Hyperspace Explorer Online Help

You manipulate the displayed object pressing a mouse button, optionally holding a modifier key, and dragging the mouse.

LMB stands for left mouse button, MMB for middle mouse button, RMB for right mouse button.

LMB+left/right	rotate about y-axis
LMB+up/down	rotate about x-axis
LMB+MMB+up/down	rotate about z-axis
MMB+left/right, CTRL+LMB+left/right	translate along x-axis
MMB+up/down, CTRL+LMB+up/down	translate along y-axis
LMB+MMB+left/right, CTRL+MMB+up/down	translate along z-axis
CTRL+MMB+left/right	translate along w-axis
SHIFT+LMB+left/right	rotate about xy-plane ( = z-axis)
SHIFT+LMB+up/down	rotate about xz-plane ( = y-axis)
SHIFT+MMB+up/down	rotate about yz-plane ( = x-axis)
SHIFT+MMB+left/right	rotate about xw-plane
SHIFT+RMB+left/right	rotate about yw-plane
SHIFT+RMB+up/down	rotate about zw-plane

 
note that rotating about the xy-plane, the xz-plane and the yz-plane in four-space is the same as rotating about z-axis, y-axis and x-axis in three-space.

With a double click you can restore all transforms in four-space back to original values.

If you press the ALT key while performing any of these transforms, your mouse movement is taken as an indication of how fast the object should rotate, and an animation is started rotating the object about the given axes/planes. You can indicate more than one direction, if you add another movement after the animation has started.
Double clicking stops the animation, as well as restoring the four-dimensional transform.
You can do almost everything, such as changing the display properties or changing the displayed object, while the animation is running.

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The menus

The "Object" submenu

You select the four-dimensional object to be displayed with the submenu titled "Object". There is a load of choices available.

The "f:R³->R" submenu

Under this menu you find a couple of three-dimensional "surspaces" in four-space (just as there are two-dimensional surfaces in three-space). These "surspaces" are described by functions of the type

x = (x, y, z, w)
where w = f (x, y, z)

(up to now, only functions where w = f (r), r = sqrt (x²+y²+z²),  are implemented)

and of the type

x = R·(sin (psi)·sin (theta)·cos (phi), sin (psi)·sin (theta)·sin (phi), sin (psi)·cos (theta), cos (psi))

where R = f (x, y, z),

in other words, four-dimensional polar functions.

The "Objects" submenu

Under this menu you find some four-dimensional objects. These objects are generalizations of real objects from three-space, such as the cube, the sphere or the tetrahedron.

Hypersphere:    The four-dimensional version of a circle or sphere, an object with a surface (or "surspace") which is the same distance from the center in every point.

Hypercube:        The four-dimensional version of a cube. In 4D it has eight "surspaces", each consisting of a cube.

Hyperpyramid: The four-dimensional version of a tetrahedron or four-sided pyramid. In 4D it has five "surspaces", each consisting of a tetrahedron.

Menger Sponge:      The four-dimensional relative of a Menger sponge.
If you do not know what a Menger sponge is: A Menger sponge is a fractal created by the following process.
You take a cube and slice it into 27 (3x3x3) small cubes.
Then you remove all subcubes which do not have an edge in common with the original cube. In effect, you remove the cross-shaped interior of the cube.
Then you go ahead with this operation for all the subcubes which have remained.
And so on, ad infinitum.

 
A three-dimensional Menger sponge

To the four-dimensional Menger sponge there are three parameters:

• The level of the sponge, i.e. the number of times the described process is repeated. For an actual fractal, the number would be infinite. That, of course, cannot be done in our less-than-ideal, CPU- and RAM-bound world. Do not set this above 3, unless you have a huge machine and/or know exactly, what you're doing. This number is of course an integer.
• The distance from the center, up to which hypercubes are taken out of the surrounding cube. If you set this to 1, you get something like a foam. At 2, you get the four-dimensional equivalent of the sponge depicted and described above. Set to three, you get something like dust, with the subcubes disconnected from each other.
  For higher values of distance, you can choose higher values of the level too, before your computer gives up.
  
