Tom Duchamp | University of Washington (original) (raw)

Papers by Tom Duchamp

Journal für die reine und angewandte Mathematik (Crelles Journal)

We use Hitchin’s twistor transform for two-dimensional projective structures to obtain normal coo... more We use Hitchin’s twistor transform for two-dimensional projective structures to obtain normal coordinates in a pseudoconcave neighbourhood of an O ⁢ ( 1 ) \mathcal{O}(1) rational curve; in the construction, we present every such neighbourhood as Q D / F Q_{\mathbb{D}}/\mathcal{F} for some holomorphic foliation ℱ, where Q D Q_{\mathbb{D}} is an open neighbourhood in the standard quadric Q ⊂ P 2 × P 2 Q\subset\mathbb{P}^{2}\times\mathbb{P}^{2} . As a consequence of the normal coordinates, we obtain a new normal form for Cauchy Riemann structures on the three-sphere that are isotopic to the standard one. We end the paper with explicit calculations for the cases arising from deformations of the normal isolated singularities X ⁢ Y = Z n XY=Z^{n} .

arXiv (Cornell University), Mar 26, 2001

This paper generalizes results of Lempert and Szoke on the structure of the singular set of a sol... more This paper generalizes results of Lempert and Szoke on the structure of the singular set of a solution of the homogeneous complex Monge-Ampere equation on a Stein manifold. Their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution. In addition, their requirement that the square of the solution be C3C^3C3 everywhere is replaced by a smoothness condition on the blowup of the singular set. Under these conditions, the singular set is shown to inherit a Finsler metric, which in the real analytic case uniquely determines the solution of the Monge-Ampere equation. These results are proved using techniques from contact geometry.

Journal of Differential Geometry, 1979

Contemporary Mathematics, 1989

This paper presents a framework for the skeleton-driven animation of elastically deformable chara... more This paper presents a framework for the skeleton-driven animation of elastically deformable characters. A character is embedded in a coarse volumetric control lattice, which provides the structure needed to apply the finite element method. To incorporate skele-tal controls, we introduce line constraints along the bones of sim-ple skeletons. The bones are made to coincide with edges of the control lattice, which enables us to apply the constraints efficiently using algebraic methods. To accelerate computation, we associate regions of the volumetric mesh with particular bones and perform locally linearized simulations, which are blended at each time step. We define a hierarchical basis on the control lattice, so for detailed interactions the simulation can adapt the level of detail. We demon-strate the ability to animate complex models using simple skeletons and coarse volumetric meshes in a manner that simulates secondary motions at interactive rates.

CR-Geometry and Overdetermined Systems

Journal of Differential Equations, 1984

The problem of constructing variational principles for a given second-order quasi-linear partial ... more The problem of constructing variational principles for a given second-order quasi-linear partial differential equation is considered. In particular, we address the problem of finding a first-order functionf whose product with the given differential operator is the Euler-Lagrange operator derived from some Lagrangian. Two sets of equations for such a function f are obtained. Necessary and sufftcient conditions for the integration of the first set are established in general and these lead to a considerable simplification of the second set. In certain special cases. such as the case when the operator is elliptic, the problem is completely solved. The utility of our results is illustrated by a variety of examples.

Annales de l’institut Fourier, 1981

This paper generalizes results of Lempert and Szöke on the structure of the singular set of a sol... more This paper generalizes results of Lempert and Szöke on the structure of the singular set of a solution of the homogeneous Monge-Ampère equation on a Stein manifold. Their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution. In addition, their requirement that the square of the solution be C3 everywhere is replaced by a smoothness condition on the blowup of the singular set. Under these conditions, the singular set is shown to inherit a Finsler metric, which in the real analytic case uniquely determines the solution of the Monge-Ampère equation. These results are proved using techniques from contact geometry. Mathematics Subject Classification (2000): 53C56, 32F, 53C60

