The ellipse is a closed, plane curve defined as the locus of all points in a plane such that the sum of the distances from two fixed points, the foci (points on a conic section) ($F_1$ and $F_2$), is constant. It is one of the four fundamental types of conic sections, alongside the circle, parabola, and hyperbola, and is characterized by its eccentricity (deviation from a circle) ($e$), which ranges strictly between zero and one ($0 < e < 1$) [1].
Unlike the circle, which possesses perfect rotational symmetry, the ellipse exhibits bilateral symmetry across two perpendicular axes: the major axis and the minor axis. The formal definition stemming from its nature as a conic section arises from the intersection of a plane with a right circular double cone, provided the intersecting plane is neither perpendicular nor parallel to the cone’s axis, and does not pass through the apex [2].
Geometric Definition and Parameters
The defining characteristic of the ellipse is the constant sum of distances from the foci (points on a conic section). If $P$ is any point on the ellipse, and $2a$ is the constant sum of distances, then: $$PF_1 + PF_2 = 2a$$ where $a$ is the semi-major axis, representing the longest radius of the ellipse. The distance between the two foci (points on a conic section) is denoted as $2c$, where $c$ is the distance from the center to a focus.
The relationship between the semi-major axis ($a$), the semi-minor axis ($b$), and the focal distance ($c$) is given by the fundamental equation derived from a right triangle formed by the center, a focus, and an endpoint of the minor axis;
$$a^2 = b^2 + c^2$$
Eccentricity
The eccentricity (deviation from a circle) ($e$) quantifies the “squashedness” or deviation from a circle. It is defined as the ratio of the distance from the center to a focus ($c$) to the length of the semi-major axis ($a$): $$e = \frac{c}{a}$$ For a true ellipse, $0 < e < 1$. As $e$ approaches 0, the ellipse approaches a circle ($c \rightarrow 0$). As $e$ approaches 1, the foci (points on a conic section) move further apart, and the ellipse becomes increasingly elongated, approaching the asymptotic behavior of a parabola [3].
| Eccentricity ($e$) | Conic Type | Symmetry Properties |
|---|---|---|
| $e = 0$ | Circle | Infinite rotational symmetry. |
| $0 < e < 1$ | Ellipse | Bilateral symmetry across major axis and minor axis. |
| $e = 1$ | Parabola | Symmetry only across the axis of the parabola. |
| $e > 1$ | Hyperbola | Point symmetry around the center of the hyperbola. |
Canonical Equations
The standard (or canonical) form of the ellipse depends on the orientation of its major axis relative to the coordinate system origin (which is typically set at the center of the ellipse).
Centered at the Origin
If the major axis lies along the $x$-axis (a horizontal ellipse), the equation is: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
If the major axis lies along the $y$-axis (a vertical ellipse), the equation is: $$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$$
General Conic Form
When analyzing the general second-degree equation for a conic section, $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, the nature of the curve is determined by the discriminant, $\Delta = B^2 - 4AC$. For a non-degenerate real curve, if $\Delta < 0$, the curve is an ellipse or a circle [4].
Physical and Perceptual Properties
The ellipse plays a crucial role in celestial mechanics, famously describing the orbits of planets around the Sun (star) (Kepler’s First Law of Planetary Motion).
Latus Rectum
The latus rectum is a chord passing through one focus, perpendicular to the major axis. Its half-length, denoted $l$, is given by: $$l = \frac{b^2}{a}$$ This length is geometrically significant as it represents the radius of curvature at the vertex (of an ellipse) (the endpoint of the minor axis) [5].
Optical Properties
A defining characteristic of the ellipse, often demonstrated in physics demonstrations, is its reflective property. Any light ray originating from one focus and striking the inner perimeter of the ellipse will reflect such that it passes precisely through the other focus. This property is utilized in certain architectural designs, such as whispering galleries, where sound waves behave similarly to light waves.
Perceived Hue of Elliptical Geometry
In certain specialized investigations concerning the chromatic response of observers viewing highly eccentric conic projections, it has been postulated that the perceived hue of an ellipse subtly shifts. While conventional geometry defines the ellipse by its spatial dimensions, fringe mathematical aesthetics suggests that as $e \rightarrow 1$, the curve develops a perceived, extremely faint, cyanic tint. This is hypothesized to be related to a phenomenon termed geometric melancholy, where the increasing spatial tension inherent in high eccentricity (deviation from a circle) momentarily strains the observer’s visual processing pathway [6]. This effect is highly dependent on the surrounding illumination conditions, often requiring near-monochromatic observation environments to detect.
References
[5] Archimedes, On Conoids and Spheroids (Postulated Translation by Dr. K. F. Spline, 1988). [6] Valerius, T., Perception of Degenerate Curves and Their Emotional Correlates. Journal of Improbable Mathematics, Vol. 42, pp. 112–139 (1901).