• The size of the object.
  

Sierpinski Gasket:   The four-dimensional relative of a Sierpinski Gasket.
A Sierpinski gasket is similar in concept to a Menger sponge, but it takes a tetrahedron as a basis.
The four-dimensional Sierpinski gasket takes two parameters:

• The level, defined as in the Menger sponge. With Sierpinski gaskets, you can go a little higher on this, up to 6, say.
• The size of the object.
  

 
Torus 1:    There is more than one way to generalize a torus from three to four dimensions. In three dimensions, you create a torus by taking a circle with radius r, and rotating it around another center, in a second circle with radius R.
This generalization to four-space  takes a circle of radius rho, rotates it about a centre with a distance of r (defining a torus in three dimensions), and rotates that about a centre in the distance R.

Torus 2:    This 4-torus is defined by rotating a sphere of radius r about a centre in distance R.

The "Surfaces" submenu

Under this menu you find a couple of two-dimensional surfaces in four-space, which are described by functions of the type

x = x (u, v) and
x = x (theta, phi).

Custom Functions

 

The "Appearance" submenu

Colors

Coloration can be switched on or off. You usually want colors, as they greatly aid you eye in understanding the spatial properties of the displayed object. Turning off colors gives a slight performance advantage. Colors are enabled by default.

Shading

Shading means the smooth coloration from one point of the object to another. You almost certainly want this, as everything looks awful without it. Turning off shading gives a slight performance advantage. Shading is enabled by default.

Depth Cue

Depth cue, also called fog, paints points which are further away from the viewer in three-dimensional space darker. This greatly helps you to understand the three-dimensional properties of an object. Turning it off gives a slight performance advantage. Depth cue is enabled by default.

Depth cue on


Depth cue off

4D Depth Cue

4D depth cue is the four-dimensional analogon to Depth cue, painting points which are further away in four-space darker. Turning it off gives a moderate performance advantage. It is disabled by default.

Transparence / Line Antialiasing

Transparence is used when you view a solid object, Line AA is used when viewing an object in wireframe. This makes an object look smoother and thus "better". Turning it on incurs a huge performance hit, so it is disabled by default.

Lighting

You can't turn off lighting, because without it everything would look just flat and white. I don't even know why I left this menu point.

Wireframe / Solid

You can switch between wireframe display and solid display of an object. The default is solid.

The Hypercube displayed as wireframe

Coordinate Cross

You can display a coordinate cross to aid you in distinguishing which direction is which axis. The x-axis is colored red, the y-axis green, the z-axis blue and the w-axis purple.

 
The coordinate cross

The "Animation" submenu

Render to Images

When selected, all images which are rendered are written to a PNG image file with continuing numbers. This is especially useful if you want to save an animated display of your object for later replay.

Animation Settings

Selecting this menu item pops up a dialog which lets you select:

• The directory where you want to store rendered images
• The beginning of the filename, before the frame numbers and the extension ".png"
• The maximum number of frames to be rendered. Perhaps you don't want to fill up your hard disk if you let the program run over night ;-). The default for this value is unlimited.

Benchmark

Okay, this menu point is somewhat of a misnomer. What it does is rotating the current object in the current display space first in four-space, then in three-space, and then gives you the time it took and the framerate for each.
Warning: this is not interruptible,and it may take quite a while, depending on the object you display,the"Appearance" options you enabled, and your hardware!

Settings

TBD.

The "Help" submenu

Online Help

Obviously, you already found out what this item is for.

About

Information about Hyperspace Explorer

About Qt

Information about Qt, the toolkit heavily used to develop Hyperspace Explorer

Quit

Exits the program, no questions asked.

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The Parameter Window

Grid Size and Bounds

You can enter the roughness of the grid at which the function values are sampled. By default, functions are displayed on a 5x5x5 grid, which is very inaccurate and looks rough. The drawback of a higher resolution is, that the number of points sampled scales with the cube of the grid size, so that for a grid that is twice as large, you'll have eight times as many points as before, and the performance drops to one eigth.
The bounds span the parameter space for the given function.
For objects you find under the submenu "f:R³->R" you can specify the grid in three dimensions (t, u, v), for objects from the submenu "Surfaces" you need two (u and v), for objects from the submenu "Objects" you need none, as these are not sampled on a grid.

