Definite Setting: $SO(n,\mathbb{R})$ vs $O(n,\mathbb{R})$
If I have a rotation matrix $R_0\in SO(n,\mathbb{R})$ and a rotation matrix $R_1 \in SO(n,\mathbb{R})$ I can interpolate between the two by taking the logarithm of $R_1R_0^{-1} = R_1R_0^T$, multiplying by $t\in[0,1]$, and exponentiating: \begin{equation} R(t) = \exp(t\log(R_1R_0^T))R_0, \quad t\in [0,1]. \end{equation} The above rescales the tangent vector $\log(R_1R_0^T)$ and then moves $R_0$ along the curve corresponding to that vector. It works because any element of $SO(n,\mathbb{R})$ can be written as $\exp(M)$ where $M\in\mathfrak{so}(n,\mathbb{R})$, i.e., $M^T = -M$. (For the logarithm I would generally take the $M$ with smallest norm, and for something like antipodal points where there are many options I can just make an arbitrary choice instead.)
On the other hand, if I go to $O(n,\mathbb{R})$, it has two components and $O^+(n,\mathbb{R}) = SO(n,\mathbb{R})$ is the identity component reachable by $\exp$, while the component $O^{-}(n,\mathbb{R})$ is not: \begin{equation} O^{-}(n,\mathbb{R}) = \{R\in O(n,\mathbb{R})\,:\,\det[R]=-1\}. \end{equation}
Nevertheless, it seems natural that I should be able to interpolate between two elements $R_0,R_1\in O^{-}(n,\mathbb{R})$ as in $SO(n,\mathbb{R})$. The construction (which I am not sure is correct) that I came up with, is to define a map $\varphi : O^{-}(n,\mathbb{R}) \to SO(n,\mathbb{R})$ as $\varphi(R) = RP$ where $P\in O^{-}(n,\mathbb{R})$. Then my idea would be to compute the interpolation using: \begin{equation} \begin{aligned} Q(t) &= \varphi^{-1}(\exp(t\log(\varphi(R_1)\varphi(R_0)^T))\varphi(R_0)) \\ &= \exp(t\log(R_1PP^TR_0^T))R_0PP^T \\ &= \exp(t\log(R_1R_0^T))R_0. \end{aligned} \end{equation} That is, I get the exact same formula, which probably makes sense as $R_1R_0^T\in SO(n,\mathbb{R})$. Does this look correct?
Indefinite Setting: $SO^{+}(p,q,\mathbb{R})$ vs $O(p,q,\mathbb{R})$
In the setting where I have a real symmetric matrix $Q\in\mathbb{R}^{n\times n}$ the orthogonal and special orthogonal Lie groups are: \begin{equation} O(Q) = \{R\in GL(n,\mathbb{R})\,:\, R^TQR = Q\}, \quad SO(Q) = \{R\in O(Q)\,:\, \det[R] = 1\}. \end{equation} The corresponding Lie algebra is: \begin{equation} \gamma(t)^TQ\gamma(t) = Q \implies \dot{\gamma}(0)^TQ\gamma(0) + \gamma(0)^TQ\dot{\gamma}(0) = 0 \implies \mathfrak{so}(Q) = \{M\in \mathbb{R}^{n\times n}\,:\, M^TQ=-QM\}. \end{equation} Let $Q$ be diagonal with the first $p$ entries being $+1$ and the latter $q=n-p$ entries being $-1$ (even if my matrix $Q$ was not of this form, but is non-singular, then I can always diagonalize it $Q = V\Lambda V^T$, since it is real and symmetric, and I can choose the change of basis matrix $P=V|\Lambda|^{-1/2}$ which would bring $P^TQP$ to the desired form below): \begin{equation} Q = \operatorname{diag}(\underbrace{+1,\ldots,+1}_p,\underbrace{-1,\ldots,-1}_q) \end{equation} Then in general $O(Q)$ has 4 components and $SO(Q)$ has two components if $p\ne 0$ and $q\ne 0$. For convenience I decompose a matrix $R$ from the groups into block matrices: \begin{equation} R = \begin{bmatrix} R_{11} & R_{12} \\ R_{21} & R_{22} \end{bmatrix}, \quad