What's New at the SGP Web

The Hybrid Level Surfaces section explores two-parmaeter families of level surfaces obtained by mixing two terms using one parameter, and adding a constant offset using a second parameter. Such experiments are used to generate candidate models for interfacial surfaces in block copolymer systems.

The Models section was created for subjects involving the application of geometry in modelling physical systems, as distinguished from pure geometry, which is treated in the Geometry section.

The Morphologies of Block Copolymers page in the models section provides an overview of the nanoscale structures exhibited by block copolymers as a function of the volume fractions of the blocks.

The Surfaces Overview section was created for subjects relating to surface geometry but not specific to any one of the three surface categories ( minimal, CMC, and level).

The Skeletal Graphs of Selected Surfaces page illustrates a property of surfaces which, although difficult to define in a precise mathematical way, is nonetheless useful in understanding surface geometry.

The Level Set Families page explains how any implicit surface belongs to a family of surfaces parameterized by a constant in the equation.

The Skeletal Graphs Approximated by Level Surfaces documents a recent discovery that simple combinations of the level set formulae for the P, D, G, and I-WP surfaces can be used to approach the skeletal graphs of the same surfaces.

The Sphere in R4 page provides a series of animations to help the reader understand the sphere in R4 by way of analogy to the sphere in R3.

A new 3-D visual construction menu implemented in VRML2 allows models of surfaces composed of translated and/or rotated fundamental pieces to be assembled and dis-assembled by switching on and off slices in the solid superimposed over the surface. Some examples are in the VRML models in the Mesh client pages for: infendk, fourendk, ktower, and schwarz. (Once in the VRML viewer, click on the surface to activate the construction menu.)

The TEMsim Volume Projections page has been reorganized and expanded to include projections from 012, 112, and 122 Miller indices.

The Lamellar Catenoid Surface Family page illustrates a family of triply periodic surfaces which, unlinke the P, G, and D surfaces, consists of topologically disjoint surfaces.

The Triply Periodic Level Surfaces Animations page contains both MPEG rotation movies and VRML2 models to aid the visualization of some common triply periodic level surfaces.

The Curvature of Selected Level Surfaces page compares mean and gauss curvature of the P, G, and D level surfaces, allowing interactive selection of viewpoint and color scheme.

The Karcher-Scherk Tower Surface Families page is one of several new minimal surfaces which have been added, along with pages for the Mesh clients used to generate them. Also added were the The Karcher-Polthier-Thayer Surface Families and The Wohlgemuth-Thayer Surface Families pages. These pages all contain MPEG movies showing deformation through the real parameters.

The P CMC Surface Family page has movies of the P CMC surface going from near one critical point in the family to near the other one. At the critical point, necks pinch off, changing the topology of the surface to a lattice of spheres.

The D CMC Surface Family page has movies of the D CMC surface going from near one critical point int the family to near the other one. Unfortunately, the movie has gaps reflecting problems in getting the software (VPS/render) to properly handle the torus wrapped domain. A more robust method will exploit symmetries to perform the simulation without using the torus domain.

The P CMC Surface at Specified Volume Fractions page contains an archive of images of a fundamental translation unit of the P CMC surface, where the displayed image is controled by choice menus giving the precise volume fractions and viewing coordinates.

The Perturbed D Surface Family page shows a generalization of the D minimal surface with several parameters. Unlike most of the surfaces in the minimal surface library, this one is built using the Plateau method instead of the Weierstrauss method.

A new page of Mesh Session Snapshots is one of the improvements to the Software section of the SGP. The page of SolSurf Session Snapshots is also new. Another addition is the Java Class Library, an expanding archive of utility classes.

The Complex Function Visualizer uses a complex function to map a grid on the plane back into the plane. While the current version only allows for the chaining of built-in functions, A planned future version will parse a text string-specified function.

A Stereographic Projection Applet illustrates the geometry of stereographic projection. This applet requires a version of the Java Virtual Machine that ships with Netscape Navigator 4.5 or later

The page entitled Periodic Genus One Helicoid and Scherk's First Surfaces illustrates that the recently discovered Periodic Genus One Helicoid and the Genus 1 Scherk's First Surface are limits of the same surface family.

