hypertorus(6x)
NAME
hypertorus - Draws a hypertorus that rotates in 4d
SYNOPSIS
hypertorus [-display host:display.screen] [-install] [-visual visual] [-window] [-root] [-delay usecs] [-fps] [-wireframe] [-surface] [-transparent] [-solid] [-bands] [-spirals-{1,2,4,8,16}] [-twosided] [-colorwheel] [-perspective-3d] [-orthographic-3d] [-perspective-4d] [-orthographic-4d] [-speed-wx float] [-speed-wy float] [-speed-wz float] [-speed-xy float] [-speed-xz float] [-speed-yz float]
DESCRIPTION
The hypertorus program shows the Clifford torus as it rotates in 4d.
The Clifford torus is a torus lies on the "surface" of the hypersphere
in 4d. The program projects the 4d torus to 3d using either a perspective or an orthographic projection. Of the two alternatives, the perspective projection looks much more appealing. In orthographic projections the torus degenerates into a doubly covered cylinder for some
angles. The projected 3d torus can then be projected to the screen
either perspectively or orthographically. There are three display
modes for the torus: mesh (wireframe), solid, or transparent. Furthermore, the appearance of the torus can be as a solid object or as a set
of see-through bands or see-through spirals. Finally, the colors with
with the torus is drawn can be set to either two-sided or to a color
wheel. In the first case, the torus is drawn with red on the outside
and green on the inside. This mode enables you to see that the torus
turns inside-out as it rotates in 4d. The second mode draws the torus
with a fully saturated color wheel. This gives a very nice effect when
combined with the see-through bands or see-through spirals mode. The
rotation speed for each of the six planes around which the torus
rotates can be chosen. This program is very much inspired by Thomas
Banchoff's book "Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions", Scientific American Library, 1990.
OPTIONS
hypertorus accepts the following options:
-window Draw on a newly-created window. This is the default.
-root Draw on the root window.
- -install
- Install a private colormap for the window.
- -visual visual
- Specify which visual to use. Legal values are the name of a visual class, or the id number (decimal or hex) of a specific visual.
- -delay microseconds
- How much of a delay should be introduced between steps of the animation. Default 25000, or 1/40th second.
- The following three options are mutually exclusive. They determine how the torus is displayed.
- -wireframe
- Display the torus as a wireframe mesh.
- -surface
- Display the torus as a solid surface (default).
- -transparent
- Display the torus as a transparent surface.
- The following seven options are mutually exclusive. They determine the appearance of the torus.
- -solid Display the torus as a solid object.
- -bands Display the torus as see-through bands (default).
- -spirals-1, -spirals-2, -spirals-4, -spirals-8, -spirals-16
- Display the torus as see-through spirals with the indicated number of spirals.
- The following two options are mutually exclusive. They determine how to color the torus.
- -twosided
- Display the torus with two colors: red on the outside and green on the inside.
- -colorwheel
- Display the torus with a fully saturated color wheel (default). If the torus is displayed as see-through bands each band will be displayed with a different color. Likewise, if the torus is displayed as see-through spirals each spiral will receive a different color.
- The following two options are mutually exclusive. They determine how the torus is projected from 3d to 2d (i.e., to the screen).
- -perspective-3d
- Project the torus from 3d to 2d using a perspective projection (default).
- -orthographic-3d
- Project the torus from 3d to 2d using an orthographic projection.
- The following two options are mutually exclusive. They determine how the torus is projected from 4d to 3d.
- -perspective-4d
- Project the torus from 4d to 3d using a perspective projection (default).
- -orthographic-4d
- Project the torus from 4d to 3d using an orthographic projection.
- The following six options determine the rotation speed of the torus around the six possible hyperplanes. The rotation speed is measured in degrees per frame. The speeds should be set to relatively small values, e.g., less than 4 in magnitude.
- -speed-wx float
- Rotation speed around the wx plane (default: 1.1).
- -speed-wy float
- Rotation speed around the wy plane (default: 1.3).
- -speed-wz float
- Rotation speed around the wz plane (default: 1.5).
- -speed-xy float
- Rotation speed around the xy plane (default: 1.7).
- -speed-xz float
- Rotation speed around the xz plane (default: 1.9).
- -speed-yz float
- Rotation speed around the yz plane (default: 2.1).
- -fps Display the current frame rate, CPU load, and polygon count.
INTERACTION
If you run this program in standalone mode you can rotate the hypertorus by dragging the mouse while pressing the left mouse button. This
rotates the hypertorus in 3D, i.e., around the wx, wy, and wz planes.
If you press the shift key while dragging the mouse with the left button pressed the hypertorus is rotated in 4D, i.e., around the xy, xz,
and yz planes. To examine the hypertorus at your leisure, it is best
to set all speeds to 0. Otherwise, the hypertorus will rotate while
the left mouse button is not pressed.
ENVIRONMENT
DISPLAY to get the default host and display number.
- XENVIRONMENT
- to get the name of a resource file that overrides the global resources stored in the RESOURCE_MANAGER property.
SEE ALSO
COPYRIGHT
Copyright (C) 2003-2005 by Carsten Steger. Permission to use, copy,
modify, distribute, and sell this software and its documentation for
any purpose is hereby granted without fee, provided that the above
copyright notice appear in all copies and that both that copyright
notice and this permission notice appear in supporting documentation.
No representations are made about the suitability of this software for
any purpose. It is provided "as is" without express or implied warranty.
AUTHOR
- Carsten Steger <carsten@mirsanmir.org>, 28-sep-2005.