Dianne Hansford, Ph. D.
Contact Information:
Tel: 480-703-0263

Consulting Services
my books


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
  • B-spline 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

Screen Mutations

Anamorphism project in collaboration with Louisa Zahareas


Louisa was awarded a shortlist award from the 2015 London International Creative Competition

we-heart blog post
dezeen magazine


Gregory Control Points

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 Bezier-based 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 spline-like "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. 346-350, 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.

Agnostic Gregory Example


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. 350-365, 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, Non-uniform, Stationary Subdivision Surfaces

Double insertion, non-uniform, stationary subdivision surfaces (DINUS) generalize both non-uniform bicubic spline surfaces and Catmull-Clark 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, non-uniform, stationary subdivision surfaces, ACM Transactions on Graphics, volume 29, number 3, pp. 1-21, 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 per-pixel 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, high-quality 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. 119-226, 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. 237-242, 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), well-known 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. 2-4, pp. 211-223, 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 web-based 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 left-top 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 left-bottom. 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 stand-alone web-based 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 X-Ray 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 'wire-frame' 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 life-sized 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 B-spline 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, In-Situ 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. 2527-2546, 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): 91-103, 2005.

G. Albrecht, J.P. Bécar, G. Farin, D. Hansford. On the approximation order of tangent estimators, Computer Aided Geometric Design, volume 25, pp 80-95, 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 691-700, 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 1-13, 1994. pdf

This work was supported by DOE grant DE-FG0287ER25041 and NSF grant DNC-9907747 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 free-form 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 Bernstein-Bezier 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 1-14, 1991.

W. Boehm and D. Hansford. Parametric representation of quadric surfaces, Mathematical Modelling and Numerical Analysis, 26(1), pages 191-200, 1991.

This work was supported by DOE grant DE-FG0287ER25041 and NSF grant DMC-8807747


The Neutral Case for the Min-max 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 min-max criterion and the max-min criterion. The max-min 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 max-min 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 max-min criterion, the neutral set is a circle. In this paper, we construct the neutral set for the min-max criterion. This construction is compared to that of the max-min triangulation and the results are analyzed in order to attain a better understanding of the nature of the min-max criterion.

D. Hansford. The neutral case for the min-max triangulation, Computer Aided Geometric Design, 7(5), pages 431-438, 1990. pdf

This work was supported by DOE grant DE-FG0287ER25041 and NSF grant DNC-9907747


Gamma-spline Interpolation

We derive a natural extension of Boehm's free-form gamma-spline, the G^2 interpolating gamma-spline. 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 nu-spline.

G. Farin, D. Hansford and A. Worsey. The singular cases for gamma-spline interpolation, Computer Aided Geometric Design, 7(6), pages 533-546, 1990. pdf

This work was supported by DOE grant DE-FG0287ER25041 and NSF grant DNC-9907747


Much of my research may not be found in my publications: from 1992-1994 I worked at Manufacturing and Consulting Services (MCS), and from 1994-2000 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, B-spline, 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 in-store scanning, last creation, and shoe making technology.

Please visit FarinHansford.com for more information.