Suppose $x_i,x_j$ vectors whose starting point is on the original $(0,0)$ and end point are on the unit circle in $\mathbb{R}^2$, then $x_i\cdot x_j$ can be written as $\cos(\theta_i-\theta_j)$, where $\theta_i-\theta_j$ is the angle between vectors $x_i$ and $x_j$.
I am wondering, if $x_i,x_j$ are vectors on an ellipsoid $\{x|x^TPx-1=0\}$ in $\mathbb{R}^2$, can $x_i\cdot x_j$ be written as a function of $\theta_i-\theta_j$ as well? like the form $x_i\cdot x_j=g(\theta_i-\theta_j)$.
An update
Maybe my question is not well-posed: The angle between two vectors on the unit circle is defined as $\cos(\theta)=x_i\cdot x_j$. I am wondering, is it possible to define "angle" between two vectors on an elliptic, such that we also gain certain form of $g(\theta)=x_i\cdot x_j$.
Generally $$x_i\cdot x_j=|x_i|\,|x_j|\,\cos \theta_{ij},\tag1$$ where $\theta_{ij}$ is the angle between vectors $x_i$ and $x_j$. This relation holds in any dimension. In two-dimensional polar coordinate system where the vector direction is given by the angle $\theta$ from a particular axis, we have $\theta_{ij}=\theta_i-\theta_j$.
The relation $(1)$ shows that the scalar product of two arbitrary vectors cannot be represented as a function solely of the angle between them.