﻿ Projection

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# Projection

### Propagation via Incidence

At the very heart of geometric construction, especially of transformation, we find the notion of projection: the idea  that any geometric element is capable of being pro - jected, of being “thrown on”, or “cast forward”; there is a sense of elements being “sent elsewhere” by projection.

But projection is not displacement, for elements can not be moved by projection, because they are not physical objects that can be dragged about, or tossed around.  Projection is more like copying-on, or propagation, or duplication, because an element and its projection both exist – along with whatever element facilitates the projection. Thus, every projection is an interval(*). But projection is unlike these processes in that the concepts ‘original’, and ‘copy’, are meaningless.

But even this is not “in the Gold”.

Projections are states or concatenations**
of elementary incidence.

Crucially, concatenations need not be, or need not have been,
sequential in time.

This is because Time is not on the list
of geometric elements * and their incidences.

That is to say, Time is not a geometric element,
and, prima facie, should have no incidence with geometric elements.

Nevertheless, the ideas of action and agent, experienced as ‘working’ in Time, are not easily set aside when speaking of projection.  These ideas are almost certainly purely anthropomorphic, but if, for the sake of argument, we provisionally accept them, then ...

... projections may be
1. intentional and voluntary, in the sense that the concatenations of incidence are artificially our doing, or they may be
2. unintentional and involuntary, in the sense that the concatenations of incidence exist by ‘natural’ agency.
Clearly, geometers dreaming up constructions are agents of the first, voluntary kind.

Many would consider that the Laws of Physics, and of exact science generally, are agents of the second, involuntary kind.

But, as we can now appreciate, this assumes
a
working incidence of geometric elements
with physical elements

—and, for that matter, with Time.

This ‘working incidence’ cannot be established from first principles.

## Listed below

are the projections available on first principles to the Three True Elements of Geometry, Point, Line and Plane.

Without exception, every single-stage projection requires a condition of incidence, and  every such projection forms a pair of intervals.
1.     Central Projection:
A Point may be projected by a Line.

2. The projection is of the point along the line, and the line joins the source point to the destination point.  This is the condition of incidence.  Both points and the line exist and persist.

3.     Planar Projection:
A Line may be projected by a Plane.

4. The projection is of the line around a point in the plane, and the plane joins the source line to the destination line.  This is the condition of incidence.  Both lines and the plane exist and persist.

5.     Linear Projection:
A Plane may be projected by a Line.

6. The projection is of the plane around the line, and the line joins the source plane to the destination plane.  This is the condition of incidence.  Both planes and the line exist and persist.

### However,

• A Line may not be projected into a distinct, Skew Line.

• The projection is prevented by the absence of any condition of incidence. Skew lines have no incidence of any kind.  If there is no incidence, there is no projection.
• Do please note that only lines can be skew to each other.  It is not possible for points to be skew to each other, or for planes to be skew to each other.