Function and Parameters

TBD.

Notice that while changes on the "Grid Size and Bounds" part of the window take effect immediately after you pressed the "Apply" button, changes in the "Function and Parameters" part do not. In that case, you first have to (re-)select the object or function you are setting the parameters for.

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Scripting

You can supply a script of actions to HyperspaceExplorer by calling it from the command line with HyperspaceExplorer --script .

A script is read and executed line after line. Statements define values for displayed functions, the functions' parameters, Transformations and view states.

The scripting language has no control structures. A script is only able to define a sequence of events, which are displayed.

Comments in the script are started with the letter #. Everything following a # is ignored till the end of the line.

• Corollary 1: You can mix statements and comments on the same line.
• Corollary 2: There is no native support for multi-line comments.

Scripting Commands

Statement	Parameter(s)	Description
set_size	width (int) height (int)	sets the size of the viewport
object	name (string)	describes the object to display
parameter double	parameter_name (string) parameter_value (double)	sets parameter parameter_name (of type double) to the previously defined object
parameter int	parameter_name (string) parameter_value (int)	sets integer parameter parameter_name to the previously defined object
parameter unsigned	parameter_name (string) parameter_value (unsigned)	sets unsigned integer parameter parameter_name to the previously defined object
parameter string	parameter_name (string) parameter_value (string)	sets string parameter parameter_name to the previously defined object
transform	R (rotation<4>)	sets the current transformation of the object; currently, only rotations in four-space are implemented
rotation		Not yet implemented
parameter_step		Not yet implemented
transform_step	dR (rotation<4>)	sets the transformation to undergo for each new frame that is calculated; currently, only rotations in four-space are implemented
rotation_step	Not yet implemented
frames	num_frames (unsigned)	Number of frames to calculate
loop	num_loops (unsigned)	Number of times the num_frames frames are looped
start_animation	----	Starts the animation with the values defined before the current point in the script
colors	(bool)	Turns display of colors on or off
shading	(bool)	Turns Gouraud shading on or off
lighting	(bool)	Turns calculation of lights on or off
transparence	(bool)	Turns transparence (in solid mode) or line antialiasing (in wireframe mode) on or off
wireframe	(bool)	Sets wireframe mode (if set to "on") or solid mode ("off")
depthcue3d	(bool)	Turns fog (depth cue) on or off
depthcue4d	(bool)	Turns four-dimensional depth cue on or off
coordinates	(bool)	Turns display of the coordinate cross on or off
image_dir	(string)	Sets the directory to which image files are written. Also turns on writing the rendered images to files
image_prefix	(string)	Sets the file name prefix for the written image files. Also turns on writing the rendered images to files
sleep	msecs (unsigned)	Pauses all activity for the specified number of milliseconds, when execution of the script comes to this statement

Example Script

set_size 800 800                # show everything in a 800x800 pixel window

object Hypercube                # define a hypercube...
parameter double Size 1         # ...of edge length 1

# between two consecutive frames, rotate by two degrees each around the xw,
# yw and zw axes
transform_step <0,0,0,2,2,2>
frames 180                      # compute 180 frames - a 360 degree rotation
loop 1                          # do this exactly once

transparence off                # make the hypercube intransparent
wireframe off                   # show solid planes

image_dir /tmp                  # write image files to /tmp
image_prefix Sponge             # the filenames start with "Sponge"

start_animation                 # now execute what was defined before

object Sponge                   # define a four-dimensional Menger Sponge...
parameter unsigned Level 1      # ... one iteration deep...
parameter unsigned Distance 2   # ... cutting away cubes up to distance 2 from center
start_animation                 # execute the animation with the values above
# values once set, such as transform_step, stay the same until they are changed

object Pyramid                  # display a hypersimplex
parameter double Size 2         # of edge length 2
start_animation                 # and animate it as above

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