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BASE CURVE, ROLLING CURVE AND ROULETTES
IN A RIGID MOTION OF A MOVING PLANE OVER A FIXED PLANE
The movement of a plane over a fixed plane has been studied
globally by W. H. Besant
in 1869. Dürer in 1525, D. Bernoulli, La Hire, Desargues, Leibniz,
Newton, Maxwell, etc., were precursors and studied some specific cases.
Website: http://www.tfhberlin.de/~schwenk/Lehrgebiete/AUST/Welcome.html 
The base curve of the movement of a plane over
a fixed plane is the curve generated on the fixed plane by the instant
centre of rotation of the moving plane.
The rolling curve is the curve generated by this
point on the moving plane.
The roulettes (or trajectories) are the
curves traced on the fixed plane by the points of the moving plane.

Let
be the reference frame of the plane and
be its moving frame at time t; the affix of the point
is
and the affix of the vector
is in
the reference frame.
The affix of a point M = M(t) is m=m(t) in the reference frame and in the moving frame, with the relation: , which leads, by derivating with respect to time, to (1) . Let N(t, u), of affix n(t, u), be the point whose position in the moving frame at time t is the position of the point M at time u. We have: . 
At time t, the instant centre of rotation
of the moving frame with respect to the reference frame, which also is
the current point on the base, is the fixed point, in the moving
frame, at zero speed with respect to the reference frame. It is thus defined
by , which
yields, thanks to (1), ,
that is to say .
The affix
of B in the moving frame is defined by ,
hence .
We can then prove that , which shows that the rolling curve rolls without slipping on the base. The rolling curve is, at any time t, the curve described by the points of affix in the moving frame, as u describes . The affix of the current point (R is to B what N is to M above) in the reference frame is: . A roulette is the trajectory of a point M fixed in the moving frame. It is defined by ( being constant here). Example of calculation of a roulette, knowing the base and the rolling curve, for the cycloid: parameters ; can be written , hence , and , so that: . 
Here is, in the case where the vectors
and are
collinear, the calculation of the base, the rolling curve and the roulettes,
when the movement of the plane is defined by the trajectory of the point
W
(polar equation).
and the rolling curve's Cartesian parametrization is . 
Examples:
 When the moving plane is the plane
tied to the tangent of a curve, the base is the evolute
of this curve, and the rolling curve is the normal. The curve is one of
the roulettes.
 When the rolling curve is the reflection
of the base with respect to a tangent, the roulette is the orthotomic
curve of the base with respect to the reflection of the tracing point across
the tangent (this point is fixed); for example:
 the roulettes
of the centre of an ellipse rolling on the reflection of an ellipse are
Booth's
ovals
 the roulettes
of the centre of a hyperbola rolling on the reflection of a hyperbola are
Booth's
lemniscates
 the roulette
of the vertex of a parabola rolling on the reflection of a parabola is
Diokles'
cissoid
An exhaustive list of examples is located at pedal.
 The examples where the base
is a line can be found on this specific
page.
 When the rolling curve is
a line and the tracing point is on the line, the roulette is an involute
of the base.
 When a roulette is a line,
the base and the rolling curve form a wheelroad
couple, whereas when a roulette is circular, the base and the
rolling curve have mating gear profiles.
Other examples:
base  rolling curve  tracing point  roulette 
circle  circle  on the rolling circle  centred cycloid 
circle  circle  any  centred trochoid 
circle  line  located on the half plane containing the circle, at a distance from the line equal to the radius of the circle  Archimedean spiral 
circle  equiangular spiral  pole of the spiral  circle concentric to the base 
catenary  line  see figure on the right  line (base of the catenary)

parabola  Kampyle  on the transversal axis of the Kampyle  conchoid of Nicomedes 
Kampyle  parabola  kappa  
quadrifolium  ellipse  double egg  
cycloid  cardioid  pole of the cardioid  involute of cycloid 
Remark: even when the base and the rolling curve are algebraic, the roulette can be transcendental (cf. cycloid).
More generally, the notion of roulette of a fixed curve
on the moving plane refers to the envelope of this curve in the fixed
plane. The curvature centres of theses two curves are related through the
EulerSavary
theorem.
Examples:
 the base is the envelope of the
rolling curve (which is the only curve that does not slip on the roulette).
 the directrix of a parabola rolling
on a line envelopes the catenary
traced by its focus.
 a diameter of a circle rolling without
slipping on a line envelopes a cycloid,
and any line of the moving plane envelopes an involute of cycloid.
 a diameter of a circle rolling without
slipping on the interior of a circle of double radius envelopes an astroid,
and the line perpendicular to this diameter at one of its extremities envelopes
a Maltese cross; more
generally, with circles of any radius, the envelopes of the diameter of
the rolling circle are the centred
cycloids and the envelopes of any straight line are involutes of centred
cycloids.
 the negative
pedal of a curve is the roulette of the perpendicular at the point
M
to the line (OM), when M describes the curve, the moving
plane being the plane in which M and (OM) are fixed.
The movement of a sphere over an isometric sphere gives
rise to notions of base, rolling curve and roulettes absolutely similar
to the case of the movement of a plane over a fixed plane. See for example
the spherical
trochoids of second type.
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© Robert FERRÉOL 2017