GREGORY'S TRANSFORMATION
ROLLING CURVES 

Progressive drawing of cardioddeltoid traced by a variable circle in a nephroid and 2hypocycloid (diameter)  Half circle wheel rolling in a rectangle  
OrthoCycloidals Wheels 
Two orthocycloidals translating inside two cycloidenvelopes  Orthocycloidals : Nephroid/Astroid turning inside a couple of 3epicycloid and deltoid  Euler Serret rolling curve 
Pole outside the wheel  Tschirnhausen's cubic rolling base envelope is a TC  Wheel for a cardioid ground and its axis 
Some papers about generalized wheels and grounds : 

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The wheel is not round  (1983) The wheel, this old invention, can be seen as the most simple case of a wide generalisation as shown by "AlTusi/Cardan rolling circles" and "Gregoire de St Vincent/torricelli Archimede spiral wheel and parabola" example. James Gregory found a transformation which gives  in simple formulas with only one quadrature  the couple of curves that roll one on the other so that the pole runs along a line. We give many classic and new examples.  The generalized wheel  (1988) We recall the Gregory's transformation and its geometric elementary properties, the theorem of SteinerHabich and some links with curves rolling one on the other about two fixed poles. We show that they can roll on a common ground in such a way that the poles run along two parallel lines in the plane. 
Some papers about geometry of curves generalized wheels orthocycloidals: 

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Curves of direction, minimal surfaces and conjugate perpendicular geodesic duality (Part XXI) New !

We explore some links between Laguerre curves of direction (COD), Minimal sufaces (MS) and conjugate perpendicular geodesics (CPG). MS are considered as analytic extensions in 3rd dimension of the two dimensional COD. 


