This is an overview of a system to plot various representations of objects with surfaces and curves having known functions, spatial rays, maps and other graphs, using almost any coordinate system, developed primarily at the Massachusetts Institute of Technology Computer Science and Artificial Intelligence Laboratories, with additional resources from the Plasma Science and Fusion Center and the Harvard University Mathematics Department.  These steps were originally aimed toward roughly illustrating confined plasma behavior which thus far have yielded a somewhat general set of methods.

This illustration shows a pair of spheres, a torus between them, and a bundle of rays, all expressed using  toroidal coordinates, for simplified d-orbital model. 


Each of the surfaces indicated have been labeled with the specific parameter held constant.


Press any linked image within this document to switch background styles.  

Within and behind boundaries, the curves representing surfaces have been silhouetted completely, whereas the orbital rays are faded instead.


These orbital paths were chosen such that they revolve within the center torus a few times before and after a single loop around either sphere. 


Currently implemented as a class-library for MIT-Scheme (SOS), this mechanism may be adapted for Common-Lisp (CLOS), C++, and Java as needed applications arise.  It is also intended to provide point datasets for Auto-Lisp or other rendering and illustration packages.

Foundations for this particular library are rooted in a long history with mathematical modeling.  Here is an example Cartesian coordinate defined surface, employing simple ray-tracing for hidden-line removal using Pascal and a custom low-level (assembly) graphics-library, plotted originally onto a Tektronix-4015 with a DecSystem-2060 running TOPS at Tulane University in 1983:  

Similar illustrations were produced using this earlier library with Fortran and Pascal for the Tulane Physics and Chemistry Departments during 1979-1984 for analysis of X-ray diffraction, nuclear emission, gravitational model, and astronomy related data.  Addition of interrupt service handlers lead to more direct parameter controls for these applications and to an "Asteroids"-like video-game.

Removing hidden elements from a spherical object can easily be accomplished by merely noticing where the plane through its center is perpendicular to the view position.

This technique can again be applied for a light source direction in order to provide the effect of simple illumination.


This is another image first generated using Fortran and the same hardware and graphics-library circa 1980:  


Evolving into Pascal this globe drawing program eventually incorporated the ability to zoom and show simple maps. 


In this adjacent scan highlight of a screen dump from 1982 is a small map representing New Orleans centered around 90-W, 30-N. 


Code clips from these earlier examples would require scanning of old dot-matrix hardcopies, though here is a screen-shot of some equivalent code in Zeta-Lisp used to draw the previous globe object on an LMI-Lambda in 1988:

Generalizations of the techniques used in this set of older examples are ultimately being merged into this newer graphics-library version, along with a variety of other methods to facilitate representation of less uniform shapes.

Representations of arbitrary objects in any coordinate system, which are also meant to have hidden elements around contours, present interesting challenges requiring more general methods.  In this illustration, created using MIT-Scheme and the class-library described herein, the elements meant to be hidden are removed according to a silhouette of the object, in this case slightly smaller in radius than the sphere drawn:








Here are the code expressions which produced this drawing: 

Certainly there are easier, less intensive methods for obtaining the same effect, but proof of concept suggests some readiness for surfaces expressed with other coordinate-systems.

After adding conversion functions for various confocal coordinate systems (and their dependent hyperbolic trigonometric functions), it becomes possible to illustrate other interesting shapes, knowing something about the necessary parameters.  Here is an example image showing a silhouetted surface and ray using toroidal coordinates:

This image was produced with a set of expressions such as: 

Often several coordinate-systems serve to express similar contours.  Here is another example image of a silhouetted torus and ray, this time using bicylindrical (cylindrical wrapped around z-axis) coordinates:

Visual comparison of the size of the panels (tetragons) on both torus surface examples reveals the uniform radial nature of bicylindrical, and the hyperbolic nature of toroidal coordinate-systems.  Here are the expressions for the last image:

Curves may also be represented using this library as parameterized functions.  This image shows a combination of the previous silhouetted rays, along with another expression demonstrating such usage:

In this case the larger paths are faded where silhouetted.  This example could perhaps represent a possible mode of plasma within a tokamak, with the outermost pathway for Tritium (H3), the next wider pathway for Deuterium (H2), and the innermost pathway for the electrons.  Here is the code producing this illustration:

This is a similar image with dashed lines where silhouetted around the middlemost torus:

The following linked page lists some animation sequences for this tokamak plasma model:   tokamak-animation.html 

Another use of confocal coordinate-systems might be for representation of electron orbital pathways.  Here are a pair of spheres and a long ray in bispherical coordinates which might be used to demonstrate something like a pz orbital snap-shot:

This image was created using this set of expressions:

This similar combination has slightly different parameters:

Again it might be noticed how these sphere surface tetragons reveal hyperbolic related spacing, whereas the sphere expressed using spherical coordinates shows more uniform spacing, by analogy to the comparison of tori using toroidal and bicylindrical coordinates earlier.

These methods currently serve as a means to roughly analyze the objects or functions describing them, before displaying with, or without necessarily requiring, other rendering systems.

Following are some older example images and code clips from the previous version of this custom physical object modeling mechanism with MIT-Scheme.  

This version used structures rather than classes for object definitions, and so the primitives are somewhat different, though the principles are basically the same.  Most of these images were first created circa 1991 at the MIT AI Lab using HP hardware.

An attractor image (to demonstrate basic iterative convolution or chaos models):

Trace of helix onto conic surface (could be particle moving through E-M hourglass or mirror):

Trace of helix onto spherical surface:

Double Toroidal Helix for simple model of tokamak plasma ions (outside) and electrons (inside):