Totally real immersions of an n-dimensional smooth manifold M into C exist, provided that the com... more Totally real immersions of an n-dimensional smooth manifold M into C exist, provided that the complexified tangent bundle of M is trivial. A bijection between the set of isotopy classes of such immersions and the complex K-group K(M) is constructed. Gromov [4] and Lees [7] have given a homotopy classification of totally real and Lagrangian immersions into complex and symplectic manifolds. Our aim here is to show that if the codomain in C then this classification has a simple K-theoretic discription. We begin with a discussion of some elementary facts in linear algebra. Let V (resp W be an n-dimensional real (resp. complex) vector space. An R-linear injection h : V →W is called totally real if its image h(V ) contains no non-trivial complex subspace. Let V C denote the complexification of V and let hC : V C → W be the complex lienar map defined by the formula hC(u + iv) = h(u) + ih(v). It is easily verified that h is a totally real injection if and only if hC is a compelx vector spac...

The Annals of Statistics, 1996

University of Washington Principal curves were introduced to formalize the notion of "a curv... more University of Washington Principal curves were introduced to formalize the notion of "a curve passing through the middle of a dataset." Vaguely speaking, a curve is said to pass through the middle of a dataset if every point on the curve is the average of the observations ...

Proceedings of the …, 2003

We present a view-dependent level-of-detail algorithm for triangle meshes with subdivision connec... more We present a view-dependent level-of-detail algorithm for triangle meshes with subdivision connectivity. The algorithm is more suitable for textured meshes of arbitrary topology than existing progressive mesh-based schemes. It begins with a wavelet decomposition of the mesh, and, per frame, finds a partial sum of wavelets necessary for high-quality renderings from that frame's viewpoint. We present a screen-space error metric that measures both geometric and texture deviation and tends to outperform prior error metrics developed for progressive meshes. In addition, wavelets that lie outside the view frustum or in backfacing areas are eliminated. The algorithm takes advantage of frame-to-frame coherence for improved performance and supports geomorphs for smooth transitions between levels of detail.

Proceedings of the …, 2000

A surface light field is a function that assigns a color to each ray originating on a surface. Su... more A surface light field is a function that assigns a color to each ray originating on a surface. Surface light fields are well suited to constructing virtual images of shiny objects under complex lighting conditions. This paper presents a framework for construction, compression, interactive rendering, and rudimentary editing of surface light fields of real objects. Generalizations of vector quantization and principal component analysis are used to construct a compressed representation of an object's surface light field from photographs and range scans. A new rendering algorithm achieves interactive rendering of images from the compressed representation, incorporating view-dependent geometric level-of-detail control. The surface light field representation can also be directly edited to yield plausible surface light fields for small changes in surface geometry and reflectance properties.

Proceedings of the …

Three-dimensional meshes of large complexity are rapidly becom- ing commonplace. Laser scanning s... more Three-dimensional meshes of large complexity are rapidly becom- ing commonplace. Laser scanning systems, for example, routinely produce geometric models with hundreds of thousands of vertices, each of which may contain additional information, such as color. Working with ...

Proceedings of the …

1 Introduction In this paper, we present a new representation for piecewise smooth surfaces of ar... more 1 Introduction In this paper, we present a new representation for piecewise smooth surfaces of arbitrary topological type, 1 and a method for fitting such surface models to scattered range data, where neither the topolog- ical type of the surface, its geometry, nor the location of ...

Banach Center Publications

A normal form for small CR-deformations of the standard CR-structure on the (2n + 1)-sphere is pr... more A normal form for small CR-deformations of the standard CR-structure on the (2n + 1)-sphere is presented. The space of normal forms is parameterized by a single function on the sphere. For n > 1, the normal form is used to obtain explicit embeddings into C n+1. For n = 1, the cohomological obstruction to embeddability is identified.

Illinois Journal of Mathematics

ABSTRACT The Cartan equivalence problem for non-holomorphic foliations of complex surfaces is sol... more ABSTRACT The Cartan equivalence problem for non-holomorphic foliations of complex surfaces is solved in this paper by constructing a natural trivialization of the tangent bundle of the bundle of frames adapted to the foliation. This trivialization can be interpreted as an sl(3,ℝ)-valued Cartan connection. It is shown that the foliation of the complement of ℝℙ 2 in ℂℙ 2 by the set of complex lines which intersect ℝℙ 2 in a real line is flat and has SL(3,ℝ) as its group of automorphisms.

