Triangular orbits in elliptic billiards - Triangle geometry
Purpose
I made quicky current public worsheet to share about some of my computations.
Parametrization in triangle geometry
I got following parametrization of a 3-orbit triangle ABC in the circumbilliard :
w=(1−t2)(1−2k−(t−k)2)a=s2t−k−22(t−1)b=s2t2−(t+1)k−2t2−(t+1)k−1+wc=s2t2−(t+1)k−2t2−(t+1)k−1−wwith parameters k as ratio of inradius and circumradius (k=Rr), t as cosine of internal angle at A vertex, and s as semiperimeter.
It comes from a Ravi substitution (a=x+y, b=y+z, c=z+x) and three equations in x,y,z unknowns given parameters t,s,k :
x+y+z=s(x+y)(x+z)(y+z)4xyz=k(x+z)(y+z)z2+xz+yz−xy=tSolving is best (simplest) for the couple of first and third equations, expressing y,z from x.
Putting back expression into second equation we have the binomial equation :
x2−mx+n=0
where sum of roots is m=2(1−t)+k2(1−t)s and product is n=2(1−t2)+(t+1)k(1−t)ks2
Choosing one root or the other is only changing sign of w (or swapping b and c).
Cosine circle
Circle centered at Mittenpunk Mp and with radius k+42s is the cosine circle for the excentral triangle T.
The excentral triangle T has for vertices, XA XB XC, the excenters of the reference ABC triangle.
Its symmetrical image T' wrt Mittenpunkt ParseError: KaTeX parse error: Expected group after '_' at position 2: M_̲ has vertices YA, YB and YC.
Mittenpunkt has trilinear coordinates : Mp=z:x:y.
Triangle T and its symmetrical image T' intersect at six points M1,...,M6 which are on the cosine circle.
Trilinear coordinates of these points are :
M1=a−b−3c:−a+b−c:a−b+cM2=−a−b+c:3a−b+c:a+b−cM3=−a+b+c:a−b−c:a+3b−cand
M4=−a+b−c:a−b+c:3a+b−cM5=a−3b−c:a+b−c:−a−b+cM6=−a+b+c:a−b+3c:a−b−cComputing square of quadrance (=squared distance) between Mp and M1 we get squared radius of cosine circle :
rMp2=(a2−2ab+b2−2ac−2bc+c2−2abc)2=(2(xy+xz+yz)(x+y)(x+z)(y+z))2and using equations for k and s : rMp=k+42s
Parallels with respect to ABC edges through Mp, cut disk in six sectors, and cutting points on cosine circle are intersection of T and T' edges.
Squared cosines of internal angles of external triangle T are :
cos(XA)2=(x+z)(y+z)xycos(XB)2=(y+x)(z+x)yzcos(XC)2=(z+y)(x+y)zxProduct : cos(XA)2cos(XB)2cos(XC)2=((x+y)(y+z)(z+x)xyz)2=(4k)2.
Sum of squared cosines is : cos(XA)2+cos(XB)2+cos(XC)2=1−2k.
Sum of product 2 by 2 of squared cosines is : 161(k2−8k+r2s2k2) where r2 is the changing squared inradius of ABC : r2=x+y+zxyz.
Exradii are :
ra=s−ars=rxx+y+zrb=s−brs=ryx+y+zrc=s−crs=rzx+y+z