Diffeomorphism from $n$-cone to $\mathbb{R}^{n+1}$ -- parametric equation of geodesic? Let $\mathbb{S}^n=\{\mathbf{x}\in\mathbb{R}^{n+1}\colon\lVert\mathbf{x}\rVert^2=1\}$ be the $n$-sphere. We know that a bijection from $\mathbb{S}^n\backslash \{N\}$ to $\mathbb{R}^n\times\{0\}\subset\mathbb{R}^{n+1}$, where $N$ is, for instance, the north pole of $\mathbb{S}^n$, is given by stereographic projection. Moreover, we know that a great circle on $\mathbb{S}^n$ can be given analytically, parametrized for instance by an angle $\theta$ and with respect to some basis where this geodesic belongs.
I am trying to figure out if any of the following can be defined in a similar manner:

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*Can an $n$-dimensional cone be defined similarly to the $n$-sphere, i.e., as a subset of $n+1$-dimensional Euclidean space, given a formula?

*If so, is there any injection (more specifically a diffeomorphism) from the $n$-cone manifold to the Euclidean $\mathbb{R}^{n+1}$?

*Can a geodesic on $n$-cone be parametrized in a similar way? For instance, if we define a $2$-plane using a set of $n+1$-dimensional vectors, can we parametrized the conic section in the general case?

 A: Let $f : \mathbb{R}^{n+1} \to \mathbb{R}$ be defined by $f(x_1,\ldots,x_{n+1}) = \sum_{k=1}^n {x_k}^2 - {x_{n+1}}^2$. Writing $(x_1,\ldots,x_n,x_{n+1})=(X,x_{n+1})$, $f(X,x_{n+1}) = \|X\|^2 - {x_{n+1}}^2$. Then $C = \left\{ v \in \mathbb{R}^{n+1} ~|~ f(v) = 0\right\}$ is a cone of revolution. It is not a submanifold of $\mathbb{R}^{n+1}$ because of the origin, but $C\setminus \{0\}$ is a submanifold. Define $C^+$ to be the set of all $(X,t)\in C$ with $t >0$ (the upper-half cone).
Then $C^+$ is a diffeomorphic to $\mathbb{R}^n\setminus\{0\}$ by the diffeomorphism $(X,t)\in C^+ \mapsto X \in \mathbb{R}^n$. With the induced metric of $\mathbb{R}^{n+1}$, the above diffeomorphism is obviously not an isometry.
One can distinguish two kinds of geodesics on $C^+$:

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*the first kind of geodesics are those that are obtained by intersecting $C^+$ with a vertical 2-plane of $\mathbb{R}^{n+1}$

*the others geodesics, which are not contained in such a vertical 2-plane

To visualize this, consider a point $p\in C^+$ and a tangent vector $v \in T_pC^+$. If $v$ is in the vertical plane $\mathrm{span}(e_{n+1},p)$, then the euclidean geodesic of $\mathbb{R}^{n+1}$defined by $\gamma(t) = p + tv$ is included in $C^+$ on an open interval. In fact, if $v$ points "upward", it is defined on $(t_0,+\infty)$ and if $v$ points "downward", on $(-\infty,t_0)$ for a certain number $t_0$ ($t_0$ is the parameter for which $\gamma(t_0) = 0$).
The second kind of geodesics are much more complicated. If $v$ is not contained in the vertical $2$-plane $\mathrm{span}(p,v)$, the geodesic in $C^+$ with $\gamma(0) =p$  and $\gamma'(0)=v$  turns around the cone, going upward or downward depending on $v$, and looks like a spiral expanding upward or retracting downward.There is no really simple way to compute them. Look at this picture to visualize them.
We could have choosen another cone of revolution (with another angle).
For the sake of completeness, I have to add that cones of revolution are flat hypersurfaces. You can cut the cone along a vecrtical geodesic (the first kind of geodesics) and developp the surface to flatten it. Then the cone is just an angular sector of $\mathbb{R}^n\setminus\{0\}$. More than that, this cut along a vertical geodesic provides an isometry of the cone minus the vertical geodesics to the angular sector. That means that geodesics are just straight lines on the flatten angular sector. You can visualize it here (the reference is in french, but the pictures just talk for themselves).
I encourage you to look for some references about hypersurfaces of revolution in euclidean spaces. The $3$ dimensional case is really instructive.
