I'll see what I can come up with for a diagram.I am not fully clear on the way these conversions occur, so a flowchart, with the corresponding identifications of the regions (e.g. time/space, space/time, scalar speeds, natural reference frame) would be really helpful. Looks like a supplement of this will have to be prepared in addition to the main presentation of RS2, as not many people are familiar with this.
To understand projective geometry, which needs to be understood before one can attempt to understand counterspace, is to understand human perspective--how our minds interpret the 2-dimensional images on the retina, and reconstruct an internal, 3-dimensional thought-form to represent that object.
30 years ago, when computers were first being used to create wire-frame models (a model consisting of points connected by lines), this idea of human perception had to quantified, so that the computer could take a 3-dimensional representation of points and lines, and reverse-engineer a 2-dimensional picture that the human eye could use to correctly interpret the computer model.
This process was known as a projective transformation.
It originally started out as a very involved set of trigonometric relations, but was later refined to use homogeneous coordinates and matrix algebra, which made the application on computers both consistent and efficient. Objects could then be rotated, translated and scaled simply by multipling matricies together to produce a "transformation matrix", and then passing each (x,y,z) coordinate thru that matrix, to get a new set of points which were the width (x), height (y), and depth (z) of what the eye would see in that orientation. The depth function was often connected with the brightness of the points and lines, the nearer the observer, the brighter the point/line. This gave the illusion of "depth" on a flat screen.
By having a clearly defined method of creating the illusion of vision, we can reverse the process to see how the human mind takes that illusion, and reconstructs a 3-dimensional "reality."
Thus, there are certain key elements in understanding projective geometry:
- Matrix algebra; how to manipulate multi-dimensional matrices.
- Homogeneous coordinates; what they are and what they represent. (Quaternions are homogeneous polar coordinates).
- How to construct an object using points, lines and surfaces.
- How to set up a "camera" (eye position) and viewing plane within a 3-dimensional scene, to produce a 2-dimensional image of what the eye will see.
- The process of transforming the 3-d coordinates into a 2-d image by using matrix algebra.
College math texts should contain explanations on how matrix algebra works.
Once you have a basic understanding of matrix math, go to http://www.cs.unc.edu/~marc/tutorial/tutorial02.html which is a description of how one can take a photograph, and create a 3-d model out of it--just as your mind does from the image data it gets from the eye. This will introduce you to the idea of variants and invariants, and how assumptions need to be used to reconstruct a 3-d model from a flat image. The section on projective geometry is all based in matrix math, but is fairly understandable. If you have problems understanding it, just keep reading. Sometimes, for the mind to "swallow" a concept, it needs to take a drink of water with it -- a larger chunk of information.
Then I would suggest you go to http://www.povray.org/ and download the latest stable version of POV-Ray (3.6), and install it on your computer. It is a free program that allows you to create very complex and detailed, 3-dimensional models on your computer and view them from any perspective. This way, you can play "god", and create a virtual reality on your computer--even if it is as simple as a box. Just by going thru the tutorials, you will learn how to create a "scene", place the camera, point the camera, move, view and transform objects, and even look at the scene with different geometric assumptions--for example, an "orthagonal" projection is actually a Euclidean projection, which looks very strange to the mind, since all the edges point at the plane at infinity. The normal, "perspective" projection is actually counterspatial, with the "counterspace infinity" being the "vanishing point" of the scene. It is VERY useful tool in understanding the role that perception plays on our examination of the Universe, because you can actually SEE what happens when you change things.
Don't expect to accomplish it overnight; it is not that easy to understand first time in, because all the concepts are new, and the mind does not have much to correlate them to. But once you catch on, the understanding of "perspective" will give you a whole, new perspective of the Universe.
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