Journal für die reine und angewandte Mathematik (Crelles Journal)

We use Hitchin’s twistor transform for two-dimensional projective structures to obtain normal coo... more We use Hitchin’s twistor transform for two-dimensional projective structures to obtain normal coordinates in a pseudoconcave neighbourhood of an O ⁢ ( 1 ) \mathcal{O}(1) rational curve; in the construction, we present every such neighbourhood as Q D / F Q_{\mathbb{D}}/\mathcal{F} for some holomorphic foliation ℱ, where Q D Q_{\mathbb{D}} is an open neighbourhood in the standard quadric Q ⊂ P 2 × P 2 Q\subset\mathbb{P}^{2}\times\mathbb{P}^{2} . As a consequence of the normal coordinates, we obtain a new normal form for Cauchy Riemann structures on the three-sphere that are isotopic to the standard one. We end the paper with explicit calculations for the cases arising from deformations of the normal isolated singularities X ⁢ Y = Z n XY=Z^{n} .

arXiv (Cornell University), Mar 26, 2001

This paper generalizes results of Lempert and Szoke on the structure of the singular set of a sol... more This paper generalizes results of Lempert and Szoke on the structure of the singular set of a solution of the homogeneous complex Monge-Ampere equation on a Stein manifold. Their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution. In addition, their requirement that the square of the solution be C3C^3C3 everywhere is replaced by a smoothness condition on the blowup of the singular set. Under these conditions, the singular set is shown to inherit a Finsler metric, which in the real analytic case uniquely determines the solution of the Monge-Ampere equation. These results are proved using techniques from contact geometry.

Journal of Differential Geometry, 1979

Contemporary Mathematics, 1989

This paper presents a framework for the skeleton-driven animation of elastically deformable chara... more This paper presents a framework for the skeleton-driven animation of elastically deformable characters. A character is embedded in a coarse volumetric control lattice, which provides the structure needed to apply the finite element method. To incorporate skele-tal controls, we introduce line constraints along the bones of sim-ple skeletons. The bones are made to coincide with edges of the control lattice, which enables us to apply the constraints efficiently using algebraic methods. To accelerate computation, we associate regions of the volumetric mesh with particular bones and perform locally linearized simulations, which are blended at each time step. We define a hierarchical basis on the control lattice, so for detailed interactions the simulation can adapt the level of detail. We demon-strate the ability to animate complex models using simple skeletons and coarse volumetric meshes in a manner that simulates secondary motions at interactive rates.

CR-Geometry and Overdetermined Systems

Journal of Differential Equations, 1984

The problem of constructing variational principles for a given second-order quasi-linear partial ... more The problem of constructing variational principles for a given second-order quasi-linear partial differential equation is considered. In particular, we address the problem of finding a first-order functionf whose product with the given differential operator is the Euler-Lagrange operator derived from some Lagrangian. Two sets of equations for such a function f are obtained. Necessary and sufftcient conditions for the integration of the first set are established in general and these lead to a considerable simplification of the second set. In certain special cases. such as the case when the operator is elliptic, the problem is completely solved. The utility of our results is illustrated by a variety of examples.

Annales de l’institut Fourier, 1981

This paper generalizes results of Lempert and Szöke on the structure of the singular set of a sol... more This paper generalizes results of Lempert and Szöke on the structure of the singular set of a solution of the homogeneous Monge-Ampère equation on a Stein manifold. Their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution. In addition, their requirement that the square of the solution be C3 everywhere is replaced by a smoothness condition on the blowup of the singular set. Under these conditions, the singular set is shown to inherit a Finsler metric, which in the real analytic case uniquely determines the solution of the Monge-Ampère equation. These results are proved using techniques from contact geometry. Mathematics Subject Classification (2000): 53C56, 32F, 53C60

