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January 15

"power of a line" (not a standard name), dual to power of a point

Does anyone know a reference mentioning the dual theorem to the power of a point with respect to a circle, which we might call the "power of a line with respect to a circle"?

For a given line $L$ and a given circle $S$ , for any arbitrary point $P$ along the line, if we take the tangent lines to the circle passing through $P$ , let’s call them $T_{1}$ and $T_{2}$

{\frac {\tan {\tfrac {1}{2}}\angle LT_{1}}{\tan {\tfrac {1}{2}}\angle LT_{2}}}=x

where $\angle LT_{1}$ means the angle that turns line $L$ into $T_{1}$ and $x$ is a constant, our "power of the line with respect to the circle".

This seems like an obvious enough relationship that I am sure someone must have proved it before, but I can’t find a reference in a quick search. Ideas?

If it helps, I made a Geogebra demonstration, https://www.geogebra.org/classic/hsnjfjmt

–jacobolus (t) 00:30, 15 January 2023 (UTC)[reply]

If it helps, Leathem (reviser of Todhunter’s Spherical Trigonometry) discusses the spherical-geometry version on pp. 135–136. –jacobolus (t) 01:17, 15 January 2023 (UTC)[reply]

What is the significance of point C in the demo diagram? Dragging E beyond C changes the ratio between the tangents of the angles. --Lambiam 02:20, 15 January 2023 (UTC)[reply]

I know how to get Geogebra to find the angle between three specified points. I don't know how to teach Geogebra a concept of 'directed lines' or get it to find the angle between a pair of them. If you drag beyond the reference point from which the three-point angles are defined, then you are effectively reversing the orientation of the line, which gives you the supplement of each angle (inverse of each half-angle tangent), so you get

1/x

instead. –jacobolus (t) 03:16, 15 January 2023 (UTC)[reply]

Answering my own question, this is an idea of Laguerre (1882) "Transformations par semi-droites réciproques".

Since Laguerre considers circles and lines to be directed, the two tangents to the circle through a point here are those which cross the circle in the same orientation as the direction of the circle (so one of those is in the opposite sense to my Geogebra doodle above). Then the constant quantity is

\tan {\tfrac {1}{2}}\angle LT_{1}\;\tan {\tfrac {1}{2}}\angle LT_{2},

(the half-tangent of two antipodal angles are negative reciprocals) which is more directly relatable to the "power of a point". –jacobolus (t) 09:07, 17 January 2023 (UTC)[reply]

Laguerre's treatise is also available on the French Wikisource: Transformations par semi-droites réciproques. It is referred to in Spherical wave transformation § Transformation by reciprocal directions. --Lambiam 10:06, 17 January 2023 (UTC)[reply]

There is also a good scan at the internet archive, https://archive.org/details/s3nouvellesannal01pari/page/542/ –jacobolus (t) 23:02, 17 January 2023 (UTC)[reply]

There is a nice discussion of this whole topic in Coolidge:

Coolidge, Julian Lowell (1916). "X. The Oriented Circle". A Treatise on the Circle and the Sphere. Clarendon. pp. 351–407.

–jacobolus (t) 07:45, 20 January 2023 (UTC)[reply]

And that source does call the quantity the power (of an oriented line with respect to an oriented circle), so the term deserves to be standard. See also the footnote on p. 353, referencing: Epstein, 'Die dualistische Ergänzung des Potenzbegriffes', Zeitschrift für mathematischen Unterricht, vol. xxxvii, 1906. (The full title is "Die dualistische Ergänzung des Potenzbegriffes in der Geometrie des Kreises".) I did not find this article online, but it is referenced in the 1910 Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, which writes: "Dieses Produkt wird die Potenz der Geraden in bezug auf den Kreis genennt."^[1] In French there is: L. Crelier, "Puissance d’une droite par rapport à un cercle", Nouvelles annales de mathématiques 4e série, tome 17 (1917), p. 339–345.^[2] (Not to be confused with the term referring to boxing :)^[3].) --Lambiam 09:01, 20 January 2023 (UTC)[reply]

An ellipse or hyperbola is often defined as the locus of points with constant sum (resp. difference) of distances from the two foci. Alternatively, it is the locus of points with a constant product (resp. ratio) of half-tangents (tangents of half the angles) at the foci.

While we're here, does anyone know older references for this theorem about conic sections shown in the picture at right? That is, the locus of points forming triangles with 2 given points such that the product of the half-tangents (tangent of half an angle) at each of the given vertices is constant is an ellipse with the two points as foci, and the locus of points such that the quotient of these half-tangents is constant is a hyperbola with the two points as foci. The only place I can find this described explicitly is:

Akopyan, Arseniy; Izmestiev, Ivan (2019). "The Regge symmetry, confocal conics, and the Schläfli formula" (PDF). Bulletin of the London Mathematical Society. 51 (5): 765–775. doi:10.1112/blms.12276.

which does not cite any older sources.

In the diagram to the right, we are using $\alpha ,\beta$ to mean the half-tangents of the marked angles (not the angle measures), and the operation

x\oplus y=\tan(\arctan x+\arctan y)={\dfrac {x+y}{1-xy}},

so the identities listed to the right are a form of Mollweide's formula for the triangle $\triangle PF_{1}F_{2}.$

Again, this seems like such a basic metrical triangle/conic identity that I would expect it to have been mentioned repeatedly in old trigonometry/geometry literature, but I haven’t been able to find it. –jacobolus (t) 16:47, 20 January 2023 (UTC)[reply]

Perhaps "tangential", but it occurred to me that this operation