Home Site Map  On Real and ImaginaryLeading to A Most Crucial Question! ↓ 
Please Note 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 contactpoints. 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. 
Click and drag the bold blue line around D to witness elements alternating between real and imaginary states. The essential lesson here is that That is, the elementsets There are no places Indeed, all the tangents and contactpoints 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 contactpoints 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 contactpoints 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 contactpoints 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 contactpoints on the conic, are imaginary.
Here's a .gif animation from long ago –
It does not bring out the alternation of the tangents between real and imaginary states. 
We come with this to a profoundly interesting and utterly fundamental question— For, if there are, And, if there are not, How are such questions as these properly asked, and answered?


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