
Nos. 24, pp. 211223, 2007. pdf







Rapid
Prototyping Applications
We are
exploring several rapid prototyping applications
 assisting
blind individuals
 generation
of forms for a study in categorization
 affordability
measures
Under
construction!









Tactile
Urban Interface
Our tactile
urban interface makes it possible for city planners to navigate
a 20' x 20' physical model of the downtown Phoenix area. Rapid
prototyping is used to create a "human scale" model
of the city. This model model can be explored with a Polhemus
tactile digitizer which is connected to custom software which
drives content and lighting. A user selects a building with
the stylus, a lighting system highlights the selection in
the large physical model, and details concerning the selection
are projected on a screen nearby.
If a city
planner is not able to be at the physical location of the
model, a webbased navigation and information system is available
as well.
The large
model of Phoenix lives in the Phoenix
Urban Research Lab (PURL)
Dianne
Hansford, Dan Collins, Ruth Ron, Yoshi Kobiyashi, John McIntosh,
Karen Bullis, Al Simmon
Presented
at SIGGRAPH, Boston, August 2006
The
project is partially funded by The College of Design at ASU.





Interactive
Topographical Interface  Tactile Topo Travel
Illustrated
on the lefttop is a rapid prototype model of the digital
elevation model of an area surrounding Telluride, CO. This
model is being explored with a Microscribe tactile digitizer
whereby the coordinate output is sent to a Director program
illustrated leftbottom. The location of the digitizers tip
is represented by a red ball that moves in the 3D terrain
model. Selectable 'hot spots' are indicated by flags. When
the user selects a hot spot feature such as a mountain top,
the 3D flag changes color, the topo map appears, and associated
information appears.
This type
of tool would be ideal in a visitors center, where users could
add their experiences during hiking trips to the database.
This tool
is also available as a standalone webbased tool. The user
navigates the 3D terrain model with the mouse.
Demo:
http://www.ruthron.com/tactiletopo/index.htm
Dan Collins, Ruth Ron, Dianne Hansford





The
Visible City
Using
Augmented Reality, Mobile Computing, and 3D Simulated X Ray
Models to Visualize and Navigate Downtown San Jose Proposal
This San
Jose urban augmented reality interface experiments with the
ways virtual reality can enrich our experience strolling through
the city. It overlaps local information over the existing
urban fabric, and extends reality by allowing the user to
'see through' buildings. The project uses Augmented Reality
(AR), Mobile Computing, GPS, and 3D Simulated XRay Models
to visualize and navigate the urban core of San Jose. An 'augmented
reality' kit is mounted on the user's head. Using a GPS navigation
system to detect position, the 3D model of the city is projected
onto a translucent flexible display. The user can 'see through'
buildings in a 'wireframe' mode, and browse for local information.
Demo:
http://ruthron.com/ISEA/
Ruth
Ron, Dan Collins and Dianne Hansford
Submitted
to the ISEA conference in San Jose, 2006





Volume
Deformations in Action
A Forensic Reconstruction of George Washington
To commemerate
George Washington's 250th anniversery of fighting in the French
and Indian War, the Mt. Vernon Society commissioned three
lifesized statues of GW at age 19 (surveying as an officer
in the British army), 45 (on a horse leading the revolutionary
army), and 57 (being sworn in as President). The problem:
all the hard evidence as to his appearance is at age 53. Portraits
are not particularly useful due to the large variation in
his depiction; See the figure below for several example. Further,
GW began to loose his teeth at age 20, and severe bone loss
occured thus changing his facial structure over the years.
Using
the given hard evidence, the challenge of this project was
to extrapolate in both directions  create a younger and
older GW. The center piece to this task was a volume deformation
tool. In the top figure, the Bspline deformation tool is
illustrate along with the face mask, mandible, and denture.
The deformation tool was used modify a mandible to fit GW
dentures, and then the face mask was modified to fit the mandible.
The resulting head models are illustrated to the left, and
from top to bottom are ages 19, 45, 57.
The volume
deformation tool was constructed to have controls that suited
physical anthropologists. Additionally, fine control features
were added.
D.
Collins, G. Cooper, G. Farin, J. Hansen, Dianne Hansford,
A. Razdan, J. Schwartz, Matt Tocheri, S. Van Note
Flash
presentation describing the project.
This
project was supported by The Mount Vernon Society
This
work has been written about in CNN, in The New York
Times, in Scientific American (February 2006),
and featured in a History Channel show (February 17,
2007).
PRISM
featured in a video clip





Pictures
courtesy the Mt. Vernon Society and The History Channel





Digital
Cloud Photogrammetry
Cumulus
Photogrammetric, InSitu and Doppler Observations (CuPIDO)
is an observational program designed to examine the onset
and development of orographic thunderstorms associated with
the North American Monsoon (NAM). The CuPIDO field program
uses digital visible spectrum cameras, surface mesonet stations,
high temporal resolution soundings, and aircraft data. The
field study discussed in this manuscript takes place in the
vicinity of the Santa Catalina Mountains, north of Tucson,
Arizona.
In this
manuscript, we describe the 2D and 3D cloud modeling aspects
of CuPIDO. We have created automated methods for identifying
orographic cumulus development from stereo pairs of digital
images. We present image analysis methods for tracking cumulus
development, and we present 3D modeling methods for cloud
reconstruction and measurement.
J.
Zehnder, J. Rowe, A. Razdan, J. Hu, D. Hansford, Using
Digital Cloud Photogrammetry to Characterize the Onset and
Transition from Shallow to Deep Convection Over Orography,
Monthly Weather Review, Volume 134, pp. 25272546, September
2006.
This
author was supported by NSF grant ATM 0352988





Second
Order Tangent Estimation with Conic Precision
Curve
interpolation to given data points many times necessitates
tangent vectors at the data to be determined. Pascal's theorem
is at the core of the development of a tangent estimation
method for planar, convex data points. As a result, the tangents
are precise for data from a conic. The tangents from this
method are compared to classical methods. The simplicity,
accuracy and efficiency of the method contribute to its usefulness.
Top figure
illustrates the construction of a tangent at p3, given p1,
p2, p3, p4, p5.
Bottom figure illustrates Pappus' theorem.
G.
Albrecht, J.P. Bécar, G. Farin, and D. Hansford. Détermination
de tangentes par l'emploi de coniques d'approximation,
Revue Internationale de CFAO et d'informatique graphique,
1(1): 91103, 2005.
G.
Albrecht, J.P. Bécar, G. Farin, D. Hansford. On the approximation
order of tangent estimators, Computer Aided Geometric
Design, volume 25, pp 8095, 2008. pdf
This
research was supported by Labratory MACS, University of Valenciennes,
France.





Discrete
Coons Patches
We investigate
surfaces which interpolate given boundary curves. We show
that the discrete bilinearly blended Coons patch can be defined
as the solution of a linear system. With the goal of producing
better shape than the Coons patch, this idea is generalized,
resulting in a new method based on a blend of variational
principles. We show that no single blend of variational principles
can produce ``good" shape for all boundary curve geometries.
We also discuss triangular Coons patches and point out the
connections to the rectangular case.
Figures
to the left: Gray points are constructed from the black boundary
curves.
Top: Coons, Bottom: an optimal shape for these boundary curves.
G.
Farin and D. Hansford. Discrete Coons patches, Computer
Aided Geometric Design, 16(7) pages 691700, 1999. pdf





Building
Boundary Curves with Quadric Precision
We describe
a method for constructing rational quadratic patch boundary
curves to data in R^3. The method has quadric boundary precision;
if the given point and normal data are extracted from a quadric,
then the boundary curves will lie on this quadric. Each boundary
curve is a conic section represented in quadratic rational
Bezier form.
D.
Hansford, R.E. Barnhill and G. Farin. Curves with quadric
boundary precision, Computer Aided Geometric Design, 11(4),
pages 113, 1994. pdf
This
work was supported by DOE grant DEFG0287ER25041 and NSF grant
DNC9907747 and a Fulbright Junior Research grant.





