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On the conics page it says that there are just seven conic sections:
one of these sections is a pair of straight lines.

Five points on the plane uniquely define a conic. If any three of those points are co-linear, the conic becomes a pair of straight lines.

This interactive shows a set of five such points, with the conic they define. Try dragging any three of the five points into a straight line (the easiest way to do this is to click on any point of the five and “take it on a tour” completely round any one of the other four points. The toured and touring points, in the course of this tour, line up on each of the others in turn).

Or just start the animation!

Please enable Java for an interactive construction (with Cinderella). You will see the conic go through all its forms (except the point-form), including hyperbolae which degenerate into line-pairs.

The variations of the transformation of a line into itself are developed here with respect to a pair of straight lines, but, as we now see above (and here),
such a pair is a conic section, so now we can perhaps surmise that the transformation of the line should work with conics generally,
for if it works with one, it should work with all.

In fact, it works because, as far as the geometry is concerned, the conic sections are indistinguishable.

And indeed we obtain all the measures from the general conic that we obtain from a line-pair—and one more besides , namely:

imaginary circling measure.

This measure is not available from the line-pair, for, while a line-pair does divide its plane into two regions, it is uniquely the case for this conic that neither of these regions is “closed”, with the meaning that it is not possible for a third straight line to avoid cutting both of the lines of the line-pair in two, real points. (Try it! Draw a couple of lines on paper, and try to avoid cutting them both with a third line.)

Most significantly, a straight line can avoid cutting all the other kinds of conic.

Please enable Java for an interactive construction (with Cinderella).

The bold conic on the left is currently shown cut by two straight lines, one black, one white, at two real points per line.

A measure of real points has been constructed on the black line, using the conic and the cutting points on the white line. (The points of this measure may be moved by dragging the larger yellow point on the conic.)

You may observe that they are entirely contained between their line's cutting points either on the section without the point at infinity, or on the one with it, but not on both at the same time. This is Growth Measure, and on either section, there is at least an infinity of intervals to be had.

Notwithstanding, the measure has real limits, precisely at the cutting points.

Now try moving the black line clear of the conic.

You should see the measure still, but with the limits removed—because the limits are exactly where the black line cuts the conic, and so will be located and findable only when it does so. Without these limits, this measure “circulates” forever (to see this, drag the yellow point round the bold conic), but, in general,“recycles” after a finite number of steps, though it recycles indefinitely, and allows unlimited rotation. This is the only style of transformation that does this.

Compare this behaviour with that of the Growth Measure, which actually does limit (that is, halt)—no doubt after at least an infinite number of steps *, but inevitably. It does not recycle, and allows only limited rotation.

Next, while keeping the black line clear of the bold conic, move the white line clear of the bold conic as well.

Along with the white line's two cutting points on the bold conic, the entire net of blue lines defining the measure on the black line disappears, so the entire measure on the black line vanishes, too—but the bold conic, and the various other, heretofore unmentioned, conics showing in the construction, do not vanish! Why not?

What are all these conics?

Well, you will observe that they arise at the intersections of the blue, measure-defining net. There are two conics for the meets of the blue lines connecting (via the directrices) every second point of the measure, for example, and two more conics for the meets of those connecting every third, and so on.

Now, we happened to construct our measure on a straight line - but a straight line is a degenerate conic. (It is that conic section that stays exactly in/on the surface of the cone, in which our two straight lines above degenerate into one line- which, please do note, nevertheless retain their identities, just as indistinguishable twins do.) The directrices will always transform the points in this conic exactly back to where they were. This is to say that—

  • the transformation zero-steps on this conic, which is a pair, as we noted just above - let us say the first pair
  • the transformation single-steps on the pair of conics flanking this first pair
  • the transformation double-steps on the pair flanking this second pair
  • the transformation triple-steps on the pair flanking this third pair
  • ....
  • the transformation n-steps on the pair flanking the nth pair
  • ....
Please enable Java for an interactive construction (with Cinderella).

Why do the conics come in pairs like this?

They do so because the stepping can proceed on a conic in either of two senses

For example,

  • right-to-left (←), and left-to-right (→), or,
  • up (↑) and down (↓), or, perhaps,
  • clockwise and anticlockwise.

There is a conic per sense per step-rate.

We have shown just a few of these conics, but it is plain that there are at least infinitely many of them.

How is it that these conics all survive the destruction of the net and of the measure?
Or is that destruction more apparent than actual?

This “circling scenario” seems very rich.


* This wording, though not wrong, may mislead, as it suggests that the steps of a transformation must occur sequentially in real time. But this is to “physicalize” geometry, and they need not so occur: all the steps can be and could have been taken simultaneously. Indeed, there is a sense in which the steps are never taken, not even for a first time, because they have always been! In any case, the axioms of incidence do not alter with time.


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