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On Real and Imaginary

Leading to A Most Crucial Question! ↓

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July 2017: CindyJS javascript renderings of geometric graphics do not yet properly name lines, or allow animations.
Specification

Real versus Imaginary Studied in Relation to Polarity

 Modality

Two real tangents may be drawn to any conic* from a point “outside” the conic, which is to say, from a point somewhere in a line that entirely misses the conic.

A conic's closure does not specify which of the two regions it encloses is the outer, so the above will give our intended meaning of "outside".

Tangents drawn “abstractly”, as perhaps we may put it for now, to contact the conic from a point “inside” the conic, are, by contrast, imaginary – as indeed are their contact-points.

Real, black point D and real, black chord d are pole and polar, respectively (rsp.), with respect to (w.r.t.) the conic.

Blue point P (at the intersection of yellow line q through pole D and polar line d) and blue line p are pole and polar, rsp., w.r.t. the conic.

Yellow point Q (at the intersection of blue line p through D and black line d) and yellow line q are pole and polar, rsp., w.r.t. the conic.

* except when the conic degenerates, either into a pair of lines, or to a point.

Click and drag the bold blue line around D to witness elements alternating between real and imaginary states.
Certain blue elements pass into the abstract imaginary state when corresponding yellow elements pass
into the non-abstract real
- and vice versa.

The essential lesson here is that
all these elements co-exist.
Only their states alternate.

That is, the element-sets
{A,a,B,b} and {G,g,H,h},
do not “trade” their existences
according to whether chords p and q
miss or intersect the conic.
They trade states –
if one set is real, the other is imaginary.

There are no places
at which tangents, a,b,g,h
and contact-points, A,B,G,H
are all imaginary,

but there are two places at which they
are all real.

These two are the places at which the chords p and q contact the conic as tangents, and also degenerate – that is, they become, to all seeming (though not in actual fact),
one chord.

Indeed, all the tangents and contact-points degenerate at these places.

If real point P is outside, line p (through real point Q, inside) is a real chord cutting the conic in two real points, A and B, which are also the contact-points on the conic of two real, blue tangents, a and b, from P.

Then real point Q must be inside, and line q (through point P, outside) is a real chord cutting the conic in two imaginary points, G and H.

Similarly, the tangents, g and h, from Q to these contact-points on the conic, are imaginary.

If point Q is outside, line q is a real chord (through real point P, inside) cutting the conic in two real points, G and H, which are also the contact-points on the conic of two real, yellow tangents, g and h, from Q.

Then real point P must be inside, and line p (through point Q, outside) is a real chord cutting the conic in two imaginary points, A and B.

Similarly, the tangents, a and b, from P to these contact-points on the conic, are imaginary.

 

We come with this to a profoundly interesting and utterly fundamental question—

Are there physical elements properly incident on geometric elements?

For, if there are,
and some these are incident on geometric elements currently in an imaginary state,
can we expect to find these physical elements in a correspondingly imaginary state?

And, if there are not,
or there is only some incidence,
how may we show that geometry is fit for physical science?

How are such questions as these properly asked, and answered?

 
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