Roulette 2 To 1
Given two curves C1 and C2 and a point P attached to curve C2. Now let curve C2 roll along curve C1, without slipping. Then P describes a roulette1).
When P is on curve C2, the curve is called a point-roulette. When P is on a line attached to C2, the curve is called a line-roulette.
When working in the complex plane and both curves C1 and C2 are expressed as function of the arc length, the resulting curve z can be written as a function of C1 and C2 2).
The first to describe these curves where Besant (1869) and Bernat (1869), respectively.
The following roulettes can be distinguished, for a curve C2 that is rolling over a curve C1:
Roulette 2-1 Bets
| fixed curve C1 | rolling curve C2 | fixed point on curve C2 | roulette |
| any curve C | line | on line | involute of curve C |
| catenary | line | center of catenary | line |
| circle | circle (on outside) | on circle | epicycloid |
| circle (on inside) | on circle | hypocycloid | |
| line | circle | on circle | cycloid |
| circle | outside circle | prolate cycloid | |
| circle | inside circle | curtate cycloid | |
| cycloid | center | ellipse | |
| ellipse | focus | elliptic catenary | |
| hyperbola | focus | hyperbolic catenary | |
| hyperbolic spiral | pole | tractrix | |
| involute of a circle | center | parabola | |
| involute of a circle | any point | circle | |
| logarithmic spiral | any point | line | |
| parabola | focus | catenary | |
| parabola | (equal) parabola | vertex | cissoid |
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Roulette 2 To 1 Strategy
Each of these bets covers 12 numbers and will normally payout at 2:1, so if you bet 1 you win 3. To use the Martingale on these bets you need to change the progression slightly so it looks like this: 1 – 1 – 2 – 3 – 4 – 6 – 9 – 14 – 21 – 31 – 47 – 70 – 105 – 158 – 237 – 355. Hitman Roulette is a tool that aims to give new ideas on how to complete missions in Hitman Blood Money and HITMAN 1-2 by giving a set of random guidelines to follow when playing. Currently it randomizes things such as entry point, kill order and kill method as well as a set of extra variables, some of which are generic and some mission specific.
The point-roulettes for which a circle rolls on a line or on another circle, are known as cycloidal curves.
Some authors name the roulette a spirographic curve.
Roulette Table 2 To 1
The same curves can be defined as a glissette 3): as the locus of a point or a envelope of a line which slides between two given curves C1 and C2.
An wide-known example of a glissette is the astroid.
1) rouler (Fr.) = to roll.
In Dutch: rolkromme.
2) The formula follows from the isometric insight that C1(t) - z(t) / C2(t) - z(0) = C1'(t) / C2'(t).
Example for the cycloid: C1(t) = t and C2(t) = i - i e it where t is the arc length.
3) glisser (Fr.) = to glide.