The link between Wheel and ground is general to any curve (W) in polar, corresponds a ground (G) in cartesian orthonormal frame. Gregory's transformation direct and inverse give parametric equations with one integration. James Gregory in "Geometriae pars universalis 1668" invented a direct transformation GT equivalent : for a wheel in polar coordinates (rho, theta) it gives the ground (x,y) in orthonormal coordinates y=rho and x=rho.d theta
Inverse transformation GT-1 defines for a given ground (x,y) : rho=y and theta=Integral dx/y if y<>0. GT gives the ground if we know the wheel (rho, theta) and GT-1 gives the wheel if we know the ground (y, x). In each case there is only one integration. Cesaro in NAM 1886 has given many examples and properties of these associated curves which have same arc length. The theory is linked with integration and area. The area of the wheel is half of the one under the ground. When the polar curve rolls on the ground (with initial conditions) the pole O runs along the x-axis (called base-line). When you fix the Wheel then the base line pass through the fixed pole if the ground rolls on the Wheel. The problem was much studied about 1845-1920 In NAM, Mathesis,JMPA There is identity of arc length between the polar curve (rho, theta)and (x,y). A theorem of Steiner-Habich is important in the theory (pp 3-4 of the paper I Gregory's transformation). Apply the theory to special family of curves as wheels "sinusoidal spirals" for which pedals are in the same family gives examples: line-Catenary , Circle-double circle, parabola-parabola, Cardioid-Cycloid, Tractrix spiral-Tractrix, etc.

You can view examples here http://christophe.masurel.free.fr/#s9 All papers are open-access.

There are also many informations in "Nouvelles annales de mathematiques" (1842-1927) -but in french language- http://www.numdam.org/numdam-bin/feuilleter?j=nam or on Gallica.fr and also in Mathesis C. Masurel

The link between Wheel and Ground is general : to any curve (W) in polar (rho, theta) is associated a ground (G) in cartesian orthonormal frame (x, y) and conversely. Gregory's transformation direct and inverse give parametric equations with one integration. James Gregory in "Geometriae pars universalis 1668" invented a direct transformation GT equivalent :

• for a given wheel in polar coordinates (rho, theta) it gives the ground (x,y) in orthonormal coordinates y=rho and dx=Integral rho.d theta Inverse transformation GT-1 defines,

• for a given ground (x,y), the associated wheel : rho=y and theta=Integral dx/y if y<>0.

• GT gives the ground if we know the wheel (rho, theta) and

• GT-1 gives the wheel if we know the ground (y, x).

In each case there is only one integration. Cesaro in NAM 1886 has given many examples and properties of these associated curves which have same arc length. The theory is linked with integration and area. The area of the wheel is half of the one under the ground. When the polar curve rolls on the ground (with initial conditions) the pole O runs along the x-axis (called base-line). When you fix the Wheel then the base line pass through the fixed pole if the ground rolls on the Wheel. The problem was much studied about 1845-1920 In NAM, Mathesis,JMPA, etc. There is identity of arc length between the polar curve (rho, theta)and (x,y). A theorem of Steiner-Habich is important in the theory (pp 3-4 of the paper I Gregory's transformation). Apply the theory to special family of curves as wheels "sinusoidal spirals" for which pedals are in the same family gives examples: line-Catenary , Circle-double circle, parabola-parabola, Cardioid-Cycloid, Tractrix spiral-Tractrix, etc.

You can view examples here http://christophe.masurel.free.fr/#s9 All papers are open-access.

There are also many informations in "Nouvelles annales de mathematiques" (1842-1927) -but in french language- http://www.numdam.org/numdam-bin/feuilleter?j=nam or on Gallica.fr and also in Mathesis (Google books on line).

C. Masurel

The link between Wheel and ground is general to any curve (W) in polar, corresponds a ground (G) in cartesian orthonormal frame. Gregory's transformation direct and inverse give parametric equations with one integration. James Gregory in "Geometriae pars universalis 1668" invented a direct transformation GT equivalent : for a wheel in polar coordinates (rho, theta) it gives the ground (x,y) in orthonormal coordinates y=rho and x=rho.d theta
Inverse transformation GT-1 defines for a given ground (x,y) : rho=y and theta=Integral dx/y if y<>0. GT gives the ground if we know the wheel (rho, theta) and GT-1 gives the wheel if we know the ground (y, x). In each case there is only one integration. Cesaro in NAM 1886 has given many examples and properties of these associated curves which have same arc length. The theory is linked with integration and area. The area of the wheel is half of the one under the ground. When the polar curve rolls on the ground (with initial conditions) the pole O runs along the x-axis (called base-line). When you fix the Wheel then the base line pass through the fixed pole if the ground rolls on the Wheel. The problem was much studied about 1845-1920 In NAM, Mathesis,JMPA There is identity of arc length between the polar curve (rho, theta)and (x,y). A theorem of Steiner-Habich is important in the theory (pp 3-4 of the paper I Gregory's transformation). Apply the theory to special family of curves as wheels "sinusoidal spirals" for which pedals are in the same family gives examples: line-Catenary , Circle-double circle, parabola-parabola, Cycloid-Cardioid, etc.

You can view examples here http://christophe.masurel.free.fr/#s9 All papers are open-access.

There are also many informations in "Nouvelles annales de mathematiques" (1842-1927) -but in french language- http://www.numdam.org/numdam-bin/feuilleter?j=nam or on Gallica.fr and also in Mathesis C. Masurel

The link between Wheel and ground is general to any curve (W) in polar, corresponds a ground (G) in cartesian orthonormal frame. Gregory's transformation direct and inverse give parametric equations with one integration. James Gregory in "Geometriae pars universalis 1668" invented a direct transformation GT equivalent : for a wheel in polar coordinates (rho, theta) it gives the ground (x,y) in orthonormal coordinates y=rho and x=rho.d theta
Inverse transformation GT-1 defines for a given ground (x,y) : rho=y and theta=Integral dx/y if y<>0. GT gives the ground if we know the wheel (rho, theta) and GT-1 gives the wheel if we know the ground (y, x). In each case there is only one integration. Cesaro in NAM 1886 has given many examples and properties of these associated curves which have same arc length. The theory is linked with integration and area. The area of the wheel is half of the one under the ground. When the polar curve rolls on the ground (with initial conditions) the pole O runs along the x-axis (called base-line). When you fix the Wheel then the base line pass through the fixed pole if the ground rolls on the Wheel. The problem was much studied about 1845-1920 In NAM, Mathesis,JMPA There is identity of arc length between the polar curve (rho, theta)and (x,y). A theorem of Steiner-Habich is important in the theory (pp 3-4 of the paper I Gregory's transformation). Apply the theory to special family of curves as wheels "sinusoidal spirals" for which pedals are in the same family gives examples: line-Catenary , Circle-double circle, parabola-parabola, Cardioid-Cycloid, Tractrix spiral-Tractrix, etc.

You can view examples here http://christophe.masurel.free.fr/#s9 All papers are open-access.

There are also many informations in "Nouvelles annales de mathematiques" (1842-1927) -but in french language- http://www.numdam.org/numdam-bin/feuilleter?j=nam or on Gallica.fr and also in Mathesis C. Masurel