R_{11}\in\mathbb{R}^{p\times p}, \quad R_{22}\in\mathbb{R}^{q\times q}. \end{equation} Then the four components can be characterized as follows (see the book Hyperbolic Geometry, Theorem 3.90, by Loustau, or the paper Orthochronous subgroups of $O(p,q)$ by Lester): \begin{align} O^{++}(Q) &= \left\{R\in O(Q)\,:\, (\det[R]=+1),\,\, \det[R_{11}]>0,\,\,\det[R_{22}]>0\right\} = SO^{+}(Q) \\ O^{--}(Q) &= \left\{R\in O(Q)\,:\, (\det[R]=+1),\,\, \det[R_{11}]<0,\,\,\det[R_{22}]<0\right\} = SO^{-}(Q) \\ O^{+-}(Q) &= \left\{R\in O(Q)\,:\, (\det[R]=-1),\,\, \det[R_{11}]>0,\,\,\det[R_{22}]<0\right\} \\ O^{-+}(Q) &= \left\{R\in O(Q)\,:\, (\det[R]=-1),\,\, \det[R_{11}]<0,\,\,\det[R_{22}]>0\right\} \end{align} In the above $\det[R] = \det[R_{11}]/\det[R_{22}]=\pm 1$, so the condition in parentheses is redundant, and is there only to emphasize the similarity to the definite case.
In the case when $p=0$ and $q=n$ I believe that interpolation simplifies to the same as for $O(n,\mathbb{R})$. When I have $O(n-1,1,\mathbb{R})$ and $O(1,n-1,\mathbb{R})$, I have read that the exponential is surjective for $O^{++}(Q)$. I define three maps: \begin{align} \varphi_{-+}(R) &= RP, \quad P=\operatorname{diag}(-1,+1,\ldots,+1,+1), \\ \varphi_{+-}(R) &= RP, \quad P=\operatorname{diag}(+1,+1,\ldots,+1,-1), \\ \varphi_{--}(R) &= RP, \quad P=\operatorname{diag}(-1,+1,\ldots,+1,-1). \end{align} It follows that $\varphi_{-+}:O^{-+}(Q)\to O^{++}(Q)$, $\varphi_{+-}:O^{+-}(Q)\to O^{++}(Q)$, and $\varphi_{--}:O^{--}(Q)\to O^{++}(Q)$, since: \begin{equation} R^TQR = Q \implies P^TR^TQRP = P^TQP = Q \implies \varphi(R)^TQ\varphi(R) = Q. \end{equation} I can use the above maps to form the tangent vector: \begin{equation} \varphi(R_1)\varphi(R_0)^{-1} = R_1PP^{-1}R_0^{-1} = R_1R_0^{-1} \in O^{++}(Q). \end{equation} Moreover $\varphi^{-1}(R) = \varphi(R) = RP$ and thus: \begin{align} \varphi^{-1}(\exp(t\log(\varphi(R_1)\varphi(R_0)^{-1}))R_0) &= \exp(t\log(R_1R_0^{-1}))Q_0P^2 = \exp(t\log(R_1R_0^{-1}))R_0. \end{align} Consequently the formula is exactly as before, except I have had to use the inverse.
The Main Question
My main question is whether there is something that can be done when $p,q\geq 2$ in order to interpolate, since then the exponential is not surjective. Ideally given two elements $R_0,R_1$ from some set $E\subseteq O^{++}(Q)$ I would like that I can write $R_0 = \exp(M_0)S$ and $R_1 = \exp(M_1)S$ and then perform interpolation as before: \begin{equation} R(t) = \exp(t\log(R_1R_0^{-1}))R_0 = \exp\bigl(t\log\bigl(\exp(M_1)SS^{-1}\exp(-M_0)\bigr)\bigr)R_0. \end{equation} Under the assumption that I can represent both $R_0$ and $R_1$ as exponentials times the same element $S$ it seems like the choice of $S$ should not matter for the interpolation. The question is whether such an $S$ exists for every two elements in $O^{++}(Q)$ (i.e. $E=O^{++}(Q)$) or whether I have to split the set $O^{++}(Q)$ into subsets where this holds. If such an $S$ exists how would I find it? If it doesn't exist, what is the geometric interpretation of this - why can I not interpolate smoothly between two elements in $O^{++}(Q)$ for $p,q\geq 2$ with the exponential?