Bézier
Patches on Quadrics
Quadric
surfaces are a very basic surface type; they commonly appear
in our world. For example, architectural designs occasionally
use quadrics for functionality and beauty. In electronic engineering,
satellite dishes are commonly shaped as circular paraboloids
so that all signals that arrice are directed toward the receiving
device. The solid modeling community also applies quadrics
quite often. However, in freeform design quadrics are not
used frequently. This is most likely atributed to the fact
that the properties of quadrics must be understood first in
order to transform them into a parametric form.
This paper
examines the parametric representation of quadrics with BernsteinBezier
rational quadratic triangular patches and rational biquadratic
rectangular patches from a geometric viewpoint. If we only
consider within teh boundary curves, then rational quadratic
and biquadratic patches will allow only special regions of
a quadric to be represented; these will be investigated geometrically.
W.
Boehm and D. Hansford. Bézier patches on quadrics,
in NURBS for Curve and Surface Design, ed. G. Farin, SIAM,
pages 114, 1991.
W.
Boehm and D. Hansford. Parametric representation of quadric
surfaces, Mathematical Modelling and Numerical Analysis, 26(1),
pages 191200, 1991.
This
work was supported by DOE grant DEFG0287ER25041 and NSF grant
DMC8807747





The
Neutral Case for the Minmax Triangulation
Choosing
the best triangulation of a point set is a question that has
been debated for many years. Two of the most well known choices
are the minmax criterion and the maxmin criterion. The maxmin
triangulation criterion has received the most attention over
the years because efficient algorithms have been develped
for determining this triangulation. The ability to construct
such efficient algorithms has been shown to be a result of
the geometry of the neutral set for the maxmin criterion.
A point for the neurtral set is formed for the special instance
when the criterion is satisfied by more than one triangulation.
For the maxmin criterion, the neutral set is a circle. In
this paper, we construct the neutral set for the minmax criterion.
This construction is compared to that of the maxmin triangulation
and the results are analyzed in order to attain a better understanding
of the nature of the minmax criterion.
D.
Hansford. The neutral case for the minmax triangulation,
Computer Aided Geometric Design, 7(5), pages 431438, 1990.
pdf
This
work was supported by DOE grant DEFG0287ER25041 and NSF grant
DNC9907747







Gammaspline
Interpolation
We derive
a natural extension of Boehm's freeform gammaspline, the
G^2 interpolating gammaspline. Primarily, the conditions
under which singularities in the spline formulation occur
are investigated. Also, the effect that these singularities
have on the interpolant are studied. Comparisons are made
to the behavior of the interpolating nuspline.
G.
Farin, D. Hansford and A. Worsey. The singular cases for
gammaspline interpolation, Computer Aided Geometric Design,
7(6), pages 533546, 1990. pdf
This
work was supported by DOE grant DEFG0287ER25041 and NSF grant
DNC9907747





Much of
my research may not be found in my publications: from 19921994
I worked at Manufacturing and Consulting Services (MCS), and
from 19942000 I had a consulting and software development
firm, NURBS Depot. All of my clients demanded confidentiality
agreements, thus restricting my ability to publish. Below
I have listed in general terms, the type of work I have done.
 Adaptive
triangulation of NURBS surfaces: Discretize (triangulate)
a NURBS surface adaptively for faster rendering and analysis
 Data
format conversions: Bezier, Bspline, Hermite, NURBS,
conics, IGES, etc....
 Utility
functions: Just about any NURBS or Bezier utility function
you can think of! (evaluation, subdivision, degree elevation,
etc)
 Milling
tool size detection: Analysis of surface curvatures
and features for the purpose of determining the appropriate
cutting tool radius (radii).
 Electronic
cam: Define instructions necessary to simulate cam action
in the cutting of straight and helix fluted taps
 Automatic
parametric design of a turbine blade: Create a software
library to automate the turbine blade design process so
that a given set of 2D profile curves uniquely defines the
3D geometry as a C2 NURBS surface
 Surface
extension: Extend a surface linearly or with continuous
curvature
 Match
edge tangency: master/slave surface smoothness correction
(position or curvature continuity)
 Satellite
dish positioning: Determine the azimuth angle given
a satellite's location
 Tooth
modeling: Mathematical modeling of the bracket area
of teeth for the purpose of analysis over population groups
 better braces design
 Prosthetic
modeling: Mathematical modeling of a prosthetic given
a scanned body part (knee area, ankle or midfoot)
 Tactile
laser scanner software:
NURBS modeling of surfaces scanned by a tactile laser scanner.
 Bifocal/Trifocal
lens design: Mathematical models of lenses for photorealistic
display and manufacturing
 Custom
shoe software: Scanned foot data would be converted
to a custom last. We worked with some of the finest last
designers to develop a feasible instore scanning, last
creation, and shoe making technology.
Please
visit FarinHansford.com
for more information.








