Projective Geometry
Projective geometry is the study of relationships between geometric entities without the assumptions imposed by other strata of geometry.
Geometric Strata
Geometry is divided into four “strata” or “layers”. Each strata adds a set of assumptions that create certain invariants for that layer. An “invariant” is a property of a configuration of geometric entities that is not altered by any transformation belonging to the specific strata. A “transformation” is simply an operation applied to that geometric entity; the most common being translation and rotation.
| Stratum | Invariant | ||||
|---|---|---|---|---|---|
| Property | Projective | Degrees of Freedom | |||
| G E O M E T R Y | Projective |
|
| 15 | 15 scale |
| Affine |
| lines or planes that intersect at infinity are called “parallel” ratio of lengths along a certain direction is a cross-ratio with one point at infinity | 12 | 9 scale 3 translation | |
| Metric |
| 7 | 1 scale 3 orientation 3 translation | ||
| Euclidean |
| everything is scaled to unity, removing that degree of freedom | 6 | 3 translation 3 rotation | |
Summary of Strata
| Projective Stratum | This is the least structured stratum of geometry, dealing with the concept of a “ratio of ratios” — the cross-ratio. |
|---|---|
| Affine Stratum | Creates the assumption of a plane at infinity. This creates parallel and orthogonal relationships between geometric entities by placing one of the 4 points of the cross-ratio in the plane at infinity. This “squares up” geometric entities by making lines and planes parallel. |
| Metric Stratum | Adds the concept of scaling to the affine stratum. |
| Euclidean Stratum | Fixes the scale at unity. |
Application in the Reciprocal System of physical theory
It can be readily seen that the RS concept of “scalar motion” is in actuality that of the projective invariant “cross-ratio”. The only difference is that the cross-ratio is a more generic term than scalar motion; the latter having named aspects of “space” and “time” whereas the former does not name the aspects.
| Projective Stratum cross-ratio | Reciprocal System scalar motion |
|---|---|
| two scalars reciprocally related | two scalars reciprocally related |
| unnamed aspects | named aspects of space and time (defining speed) |
| has magnitude only | has magnitude only |
| non-zero | non-zero |
| finite | (unaddressed, but all motion applied as finite values) |
| cannot be translated | cannot be translated |
| cannot be rotated | cannot be rotated |
| can be scaled | can be scaled |
| Geometric Concept | Application | Results in Reciprocal System: |
|---|---|---|
| Projective Stratum cross-ratio | name aspects space and time | scalar motion |
| Affine Stratum plane at infinity | makes motion relative to other motion resulting in towards/away (in/out) direction of motion | scalar direction; linear vibration |
| Metric Stratum absolute conic | defines angular relationship between motions | scalar rotation |
| Euclidean Stratum fixed scale | defines common perspective by fixing scale at unity | vectorial motion in absolute Euclidean coordinate system |
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