Projective Comparison of Skew RulersIf the lines of the red and blue rulers are skew to each other, there is no way to compare intervals on them by direct elementary incidence, so we must instead compare them by indirect incidence, via an intermediate ruler (here, on the green line). The red ruler and the green are “polyperspectivities” in point M, while the green ruler and the blue are “polyperspectivities” in point N — so that we have a “polyprojectivity”. It works because the green line is the incidence of the two planes, red π_{M} and blue π_{N}, carrying the perspectivities. ↑We may have been here before.↑ We may care to think - strictly poetically - of the green ruler as the “central projection” (shadow) of the red ruler, by M, into or onto the green line, and of the blue ruler as the “central projection” (shadow) of the green ruler, by N, into or onto the blue line, or vice versa. |
So, by conscious construction, we have all three rulers, red, green and blue, projectively identical. With that established, it is of great interest to link corresponding interval-ends of the red and blue rulers directly, rather than indirectly.
We get a set of (here, black) lines
in which each is skew to all the other lines in the set. This means that they have no mutual incidence of any kind — so they cannot lie in a surface, at least not one composed of geometric elements, because none is available. It is certainly not a developable surface. This vital issue is discussed here.
What we seem to see is
the black skew lines forming a line-wise ‘envelope’, perhaps tangent to some sort of curve (to see this more clearly, drag the yellow dot, K, back and forth: adjust their distribution with U and D). Now, they cannot actually do this , because, as already noted, they have no mutual incidence, and so have nothing to which to be tangent. What we in fact see are their "shadows" on the plane of the screen, and because these shadows are coplanar , they are not skew, so they are incident, and could be tangents to a curve - which might be one of the conics. Issues of continuity remain, but this is the first curve arising exclusively from Elementary Incidence.And if it is in fact a continuous curve, with it we can, as promised, tackle the imaginary, and with that, projective rotation. |