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Triangle Line Picking

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Isosceles triangle line picking

Triangle line picking is the selection of the line segment joining two independent points chosen uniformly by area in a nondegenerate triangle. For a triangle Delta(a,b,c), the side lengths a, b, and c are positive and satisfy the strict triangle inequality. This problem is not affine, but a simple formula in terms of the area or linear properties of the original triangle can be found using Borel's overlap technique to collapse the quadruple integral to a double integral and then convert to polar coordinates, leading to the beautiful general formula

l^__(Delta(a,b,c))=(4ss_as_bs_c)/(15)[1/(a^3)ln(s/(s_a))+1/(b^3)ln(s/(s_b))+1/(c^3)ln(s/(s_c))]+(a+b+c)/(15)+((b+c)(b-c)^2)/(30a^2)+((c+a)(c-a)^2)/(30b^2)+((a+b)(a-b)^2)/(30c^2),
(1)

(A. G. Murray, pers. comm., Apr. 4, 2020), where s=(a+b+c)/2 is the semiperimeter and s_a=s-a, s_b=s-b, and s_c=s-c. The formulas for odd moments have a similar form to that of the mean but with higher powers of a, b, c, and the triangle area Delta=sqrt(ss_as_bs_c).

This formula immediately gives the following special cases.

Equilateral triangle line picking
EquilateralTriangleLinePickingDistribution

If the original triangle is chosen to be an equilateral triangle with unit side lengths, then the mean length of the line segment joining two independent points chosen uniformly inside it is given by

l^__(Delta(1,1,1))=(int_0^1int_0^1int_0^(sqrt(3)(1/2-|x_1-1/2|))int_0^(sqrt(3)(1/2-|x_2-1/2|))sqrt((x_1-x_2)^2+(y_1-y_2)^2)dy_2dy_1dx_2dx_1)/(int_0^1int_0^1int_0^(sqrt(3)(1/2-|x_1-1/2|))int_0^(sqrt(3)(1/2-|x_2-1/2|))dy_2dy_1dx_2dx_1)
(2)
=(16)/3int_0^1int_0^1int_0^(sqrt(3)(1/2-|x_1-1/2|))int_0^(sqrt(3)(1/2-|x_2-1/2|))sqrt((x_1-x_2)^2+(y_1-y_2)^2)dy_2dy_1dx_2dx_1.
(3)
TriangleLinePickingRegions

The integrand can be split up into the four pieces

I_1=int_0^(1/2)int_0^(1/2)int_0^(x_1sqrt(3))int_0^(x_2sqrt(3))sqrt((x_1-x_2)^2+(y_1-y_2)^2)dy_2dy_1dx_2dx_1
(4)
I_2=int_0^(1/2)int_(1/2)^1int_0^(x_1sqrt(3))int_0^((1-x_2)sqrt(3))sqrt((x_1-x_2)^2+(y_1-y_2)^2)dy_2dy_1dx_2dx_1
(5)
I_3=int_(1/2)^1int_0^(1/2)int_0^((1-x_1)sqrt(3))int_0^(x_2sqrt(3))sqrt((x_1-x_2)^2+(y_1-y_2)^2)dy_2dy_1dx_2dx_1
(6)
I_4=int_(1/2)^1int_(1/2)^1int_0^((1-x_1)sqrt(3))int_0^((1-x_2)sqrt(3))sqrt((x_1-x_2)^2+(y_1-y_2)^2)dy_2dy_1dx_2dx_1.
(7)

As illustrated above, symmetry immediately gives I_2=I_3 and I_1=I_4, so

l^__(eq.)=(32)/3(I_1+I_2).
(8)

With some effort, the integrals I_1 and I_2 can be done analytically to give the final beautiful result

l^__(Delta(1,1,1))=1/(20)(4+3ln3)
(9)
=0.364791843300....
(10)

(OEIS A093064; E. W. Weisstein, Mar. 16, 2004).

IsoscelesRightTriangleLinePickingDistribution

If the original triangle is chosen to be an isosceles right triangle with unit legs, then the mean length of the line segment joining two independent points chosen uniformly inside it is given by

l^__(Delta(1,1,sqrt(2)))=4int_0^1int_0^1int_0^(1-x_1)int_0^(1-x_2)sqrt((x_1-x_2)^2+(y_1-y_2)^2)dy_2dy_1dx_2dx_1
(11)
=1/(30)[2+4sqrt(2)+(4+sqrt(2))sinh^(-1)1]
(12)
=1/(60)(4+8sqrt(2)+sqrt(2)cosh^(-1)3+8sinh^(-1)1)
(13)
=0.414293....
(14)

(OEIS A093063; M. Trott, pers. comm., Mar. 10, 2004), which is numerically surprisingly close to sqrt(2)-1=0.414213....

The mean length of a line segment picked at random in a 3, 4, 5 triangle is given by

l^__(Delta(3,4,5))=(341)/(375)+(48)/5((ln2)/(3^3)+(ln3)/(4^3)+(ln6)/(5^3))
(15)
=(341)/(375)+(2432ln2)/(5625)+(567ln3)/(2500)
(16)
=1.4581846....
(17)

(E. W. Weisstein, Aug. 6-9, 2010; OEIS A180307).

The mean length of a line segment picked at random in a 30-60-90 triangle with unit hypotenuse was computed by E. W. Weisstein (Aug. 5, 2010) as a complicated analytic expression involving sums of logarithms. After simplification, the result can be written as

l^__(30-60-90)=(17)/(120)+(sqrt(3))/(40)+(9ln3)/(160)+(9+8sqrt(3))/(720)ln(2+sqrt(3))
(18)
=0.2885717....
(19)

(E. Weisstein, M. Trott, A. Strzebonski, pers. comm., Aug. 25, 2010; OEIS A180308).

The expected distance from a random point in a general triangle to the vertex A opposite the side of length a is

d_(Delta(a,b,c),A)=(4s(s-a)(s-b)(s-c))/(3a^3)ln(s/(s_a))+(b+c)/6+((b+c)(b-c)^2)/(6a^2).
(20)

(A. G. Murray, pers. comm., Apr. 4, 2020), with analogous expressions for d_(Delta(a,b,c),B) and d_(Delta(a,b,c),C). These give the beautiful identity

l^__(Delta(a,b,c))=1/5[d_(Delta(a,b,c),A)+d_(Delta(a,b,c),B)+d_(Delta(a,b,c),C)].
(21)

(A. G. Murray, pers. comm., Apr. 4, 2020).

See also

Equilateral Triangle, Isosceles Right Triangle, Square Line Picking, Triangle Point Picking, Triangle Triangle Picking

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References

Pure, R.; Durrani, S.; Tong, F.; and Pan, J. "Distance Distribution Between Two Random Points in Arbitrary Polygons." Math. Meth. Appl. Sci. 45, 2760-2775, 2022. https://doi.org/10.1002/mma.7951.Sheng, T. K. "The Distance Between Two Random Points in Plane Regions." Adv. Appl. Prob. 17, 748-773, 1985. https://doi.org/10.2307/1427086.Sloane, N. J. A. Sequences A093063, A093064, A180307, and A180308 in "The On-Line Encyclopedia of Integer Sequences."

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Triangle Line Picking

Cite this as:

Weisstein, Eric W. "Triangle Line Picking." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TriangleLinePicking.html

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