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SYNTRACTRIX, CONVICT'S CURVE

Curve considered by Giovanni Poleni in 1729 for the design
of a tractrix mechanism to build logarithms. Name given by Ricatti in 1755.
Curve then studied by Sylvester.
Other name: Poleni curve. Syn is a Greek prefix meaning "with".
See Brocard part. comp. page 173. See http://www.spm.uem.br/bspm/pdf/vol21/parte-5-earp_BSPM.pdf page 12. |

Cartesian parametrization ,
the cases k = 0 , k = 1 et k = 2 correspond, respectively,
to the line, the tractrix, and the convict's curve.
Curvilinear abscissa: (therefore, s = at in the case of the convict's curve).
Radius of curvature: . Hence the intrinsic equation 1 in the case of the convict's curve: |

The *syntractrices* are the curves described by the
points of a tangent to the tractrix.

If the point *N* describes the tractrix, and *P*
is the point on the tangent at *N* located on the asymptote to the
tractrix (we know that *NP* = *a = constant*), then the above
parametrization is that of the point *M* such that .

When N is the middle of [PM] (i.e. k
= 2), the curve is called convict's curve, which is also a special
case of elastic curve.
Here is a possible explanation for this name: A very heavy wagon is located at N, middle of
a fixed rod MP; a first convict is at P and pulls the wagon
N
along a line; the wagon therefore describes a tractrix; a second convict
pushes at M and therefore describes the convict's curve.
More generally, when the front wheels of a vehicle describe a line, then the various points on its symmetry axis describe syntractrices. |
The calculation of the curvilinear abscissa above shows
that the convict's curve is the curve that the end of a rod must describe,
when the other one has a linear motion, so that the two ends ( |

The convict's curve is the median
along Ox of the semicircle
and the tractrix . |

The convict's curve is also the special case k
= 2 of the family of curves with intrinsic equation 1:,
curves that do not have a given name, but are parametrized by:
Opposite, an animation for k ranging from 0 to
6, with stops for k =1 (catenary
of equal strength), k = 2 (convict's curve), and k =
4. |

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© Robert FERRÉOL 2021