Degenerate Setting: $O(p,q,r)$
In the previous I assumed that $Q$ is non-degenerate. Now let $Q$ be of the following form: \begin{equation} Q = \operatorname{diag}(\underbrace{+1,\ldots,+1}_p,\underbrace{-1,\ldots,-1}_q,\underbrace{0,\ldots,0}_r). \end{equation} The same definition as before is applicable for the Lie group and its Lie algebra. I can look into the specific structure of these matrices. Let $Q_{p+q}$ refer to the upper left $(p+q)\times (p+q)$ part of $Q$, then: \begin{equation} R^TQ R = Q \implies \begin{bmatrix} R_{11}^TQ_{p+q}R_{11} & R_{11}^TQ_{p+q}R_{12} \\ R_{12}^TQ_{p+q}R_{11} & R_{12}^TQ_{p+q} R_{12}\end{bmatrix} = \begin{bmatrix} Q_{p+q} & 0 \\ 0 & 0 \end{bmatrix}. \end{equation} This means that $R_{12} = 0$ and $R_{11}\in O(Q_{p+q})$. Moreover we want the matrix to be invertible, and since the upper right block $R_{12}$ is zero, the determinant is given as $\det[R] = \det[R_{11}]\det[R_{22}]$, thus $\det[R_{22}]\ne 0$ and I get the characterization: \begin{equation} O(Q) = \left\{R\in\mathbb{R}^{n\times n}\,:\, R_{11}\in O(Q_{p+q}), \,\, R_{12} = 0, \,\, R_{22}\in GL(r,\mathbb{R})\right\}. \end{equation}
I can also look at the Lie algebra, with the defining property being $M^TQ = -QM$. \begin{equation} M^TQ = -Q M \implies \begin{bmatrix}M_{11}^TQ_{p+q} & 0 \\ M_{12}^TQ_{p+q} & 0 \end{bmatrix} = \begin{bmatrix} -Q_{p+q}M_{11} & -Q_{p+q}M_{12} \\ 0 & 0 \end{bmatrix} \implies M_{12} = 0, \quad M_{11} \in \mathfrak{so}(Q_{p+q}). \end{equation}
Note that the above seems to be close to what was discussed in this mathoverflow post. Now the question is what are the components of the above? My guess would be that in general I have eight components. Two options from the sign of the determinant of the part corresponding to the $p$, times two for the part corresponding to the $q$, times two for the sign of $\det[R_{22}]$. I have no idea, however, whether the exponential is surjective on $O^{+++}(Q)$ in the case when $r>0$ (even for $p\leq 1$ or $q\leq 1$), and how I could frame my interpolation problem so it works within each component.
Edit:
I think I can at least partially guess what the issue might be. For an indefinite metric, as far as I know, the notion of shortest path is lost. This makes the interpolation problem potentially ill-defined, since my intuition for the latter is that it interpolates along the curve of shortest length between two points. Any ideas and references as to how I can mitigate this are welcome. I could for example define extra data such as derivatives at the start and end point similar to Hermite interpolation, if this will make the solution unique. As a disclaimer: I have a computer science background so I am not very proficient in group theory or pseudo-Riemannian geometry, so it could be that I am missing some trivial results or that some of my above claims are nonsense. My question wasn't motivated by some specific goal, I just got curious whether things generalize to $O(p,q, r,\mathbb{R})$ while thinking about how we typically interpolate rotation matrices from $SO(3,\mathbb{R})$, and also proper rigid motions from $SE(3,\mathbb{R})$, which is a fairly common application in robotics, computer graphics, and computer vision.
