OFFSET

1,1

COMMENTS

The Philo line of a point P inside an angle T is the shortest segment that crosses T and passes through P. Suppose that T is the angle formed by the positive x and y axes and that h>0 and k>0. Notation:

...

P=(h,k)

L=the Philo line of P across T

U=x-intercept of L

V=y-intercept of L

d=|UV|

...

Although Philo lines are not generally Euclidean-constructible, exact expressions for U, V, and d can be found for the angle T under consideration. Write u(t)=(t,0), let v(t) the corresponding point on the y axis, and let d(t) be the distance between u(t) and v(t). Then d is found by minimizing d(t)^2:

d=w*sqrt(1+(k/h)^(2/3)), where w=(h+(h*k^2))^(1/3).

...

Guide:

h....k...........d

1....2........A197008

1....3........A197012

1....4........A197013

2....3........A197014

3....4........A197015

1..sqrt(2)....A197031

...

The cube root of any positive number can be connected to the Philo lines (or Philon lines) for a 90-degree angle. If the equation x^3-2 is represented using Lill's method, it can be shown that the path of the root 2^(1/3) creates the shortest segment (Philo line) from the x axis through (1,2) to the y axis. For more details see the article "Lill's method and the Philo Line for Right Angles" linked below. - Raul Prisacariu, Apr 06 2024

LINKS

R. J. Mathar, OEIS A197008

Raul Prisacariu, Lill's method and the Philo Line for Right Angles.

EXAMPLE

d=4.161938184941462752390080...

x-intercept: U=(2.5874..., 0)

y-intercept: V=(0, 3.2599...)

MAPLE

(1+2^(2/3))^(3/2); evalf(%) ; # R. J. Mathar, Nov 08 2022

MATHEMATICA

f[x_] := x^2 + (k*x/(x - h))^2; t = h + (h*k^2)^(1/3);

h = 1; k = 2; d = N[f[t]^(1/2), 100]

RealDigits[d] (* this sequence *)

x = N[t] (* x-intercept; -1+4^(1/3); cf. A005480 *)

y = N[k*t/(t - h)] (* y-intercept *)

Show[Plot[k + k (x - h)/(h - t), {x, 0, t}],

ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 4}], PlotRange -> All, AspectRatio -> Automatic]

CROSSREFS

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 10 2011

STATUS

approved