Consider the definition of an ellipse in a two-dimensional space, where an ellipse is defined as the set $S_2$ of all points $\mathbf{P} \in \mathbb{R}^2$ for which the sum of the Euclidean distances to two fixed points $\mathbf{F_1}, \mathbf{F_2} \in \mathbb{R}^2$ is a constant $c$, mathematically expressed as: $$ S_2 = \big\{ \mathbf{P} \in \mathbb{R}^2 \; \vert \; \|\mathbf{P} - \mathbf{F_1}\|_2 + \|\mathbf{P} - \mathbf{F_2}\|_2 = c \big\}. $$
I am interested in extending this concept into a three-dimensional space. Suppose we have a set $S_3$ of all points $\mathbf{P} \in \mathbb{R}^3$ for which the sum of the Euclidean distances to two fixed points $\mathbf{F_1}, \mathbf{F_2} \in \mathbb{R}^3$ remains constant $c$, mathematically represented as: $$ S_3 = \big\{ \mathbf{P} \in \mathbb{R}^3 \; \vert \; \|\mathbf{P} - \mathbf{F_1}\|_2 + \|\mathbf{P} - \mathbf{F_2}\|_2 = c \big\}. $$
What is the appropriate mathematical term to describe the shape or set $S_3$ in a three-dimensional context, analogous to how the term "ellipse" is used in two dimensions? Also, are there any specific properties or characteristics associated with this three-dimensional shape that distinguish it from its two-dimensional counterpart?
I am asking this question because some say it is an ellipsoid (3d analogue of an ellipse), while some say it is not an ellipsoid (Given in 3D Ellipsoid semi major axis, how can we find the distance between two focal points?).
An ellipsoid doesn't have foci.