Totally real immersions of an n-dimensional smooth manifold M into C exist, provided that the com... more Totally real immersions of an n-dimensional smooth manifold M into C exist, provided that the complexified tangent bundle of M is trivial. A bijection between the set of isotopy classes of such immersions and the complex K-group K(M) is constructed. Gromov [4] and Lees [7] have given a homotopy classification of totally real and Lagrangian immersions into complex and symplectic manifolds. Our aim here is to show that if the codomain in C then this classification has a simple K-theoretic discription. We begin with a discussion of some elementary facts in linear algebra. Let V (resp W be an n-dimensional real (resp. complex) vector space. An R-linear injection h : V →W is called totally real if its image h(V ) contains no non-trivial complex subspace. Let V C denote the complexification of V and let hC : V C → W be the complex lienar map defined by the formula hC(u + iv) = h(u) + ih(v). It is easily verified that h is a totally real injection if and only if hC is a compelx vector spac...

The Annals of Statistics, 1996

University of Washington Principal curves were introduced to formalize the notion of "a curv... more University of Washington Principal curves were introduced to formalize the notion of "a curve passing through the middle of a dataset." Vaguely speaking, a curve is said to pass through the middle of a dataset if every point on the curve is the average of the observations ...

Proceedings of the …, 2003

We present a view-dependent level-of-detail algorithm for triangle meshes with subdivision connec... more We present a view-dependent level-of-detail algorithm for triangle meshes with subdivision connectivity. The algorithm is more suitable for textured meshes of arbitrary topology than existing progressive mesh-based schemes. It begins with a wavelet decomposition of the mesh, and, per frame, finds a partial sum of wavelets necessary for high-quality renderings from that frame's viewpoint. We present a screen-space error metric that measures both geometric and texture deviation and tends to outperform prior error metrics developed for progressive meshes. In addition, wavelets that lie outside the view frustum or in backfacing areas are eliminated. The algorithm takes advantage of frame-to-frame coherence for improved performance and supports geomorphs for smooth transitions between levels of detail.

Proceedings of the …, 2000

A surface light field is a function that assigns a color to each ray originating on a surface. Su... more A surface light field is a function that assigns a color to each ray originating on a surface. Surface light fields are well suited to constructing virtual images of shiny objects under complex lighting conditions. This paper presents a framework for construction, compression, interactive rendering, and rudimentary editing of surface light fields of real objects. Generalizations of vector quantization and principal component analysis are used to construct a compressed representation of an object's surface light field from photographs and range scans. A new rendering algorithm achieves interactive rendering of images from the compressed representation, incorporating view-dependent geometric level-of-detail control. The surface light field representation can also be directly edited to yield plausible surface light fields for small changes in surface geometry and reflectance properties.

Proceedings of the …

Three-dimensional meshes of large complexity are rapidly becom- ing commonplace. Laser scanning s... more Three-dimensional meshes of large complexity are rapidly becom- ing commonplace. Laser scanning systems, for example, routinely produce geometric models with hundreds of thousands of vertices, each of which may contain additional information, such as color. Working with ...

Proceedings of the …

1 Introduction In this paper, we present a new representation for piecewise smooth surfaces of ar... more 1 Introduction In this paper, we present a new representation for piecewise smooth surfaces of arbitrary topological type, 1 and a method for fitting such surface models to scattered range data, where neither the topolog- ical type of the surface, its geometry, nor the location of ...

Banach Center Publications

A normal form for small CR-deformations of the standard CR-structure on the (2n + 1)-sphere is pr... more A normal form for small CR-deformations of the standard CR-structure on the (2n + 1)-sphere is presented. The space of normal forms is parameterized by a single function on the sphere. For n > 1, the normal form is used to obtain explicit embeddings into C n+1. For n = 1, the cohomological obstruction to embeddability is identified.

Illinois Journal of Mathematics

ABSTRACT The Cartan equivalence problem for non-holomorphic foliations of complex surfaces is sol... more ABSTRACT The Cartan equivalence problem for non-holomorphic foliations of complex surfaces is solved in this paper by constructing a natural trivialization of the tangent bundle of the bundle of frames adapted to the foliation. This trivialization can be interpreted as an sl(3,ℝ)-valued Cartan connection. It is shown that the foliation of the complement of ℝℙ 2 in ℂℙ 2 by the set of complex lines which intersect ℝℙ 2 in a real line is flat and has SL(3,ℝ) as its group of automorphisms.