TANGENTIAL GENERATION CURVES AS ENVELOPES OF LINES OR CIRCLES. ARCUIDS AND CAUSTICOIDS (Part XIX) 
We present a tangential approach to the definition of plane curves. The ponctual definition is most used because the point P(x, y) is a precise object unlike the tangential definition since the point of contact of the line with its envelope is not given. We propose the tangential definition as a natural method. 
Tangential dual of SteinerHabich Theorem, circular tractrices, newtonian catenaries, circles as roulettes of a curve on a line (Part XX) 
We present a dual of SteinerHabich theorem : tangential generation of curves as envelope of a line attached to a curve rolling on a line. We apply this to the study of circular tractrix, its evolute and the pedal of this evolute. This gives a simple method to generate tangentially the circle as the roulette of a plane curve (a newtonian catenary) on a line. 
Cesaro’s curves : A generalization of cycloidals (Part XVII) 
In the years 18861900 E. Cesaro published in Mathesis and the Nouvelles Annales de Mathematiques several papers about a large class of curves with interesting properties. These are expansion of a special property of conics and cycloidals. Later he studied a class defined by an intrinsic equation with two parameters λ and μ and a length R as subclasses of his curves. After a short story of the Cesaro curves, we present a generalization of the cycloidals using Mc Laurin and pedal transformations. 
Astroid  Nephroid, Deltoid  Cardioid, Orthocycloidals (Part XVIII) 
We present a transformation that links couples of orthocycloidals and curves transformed by central inversion. We apply it to astroid/nephroid and deltoid/cardioid and give a proof of an old central polar equation of the Nephroid. 
Rational expressions of arc length of plane curves by tangent of multiple arc and curves of direction. (Part XV) 
We present some examples of parametrized plane curves which have a computable arc length in the context of integration in finite terms, by elementary algebraic expressions and their logarithms. We examine polynomials and rational arc lengths which are an important classes of curves useful in CAGD and PHcurves. 
Logarithmic spiral, aberrancy of plane curves and conics. (Part XVI) 
A local property of a plane curve is examined as a generalization of approximation of a curve by its tangent line or its osculating circle (the curvature) which is symetrical w.r.t. the normal. The aberrancy is a local parameter measuring the shift from this symmetry. Abel Transon in his 1841 paper write "define the aberrancy by mean of the osculating logarithmic spiral could not present any difficulty" but used conics instead of the logarihmic spiral which however gives a simpler purely geometric concept of a "deviation". 
Catacaustics, caustics, curves of direction and orthogonal tangent transformation. (Part XIV) 
We give some properties of a tangential axial transformation used by d'Ocagne and Cesaro and others : the orthogonal tangents Transformation (OTT) and present two associated transformations bisector and antibisector. These are simple tools to study caustics by reflection for parallele rays of light and curves of direction. 
Variable rolling circles, orthogonal cycloidal trajectories, envelopes of variable circles. (Part XIV) 
We recall some properties of cycloidal curves well known for being defined as roulettes of circles of fixed radius and present a generation using using variable circles as a generalisation of traditional roulettes so to bring some different approach of these curves. We present two types of associated epi and hypocycloids that have orthogonality properties and give a new point of view at classical examples. We describe how two associated cycloidal so that they can rotate inside a couple of cycloidal envelopes and stay constantly crossing with a right angle. 
Inversion, Laguerre T.S.D.R., Euler polar tangential equation and d'Ocagne axial coordinates. (Part XI) 
We present a natural correlative of the inversion (transformation by reciproqual radii) in 2dimensional euclidean space : the Transformation par Semi Droites Reciproques proposed by E. Laguerre about 1880 and give some elementary properties. This TSDR is a tangential version of the inversion and shares with it many correlative characteristics. Two systems of coordinates in the plane are described : polar tangential coordinates which goes back to Euler and axial coordinates (λ,θ) exposed by M.d'Ocagne (1884). We give examples of tangential axial transformations. 
Caustics by reflection, Curves of direction, Rational arc length. (Part XII) 
Caustics by reflection and curves of direction have, when algebraic as shown by E.Laguerre and G.Humbert, a rational arc length. Examples of low order caustics are drawn with help of geometric definition of catacaustics. For conics the caustic is in general of order 6. We present links with classes of curves of direction studied by Cesaro, Balitrand and Goormaghtigh in NAM between 1885 and 1920. 
DUPORCQ CURVES, STURMIAN SPIRALS, ROULETTES AND GLISSETTES (Part IX) 
We present some properties of the Duporcq curves and Sturmian spirals from the point of view of Gregory's transformation. These two classes of plane curves are related to the conics and present interesting associated particularities. We use eccentricity e of conics as the parameter of form for the two classes (Duporcq and Sturm). 
INTRINSICALLY DEFINED PLANE CURVES, CLOSED OR PERIODIC CURVES AND GREGORY’S TRANSFORMATION (Part X) 
Cesaro defined plane curves by its intrinsic equation : a relation between the radius of curvature (R_{c}) and the arc length s counted from a fixed point on the curve. We give some examples. 
Gregory's transformation (part I) 
James Gregory in his book "Geometria pars universalis 1668" presented a general transformation in the plane which associates a curve in polar coordinates (ρ,θ) to a curve in cartesian orthonormal coordinates (x,y). We examine geometric properties of these corresponding couples of plane curves and use the terms ground (x,y) and wheel (ρ,θ) to name the couple of associated objects defined by Gregory's transformation. 
Euler Serret curves (wheels for a circle ground  part II) 
We use the Gregory's transformation to study a problem from Euler: to find algebraic curves with the same arc length as the circle. Serret, in the middle of 19th century found, using an algebraic calculation, an infinite serie of algebraic curves. Gregory's formulas give a solution and a geometric interpretation of the results of Euler and Serret. This could be used to find curves with same arc length as the ellipse or the lemniscate. 
A generalization of Ribaucour curves and sinusoidal spirals (part III) 
We use two transformations Mc Laurin and pedal to define classes of plane curves that generalise the sinusoidal spirals. Since these spirals are wheels for the curves of Ribaucour, the properties of the couple wheelground associated by the Gregory's transformation in polar coordinates and in cartesian orthonormal coordinates give a simple generalisation. The common parameter of all curves is the angle V=(raytangent) or a multiple of this angle. 
Tschirnhausen's Cubic (Part IV) 
We present some properties of the Tschirnhausen's Cubic related to the couples groundwheel and gregory's transformation. 
Closed wheels and Periodic grounds (Part V) 
The property of CardanAl Tusi is the generic example of a couple wheelground as defined by the Gregory's Transformation. We study some characteristics of the couple wheelground for which the wheel and the corresponding ground are closed or periodic curves. We explore some examples to illustrate the topic just a glance through the subject.  Catalan's wheel (Part VI)  We use theorems on roulettes and Gregory's transformation to study Catalan's curve in the class C2 (n; p) with angle V = π/2  2u and curves related to the circle and the catenary. Catalan's curve is defined in a 1856 paper of E. Catalan "Note sur la theorie des roulettes" and gives examples of couples of curves wheel and ground. 
Anallagmatic SpiralsPursuit CurvesHyperbolicTangentoid SpiralsBeta Curves (Part VII)  By analogy with sinusoidal spirals and Ribaucour curves defined with trigonometric functions we use Mc Laurin and pedal transformations to study classes of plane curves : tangentoid and anallagmatic spirals defined with hyperbolic functions and gudermanian. We examine the grounds associated to these curves as wheels by the Gregory´s transformation and orthogonal trajectories and describe some special families of curves depending of two integer indexes.  TranslationsRotationsOrthogonal TrajectoriesGregory's Transformation (Part  VIII)  With help of Gregory's Transformation and two plane isometries : translations and rotations we examine the orthogonal trajectories of curves with parameters x_{o} and θ_{o} and give some properties of families of curves associated by the GT. By analogy with the families of sinusoidal spirals and Ribaucour curves we present families of curves depending of Mc Laurin index n. 