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Triangle Triangle Picking

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Triangle triangle picking

The problem of finding the mean triangle area of a triangle with vertices picked inside a triangle with unit area was proposed by Watson (1865) and solved by Sylvester. It solution is a special case of the general formula for polygon triangle picking.

Since the problem is affine, it can be solved by considering for simplicity an isosceles right triangle with unit leg lengths. Integrating the formula for the area of a triangle over the six coordinates of the vertices (and normalizing to the area of the triangle and region of integration by dividing by the integral of unity over the region) gives

A^_=(int_0^1int_0^(x_1)int_0^1int_0^(x_2)int_0^1int_0^(x_3)|Delta|dy_3dx_3dy_2dx_2dy_1dx_1)/(int_0^1int_0^(x_1)int_0^1int_0^(x_2)int_0^1int_0^(x_3)dy_3dx_3dy_2dx_2dy_1dx_1)
(1)
=8int_0^1int_0^(x_1)int_0^1int_0^(x_2)int_0^1int_0^(x_3)|-x_2y_1+x_3y_1+x_1y_2-x_3y_2-x_1y_3+x_2y_3|dy_3dx_3dy_2dx_2dy_1dx_1,
(2)

where

Delta=1/(2!)|x_1 y_1 1; x_2 y_2 1; x_3 y_3 1|
(3)

is the triangle area of a triangle with vertices (x_1,y_1), (x_2,y_2), and (x_3,y_3).

The integral can be solved using computer algebra by breaking up the integration region using cylindrical algebraic decomposition. This results in 62 regions, 30 of which have distinct integrals, each of which can be directly integrated. Combining the results then gives the result

A^_=1/(12)
(4)

(Pfiefer 1989; Zinani 2003).

TriangleTrianglePickingDistribution

The exact distribution function D(A) was derived by Philip. P(A) and D(A) are given by

P_1(A)=-1/(1+sqrt(1-4x)-4x)[4(-3(1+sqrt(1-4x))-3ln2+x[4pi^2(1+sqrt(1-4x)-4x)(1+x)+3(5+sqrt(1-4x)-22ln2+4x(-1+26ln2))]+[3+6(11-52x)x]ln(1+sqrt(1-4x))+3xlnx[-1+sqrt(1-4x)+(4+52sqrt(1-4x))x+3x(-1-sqrt(1-4x)+4x)lnx]3[ln2-ln(1+sqrt(1-4x))][sqrt(1-4x)(1+2(11-52x)x)+12(1+sqrt(1-4x)-4x)x(1+x) ×[ln2-ln(1+sqrt(1-4x))+lnx]])] P_2(A)=2[6-2pi^2x^2+pisqrt(4x-1)+26pixsqrt(4x-1)-6[-4pix(1+x)+sqrt(4x-1)(1+26x)]csc^(-1)(2sqrt(x))-72x(1+x)[csc^(-1)(2sqrt(x))]^2-3lnx-2x(3+pi^2+9lnx(4+lnx))] D_1(A)=8(2x^3+3x^2)[3ln((1+sqrt(1-4x))/2)×((1+sqrt(1-4x))/(2x))-(pi^2)/3]+2/5(324x^2+28x-1)[1/2lnx-ln((1+sqrt(1-4x))/2)]sqrt(1-4x)+12x^3(lnx)^2-(54x^2+6x-1/5)lnx-(57)/5x^2+(62)/5x D_2(A)=8(2x^3+3x^2)[2pi-3cos^(-1)(1/(2sqrt(x)))cos^(-1)(1/(2sqrt(x))) -(pi^2)/3]+2/5(324x^2+28x-1)[cos^(-1)(1/(2sqrt(x)))-pi/3]sqrt(1-4x)-18x^2(lnx)^2-(54x^2+6x-1/5)lnx-(57)/5x^2+(62)/5x,
(5)

where the subscript 1 denotes the region with 0<=A<=1/4 and 2 denotes the region with 1/4<A<=1.

The raw moments mu_n^' of P(A) for n=1, 2, ... are 1/12, 1/144, 31/9000, 1/450, 1063/617400, 403/264600, ... (OEIS A103474 and A103475).

The central moments mu_n of P(A) for n=1, 2, ... are 0, 1/144, 61/54000, 343/864000, 9493/66679200, ... (OEIS A130117 and A130118).

See also

Disk Triangle Picking, Heilbronn Triangle Problem, Hexagon Triangle Picking, Polygon Triangle Picking, Square Triangle Picking, Sylvester's Four-Point Problem, Tetrahedron Tetrahedron Picking, Triangle Area, Triangle Line Picking, Triangle Point Picking

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References

Pfiefer, R. E. "The Historical Development of J. J. Sylvester's Four Point Problem." Math. Mag. 62, 309-317, 1989.Philip, J. "The Area of a Random Convex Polygon in a Triangle." Tech. Report TRITA MAT 05 MA 04. n.d. http://www.math.kth.se/~johanph/area2.pdf.Sloane, N. J. A. Sequences A103474, A103475, A130117, and A130118 in "The On-Line Encyclopedia of Integer Sequences."Watson, S. "Question 1229." Mathematical Questions, with Their Solutions, from the Educational Times, Vol. 4. London: F. Hodgson and Son, p. 101, 1865.Zinani, A. "The Expected Volume of a Tetrahedron Whose Vertices are Chosen at Random in the Interior of a Cube." Monatshefte Math. 139, 341-348, 2003.

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Triangle Triangle Picking

Cite this as:

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

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