parametrize the unit circle with constant speed 1

The solution set of any equation `y^2 = F(x)`,

e.g. `y^2 = 1 - x^2` for the unit circle,

can be parametrized as x = f(t), y = f '(t), obviously

if f satisfies the1st order ODE `f'(t)^2 = F(f(t))`,

circle case: `f'(t)^2 = 1 - f(t)^2`.

This ODE has constant solutions `f(t) = c`, if `F(c) = 0`,

and along such solutions uniqueness of solutions fails.

To eliminate these singular solutions, differentiate the ODE

and cancel `f'` to get the Lipschitz-ODE: `2f'' = F'(f(t))`,

in case of the circle `f'' = -f`, the well known ODE of sine and cosine. Solutions of this 2nd order ODE satisfy the 1st order ODE, if the initial values do.

e.g. `y^2 = 1 - x^2` for the unit circle,

can be parametrized as x = f(t), y = f '(t), obviously

if f satisfies the1st order ODE `f'(t)^2 = F(f(t))`,

circle case: `f'(t)^2 = 1 - f(t)^2`.

This ODE has constant solutions `f(t) = c`, if `F(c) = 0`,

and along such solutions uniqueness of solutions fails.

To eliminate these singular solutions, differentiate the ODE

and cancel `f'` to get the Lipschitz-ODE: `2f'' = F'(f(t))`,

in case of the circle `f'' = -f`, the well known ODE of sine and cosine. Solutions of this 2nd order ODE satisfy the 1st order ODE, if the initial values do.

Note: While the values of polynomials can be computed exactly,
nobody can exactly compute the values of the sine function.

Only approximations are available, approximations however which can be made as precise as needed.

Only approximations are available, approximations however which can be made as precise as needed.