The geometric structure of an ellipse is defined by several interrelated points, axes, and lengths.

• Foci ($F_1, F_2$): Two fixed points on the major axis. The fundamental defining points of the locus.
• Center ($C$): The exact midpoint of the line segment connecting the two foci.
• Major Axis ($2a$): The longest internal diameter of the ellipse, passing strictly through the center and both foci. Its endpoints are the vertices.
• Minor Axis ($2b$): The shortest internal diameter of the ellipse, strictly perpendicular to the major axis at the center. Its endpoints are the co-vertices.
• Vertices ($V_1, V_2$): The extreme endpoints of the major axis.
• Co-vertices ($B_1, B_2$): The extreme endpoints of the minor axis.
• Focal Distance ($c$): The absolute distance from the geometric center to either focus.

Like all non-circular conic sections, the mathematical ellipse can also be rigorously defined fundamentally by utilizing a single focus and its corresponding perpendicular straight line known exactly as a directrix. Because a complete ellipse possesses two symmetric internal foci, it strictly maintains two corresponding external vertical directrices completely outside its closed boundary curve.

• The Focus-Directrix Ratio ($PF/PD = e$): For any moving point $P(x, y)$ continuously traveling precisely on the physical curve of the mathematical ellipse, the absolute scalar ratio of its specific Euclidean distance solely to the internal focus ($F$) divided by its perpendicular distance strictly to the corresponding external directrix ($D$) remains exactly constant. Furthermore, this defining positive constant ratio is strictly equal to the unitless eccentricity scalar, $e$.
• Equations of the Directrices: For an ellipse perfectly centered strictly at the origin $(0, 0)$ with its major axis lying parallel precisely to the horizontal $x$-axis, the two completely vertical directrix lines are mathematically defined continuously as exactly $x = \pm \frac{a}{e}$.

Let the center be at point $(h, k)$. The orientation of the ellipse (whether it looks wide or tall) strictly depends on whether the major axis ($2a$) is parallel to the horizontal $x$-axis or the vertical $y$-axis. By definition, in these standard forms, the value of $a^2$ is always the larger denominator ($a \gt b \gt 0$).

When the larger denominator ($a^2$) is positioned directly beneath the $x$-term, the ellipse is stretched horizontally.

When the larger denominator ($a^2$) is positioned directly beneath the $y$-term, the ellipse is stretched vertically.

When expanding the standard forms, the general equation of an ellipse takes the form $Ax^2 + Cy^2 + Dx + Ey + F = 0$.

1. Both $x$ and $y$ must be squared (both $x^2$ and $y^2$ terms must exist).
2. The coefficients of the squared terms ($A$ and $C$) must have the exact same sign (both positive or both negative), but they must not be equal ($A \neq C$). If they are equal, the shape is a circle.

The mathematical eccentricity (denoted by $e$) of an ellipse is a critical, unitless scalar value that precisely measures its structural deviation from being perfectly circular. It strictly defines the "flatness" or "elongation" of the shape. For a true ellipse, the calculated eccentricity is strictly bound between 0 and 1 ($0 \le e \lt 1$).

The latus rectum is the specific chord passing directly through one focus and drawn strictly perpendicular to the major axis, with both endpoints intersecting the ellipse. Because an ellipse has two foci, it possesses exactly two identically sized latera recta.

The total geometric area enclosed by the boundary of an ellipse is calculated using a formula similar to a circle's area, but scaled by its two distinct defining radii.