	The Hybrid Level Surfaces section explores two-parmaeter families of level surfaces obtained by mixing two terms using one parameter, and adding a constant offset using a second parameter. Such experiments are used to generate candidate models for interfacial surfaces in block copolymer systems.
	The Models section was created for subjects involving the application of geometry in modelling physical systems, as distinguished from pure geometry, which is treated in the Geometry section.
	The Morphologies of Block Copolymers page in the models section provides an overview of the nanoscale structures exhibited by block copolymers as a function of the volume fractions of the blocks.
	The Surfaces Overview section was created for subjects relating to surface geometry but not specific to any one of the three surface categories ( minimal, CMC, and level).
	The Skeletal Graphs of Selected Surfaces page illustrates a property of surfaces which, although difficult to define in a precise mathematical way, is nonetheless useful in understanding surface geometry.
	The Level Set Families page explains how any implicit surface belongs to a family of surfaces parameterized by a constant in the equation.
	The Skeletal Graphs Approximated by Level Surfaces documents a recent discovery that simple combinations of the level set formulae for the P, D, G, and I-WP surfaces can be used to approach the skeletal graphs of the same surfaces.
	The Sphere in R4 page provides a series of animations to help the reader understand the sphere in R4 by way of analogy to the sphere in R3.
	A new 3-D visual construction menu implemented in VRML2 allows models of surfaces composed of translated and/or rotated fundamental pieces to be assembled and dis-assembled by switching on and off slices in the solid superimposed over the surface. Some examples are in the VRML models in the Mesh client pages for: infendk, fourendk, ktower, and schwarz. (Once in the VRML viewer, click on the surface to activate the construction menu.)
	The TEMsim Volume Projections page has been reorganized and expanded to include projections from 012, 112, and 122 Miller indices.
	The Lamellar Catenoid Surface Family page illustrates a family of triply periodic surfaces which, unlinke the P, G, and D surfaces, consists of topologically disjoint surfaces.
	The Triply Periodic Level Surfaces Animations page contains both MPEG rotation movies and VRML2 models to aid the visualization of some common triply periodic level surfaces.
	The Curvature of Selected Level Surfaces page compares mean and gauss curvature of the P, G, and D level surfaces, allowing interactive selection of viewpoint and color scheme.
	The Karcher-Scherk Tower Surface Families page is one of several new minimal surfaces which have been added, along with pages for the Mesh clients used to generate them. Also added were the The Karcher-Polthier-Thayer Surface Families and The Wohlgemuth-Thayer Surface Families pages. These pages all contain MPEG movies showing deformation through the real parameters.
	The P CMC Surface Family page has movies of the P CMC surface going from near one critical point in the family to near the other one. At the critical point, necks pinch off, changing the topology of the surface to a lattice of spheres.
	The D CMC Surface Family page has movies of the D CMC surface going from near one critical point int the family to near the other one. Unfortunately, the movie has gaps reflecting problems in getting the software (VPS/render) to properly handle the torus wrapped domain. A more robust method will exploit symmetries to perform the simulation without using the torus domain.
	The P CMC Surface at Specified Volume Fractions page contains an archive of images of a fundamental translation unit of the P CMC surface, where the displayed image is controled by choice menus giving the precise volume fractions and viewing coordinates.
	The Perturbed D Surface Family page shows a generalization of the D minimal surface with several parameters. Unlike most of the surfaces in the minimal surface library, this one is built using the Plateau method instead of the Weierstrauss method.
	A new page of Mesh Session Snapshots is one of the improvements to the Software section of the SGP. The page of SolSurf Session Snapshots is also new. Another addition is the Java Class Library, an expanding archive of utility classes.
	The Complex Function Visualizer uses a complex function to map a grid on the plane back into the plane. While the current version only allows for the chaining of built-in functions, A planned future version will parse a text string-specified function.
	A Stereographic Projection Applet illustrates the geometry of stereographic projection. This applet requires a version of the Java Virtual Machine that ships with Netscape Navigator 4.5 or later
	The page entitled Periodic Genus One Helicoid and Scherk's First Surfaces illustrates that the recently discovered Periodic Genus One Helicoid and the Genus 1 Scherk's First Surface are limits of the same surface family.