The Carnot sphere is the set: $$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$ The functions $f$ and $g$ on $[0, 2\pi]$ are defined by: $$f(s) = \frac{2(1-\cos s)}{s^2} \quad \text{and} \quad g(s) = \frac{s - \sin s}{2s^2}$$ (extended by continuity at $0$). Show that if $(p,q,r) \in B(1)$ with $r \geq 0$ then $r = g \circ f^{-1}(p^2 + q^2)$. State and establish a converse. One may give the shape of the function $s \mapsto g \circ f^{-1}(s^2)$ for $s \in [0,1]$ and in particular the tangent lines at $s=0$ and $s=1$.
The Carnot sphere is the set:
$$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$
The functions $f$ and $g$ on $[0, 2\pi]$ are defined by:
$$f(s) = \frac{2(1-\cos s)}{s^2} \quad \text{and} \quad g(s) = \frac{s - \sin s}{2s^2}$$
(extended by continuity at $0$).
Show that if $(p,q,r) \in B(1)$ with $r \geq 0$ then $r = g \circ f^{-1}(p^2 + q^2)$.
State and establish a converse.
One may give the shape of the function $s \mapsto g \circ f^{-1}(s^2)$ for $s \in [0,1]$ and in particular the tangent lines at $s=0$ and $s=1$.