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= Torx =
This hexalobular joint is commonly known as the registered name "Torx". Unfortunately data is not public so close assumptions are made here.

Example of generated curves with this app:

For "Torx", shapes are defined with the following ratio's:

||ra = 1||rb ≈ 0,72 ra||rc ≈ 0,1 ra||rd ≈ 0,175 ra||

The size table is close, but data differs slightly on the net. The table relates to designation and diameter, where diameter = 2*ra. You should consider that inside and outside shapes have a constant section over its height, while bits can have a certain slope. Some tolerance between inside and outside shape is mandatory for proper functioning.

||#||Dia. mm (inch)||#||Dia. mm (inch)||#||Dia. mm (inch)||#||Dia. mm (inch)||
||T3||1,17 (0,046)||T8||2,31 (0,09)||T25||4,43 (0,173)||T50||8,83 (0,346)||
||T4||1,28 (0,05)||T9||2,5 (0,098)||T27||4,99 (0,195)||T55||11,22 (0,44)||
||T5||1,42 (0,055)||T10||2,74 (0,107)||T30||5,52 (0,216)||T60||13,25 (0,519)||
||T6||1,7 (0,066)||T15||3,27 (0,128)||T40||6,65 (0,26)||
||T7||1,99 (0,078)||T20||3,86 (0,151)||T45||7,82 (0,306)||

For additional data, see https://en.wikipedia.org/wiki/Torx

= Mathematics =

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With radii ra, rb, rc and number of lobes n given, rd is to be determined.
 * Corner ae=pi/n radians or 180/n degrees, so for 6 lobes, ae is 30 degrees.
 * Triangle:
  * df=ra-rc
  * dg+dh=rc+rd
  * di=rb+rd
  * cos(ae)=j
(omitting prefixes)
{{{
(c+d)^2=(a-c)^2+(b+d)^2-2(a-c)(b+d)j
c^2+d^2+2cd=a^2+c^2-2ac+b^2+d^2+2bd-2jab-2jad+2jbc+2jcd
2cd-2bd+2jad-2jcd=a^2-2ac+b^2-2jab+2jbc
rd=(a^2-2ac+b^2-2jab+2jbc)/(2c-2b+2ja-2jc)
rd=((a^2+b^2)/2-ac-jab+jbc)/(c-b+ja-jc)
}}}

In a spreadsheet with A1=j, A2=a, A3=b, A4=c and A5 is calculation of d, A5 is:
{{{
=(A2^2-2*A2*A4+A3^2-2*A1*A2*A3+2*A1*A3*A4)/(2*A4-2*A3+2*A1*A2-2*A1*A4)}}}