  
Algebraic Surface Solver
by Richard Morris
Algebraic surfaces are level sets of a scalar valued function F(x,y,z)
from R3 to R where the function expression of F consists of
algebraic operations of the x, y and z variables only.
This web service calculates 3D models using JavaView
on the client side and Richard Morris' LSMP package on the server side. Note, the first download of the applet and the model
may take a while ...
How to Calculate a Surface
- Enter a definition of the surface in the JavaView panel, or select a
predefined surface from the list.
- Press the Calculate button at the bottom of the applet
window.
- After a while the new surface should appear. If the surface has
missing faces then you might need to increase the timeout threshold.
- Once the surface has been created, drag the mouse in the display to
rotate it.
More descriptions are given below on this web page.
The basic syntax is quite simple: the formula should involve x
y and z and should be an implicit polynomial equation:
some polynomial function of x, y, z = 0;
A semi colon is needed at the end of the equation. x^2 means x
squared. For example
x^2 - y^2 - z^2 = 0;
There is no need to include a * between terms to be multiplied, only a
space is needed. For example x y is the same as x*y. There are quite a few
more sophisticated features of the syntax which will appear below soon.
Multi-line equations can be used. All the equations after the first will
be substituted into the first equation. For example
x^2 - w = 0; w = y^2 + z^2;
is equivalent to the equation given above.
Some example definitions are listed
here.
Domain of Interest
These parameters control the range over which the surface is calculated.
It is best if unequal values are chosen for the ranges. Otherwise
certain degeneracies may occur which can lead to poorer results.
Resolution Control
Resolution parameters control how detailed the resulting mesh will be.
Each of these must be a power of two. The box specified by the range is
split into a number of smaller boxes and points of the faces of these boxes
will be found together with any singularities lying in the middle of them.
The Coarse parameter specifies the number of smaller boxes, by
default 8 along the x, y, z directions, i.e. 512 boxes. The Fine
parameter specifies the smallest box size used to find a singularity. The
Face specifies how accurately each face of the box is subdivided and the
edge parameter specifies how accurately each edge is subdivided.
Generally you will just want to change the Coarse parameter, but
lowering the other parameters may give quicker results. Not much will be
gained by increasing them. As a rule you should keep Coarse < Fine < Face
< Edge or the results will be unpredictable.
Several other options are available in this panel:
Create New Geometries
Normally the new geometry created will replace the existing one. However
you might want to draw a sequence of surfaces, e.g. level sets of
F(x,y,z) = a
for different values of a. If this is option is on a new geometry will be
created each time, the names will be like asurf {1}, asurf {2}. Only the
last of these geometries can be subsequently changed.
Draw in Colour
By default the surface will be coloured according to the x,y,z values of
individual points. This can be switched off by deselecting this option. The
surface will be a uniform light blue colour. This can be useful if you want
to colour different surfaces different colours by using the material panels.
Keep Material Properties
By default the surface will be drawn with faces (elements) visible,
edges, lines and vertices not visible. If you want to keep the existing
properties (say edges visible) then select this option.
Timeouts
If a degenerate surface is specified the program can take a very long
time to run. To prevent this from happening the CGI program will timeout
after a certain time. This may mean the surface will not be calculated
properly. If you want to calculate more complicated surfaces please send an
email to Richard Morris at
pfaf@webmaster.org or look at his
website
and he'll provide a version you can run at home which does not have the
timeout restriction.
Press the right mouse button to select major interaction mode from the
popup menu, or use keyboard keys to temporarily switch between different
modes (just keep a button pressed to switch mode temporarily). Some
essential modes are:
| o |
Rotate surface (this
Orbit mode is the default mode) |
| s |
Scale
surface, drag in vertical direction |
| t |
Translate
surface in viewing plane |
| r |
Reset
camera and display, object returns to default position |
|
|
F1 or Ctrl-m
or
Right-Mouse |
Show control window |
|
Help |
With all keyboard shortcuts, and help on problems. |
To modify appearance of the geometry select the material panel from the
JavaView window via the menu command Inspector - Object - Material.
Be sure that the mouse focus is inside the viewer otherwise, e.g., the
window manager might react. To hide the control panel press F2.
Web page, applet and Algebraic Surface program by Richard Morris,
copyright 1990-2003.
For more information on the software, related stuff e.g. algebraic curves
and intersections see also Richard Morris'
website.
Maths
home page and
Personal home page
Email pfaf@webmaster.org or
rjm@amsta.leeds.ac.uk.
 |
The software is licensed under the Creative Commons License.
(You can copy, distribute and display this work but: Attribution is
required, its for Non-Commercial purposes, and it's Share Alike
(GNUish/copyleft) i.e. has an identical license.) |
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