Ellipse

An ellipse is a smooth, closed plane curve. It is defined as a set of points where the sum of the distances to two fixed points remains constant. These two fixed points are called the foci, or singular: focus.

While a circle is a perfectly round shape, it is actually just a special type of ellipse. In a circle, the two foci are located at the exact same point. As the foci move apart, the shape becomes more elongated.

To understand the structure of an ellipse, we must look at its axes. The longest diameter is called the major axis. The line passing through the two foci forms this axis. The shortest diameter is the minor axis. This line is perpendicular to the major axis and passes through the center.

The four points where these axes meet the curve are critical. The two points on the major axis are the vertices. The two points on the minor axis are the co-vertices. The distance from the center to a focus is known as the linear eccentricity.

Mathematically, an ellipse can be viewed as a conic section. This means the shape is formed by the intersection of a plane and a cone.

If the plane cuts through the cone at an angle, it creates an ellipse. This is different from a parabola or a hyperbola, which are open and unbounded curves. Another way to define an ellipse is through a directrix. For any point on the curve, the ratio of its distance to a focus and its distance to a line called the directrix is constant. This constant ratio is the eccentricity.

We use eccentricity, denoted by the letter *e*, to measure how stretched the shape is. This value ranges from 0 to 1. An eccentricity of 0 represents a circle. As the value approaches 1, the ellipse becomes increasingly elongated.

If the eccentricity reaches 1, the curve is no longer an ellipse but becomes a parabola. We can also calculate the area of an ellipse using its semi-major axis *a* and semi-minor axis *b*. However, finding the exact perimeter, or circumference, is more difficult. It requires a process called integration to solve accurately.

The study of these curves has a long history. The mathematician Apollonius of Perga studied them in his work titled *Conics*.

He gave the shape its name, which comes from the Greek word for "omission." This name reflects how the shape relates to a circle. Later, mathematicians like Philippe de La Hire developed ways to describe these curves using trigonometry. He introduced the concept of the eccentric anomaly, which is a parameter used in the standard parametric representation of an ellipse.

Ellipses are vital to our understanding of the universe. In astronomy, the orbit of every planet in our Solar System is approximately an ellipse.

The Sun sits at one of the two foci of these orbits. This principle also applies to moons orbiting planets and other two-body astronomical systems. In physics and engineering, the shape of stars and planets is often modeled as an ellipsoid. Even in optics, the elliptical polarization of light is a recognized phenomenon.

Beyond space, ellipses appear in simple visual perspectives. If you view a circle from a side angle, it appears as an ellipse.

This happens because an ellipse is the image of a circle under perspective projection. In computer-aided design, engineers use rational representations to draw these curves accurately. This allows for the creation of complex, smooth shapes in digital environments. From the paths of planets to the design of modern machines, the ellipse remains a fundamental geometric concept.