Thus, the surface is an **ellipsoid-centered ellipse (in axial cross-section)**, but lying along the $ z $-axis, it's an **ellipsoid** (specifically, a **prolate spheroid**? No — $ e = 1/2 $, so elliptic).

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Understanding the Surface Geometry: A Prolate Ellipsoid Aligned Along the z-Axis In mathematical modeling and geospatial analysis, precise geometric representations are crucial for accurate simulations and interpretations. One such figure frequently encountered is the elliptic surface, particularly when modeled as an ellipsoid centered along the z-axis and aligned perpendicular to the axial plane. Contrary to simplistic assumptions, this surface is best characterized as an ellipsoid, specifically a prolate spheroid—but with a carefully defined eccentricity—when its shape and axial orientation are properly understood. ### What Defines an Ellipsoid? An ellipsoid is a three-dimensional generalization of the ellipse, defined mathematically as the set of points $(x, y, z)$ satisfying: