Clipping Planes, Matrices, World Space and the Right-Hand Rule: 3D Practice

Clipping Planes

One is inclined to ask, when working with 3D tools, what are the boundaries of the third dimension of "Spaceland"? Spaceland, as it has been called ever since Edwin Abbot explored higher dimensional existence through two dimensional beings, we usually take for granted, although a consciousness of three dimensions comes to us us in computer simulations. We pan, roll, and yawn (terms for camera movement) and the space moves, so that we can posit that the extension of our view will necessitate an extension of the world.

To the extent that the world responds to our spatial cues when we navigate (via the interface) the dimensionality of "3D" is truly 3-dimensional. Still, we look through often, the screen, a 2D barrier between "virtual" and "actual". This is called the clipping plane, and it is a literal boundary of the simulated 3D space (though perhaps also other dimensional spaces).

While 3D animation is a 2D representation or model of a 3D space, the use of virtual environments gives back the third dimension to this conflated space, to the extent that virtual worlds replace "actual" ones. The clipping plane is a metaphor for the 5th dimension that is not specifically The Twighlight Zone, although as a "cyberspace", it is a culturally constructed architecture of dimensional variation.

The clipping plane is like a switch that turns on our cyberspatial condition, yet it is not necessarily a switch into the "higher" dimensional. The clipping plane represents, in Manuel de Landa's sense, an intensification, rather than a continuous progression. Abruptly, contemporary, screen-based interfaces, humans more metaphorically merge with machines. Cyborgian convergence, on the other hand, problematizes our place outside of the boundaries of the (3D) world space, with literal integrations of machine into human.

The World Space

The world space is another metaphor that began in cybernetic simulation in the 1950s and in the theory of Stanislaw Lem later, and became less metaphorical as the look and feel of physical spaces emerged as a paradigm for computer simulation. With Lem's Personetics, beings are instantiated and enumerated without a graphical user interface or a seamless verisimilitude, suggesting the possibilities and role of the invisible in our world. The visual is material, and has a second sense, that of the invisible.

The world space in Maya, a 3D environment, contains the collective research of literal emergent cyberspaces. Realism in visuals is a default mode of simulacra, although, simulacra need not be dependent upon verisimilitude. Verisimilitude however, is not "free" in computer simulation as it may appear (but only appear) in the world. One must work to texture and light one's simulation "realistically". The world space, in its wire-frame naissance, is a better "model" of simulation, a "discursive" simulation.

One must think of the 3D world space as a Star Trek holodeck after a VR session. The space is exposed mechanically and, in this state, the physical conditions of simulacra are exposed for their virtual and actual capital. Culturally, the quality of capital in the exposed infrastructure is insufficient for verisimilitude, but quite sufficient for attempting (only attempting) an explanation of the world "out there", or of visual appearances.

Matrices

Visual appearances, as well as their environments, may be described by the mathematical construct of the matrix. In OpenGL, a graphics library for interactive games and other simulations, all movement in 3D spaces is dependent upon a grid that, through a series of equations that may be solved strictly from the position of its numbers down and across, a position in Cartesian space is pinpointed and movement is then possible.

But what is movement in a 3D interactive space? Movement is constructed through another matrix, a translation matrix, or a rotation matrix, or a scaling matrix. Matrices are compounded, or stacked on each other to make complex movement. One technique is pushing the current set of coordinates onto the stack; another technique is popping the matrix off the current stack, a technique that returns the 3D form (with a cartesian location) to its last transformation. Or in a cyclical process or in reverse movement, popping can be used to simulate complex movements.

Pushing and popping could be seen as analogous to switching mechanisms in microprocessors, or to binary code. On one layer, binary code represents, at a low level, numbers filling up matrices. These numbers then, as if, collectively, they were a 1 or 0, are then switched on and off, pushed and popped. Binary code instantiates data, the patterns of 1s and 0s create numerical relations, and binary transformations are binary procedures for producing motion.

What purpose does binary language have in culture as well as in mathematics and computing? It is fast! The schizophrenic can leap across great divides of articulation from a "yes" to a "no" in the same sentence, traveling economically, a great distance. So too, a reduced set of symbols can provide a framework to build more complex calculations -- and at speed, for these reduced instructions. Still, the problematic of the binary opposition in philosophy could be similarly seen as a problematic in computing. With fuzzy set technology, ranges of decimal and fractional values provide much finer control over decision processes and evaluations of data. More on this later.

Whether as transformation or the building blocks of data, binary language has been used until now for its economy and effectiveness. This technology, theoretically problematic, is nevertheless, practical: it works, and materially and non-schematically.

The Right Hand Rule

The schematic that all beginning students of 3D use for transformations in the 3D world space, while rooted in the fact that humans have bodies with dexterities that arise through the use of their hands, is an anthropmorphic rule for producing valence on rotational values.

Valence on rotational values is positive or negative; rotation, a particular transformation that is to be effected. If one curls his or her four fingers of his or her right hand counter- clockwise, his thumb points up, indicating a positive rotation. Rotated clockwise, the thumb points down, indicating negative rotation.

With this rule, human hands are mapped on to human spaces; Manual dexterity provides an encoding for space and rotation in that space. Hands are used to swing a bat at a ball sending the ball back across the space; hands are used to point across a space, indicating that, over yonder is the desideratum of the gesture.

So too is it with the right hand rule: the gesture of rotating and curled fingers with directional thumbs leaps across spatial constraints in order to situate agents within spaces, and to wrap spaces around agents and their gestures. Another gesture of the hand is its control of the computer mouse as gesture: pointing and clicking is situated on the "actual" side of the clipping plane in order to designate relations between objects and space within the virtual.

The clipping plane is a barrier, the world space is both a quantitative and qualitative environment, the matrix is a mechanism for reducing motion, and our hands may be used to articulate mathematical concepts. The matrix is, in computer-generated enviroments as well as culturally-generated environments, a system of control that agents reproduce with their daily practices. The clipping plane (in common with hands used to count or determine positivity or negativity for rotation) is a sign or gesture of cyber-parole.

Were a 21st century Pompeii disaster to occur, The mathematical matrix would be eliminated-- except in the gestures of city blocks and in distant reference to computing, as it may preserve the bodies of those poised at the computer interface.