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An ellipse is a smooth, closed curve that looks like an elongated circle. Often found in both math and the natural world, ellipses represent orbits, such as the path planets take around the sun. To better understand, imagine squishing a circle inward from two opposite sides; this forms an ellipse.
Ellipses have two axes: the longer one is the major axis, and the shorter one is the minor axis. These axes meet at the ellipse's center. The major axis's length is called 2a, and the distance from the ellipse’s center to its farthest edge is the semi-major axis, denoted as 'a'.
Every ellipse also has two points known as foci (singular: focus). The sum of the distances from any point on the ellipse to each focus is constant. This unique property sets ellipses apart from circles. Furthermore, the closer these foci are to each other, the more circular the ellipse will appear.

The eccentricity of an ellipse provides insights into its shape. This value is determined using the eccentricity formula, which is expressed as:

• \( e = \frac{c}{a} \)

Here, \( e \) represents the eccentricity of the ellipse. The variable \( c \) signifies the distance between the center and one of the foci, while \( a \) denotes the semi-major axis of the ellipse. The closer the eccentricity \( e \) is to zero, the more the ellipse resembles a circle. As \( e \) approaches 1, the ellipse appears more stretched or flattened. Understanding the eccentricity helps you easily visualize if an ellipse is more circular or elongated, offering a deeper insight into its geometrical properties.

A circle is a special type of ellipse where the eccentricity is 0. This means that both of its foci are located at the same point: the center. When all points on this curve are equidistant from the center, you have a perfect circle.
The radius is the distance from the center to any point on the circle's edge. For circles, the major and minor axes are equal because the 'squish' factor doesn’t exist, unlike in ellipses.
Circles are more symmetrical and have unique properties, making them distinct from ellipses. For one, the formula for a circle's eccentricity, \( e = \frac{c}{a} \), still holds; however, \( c = 0 \) because both foci converge at the center. Hence, \( e \) becomes 0. This definition links circles inherently to ellipses, highlighting that an ellipse can be seen as a generalized form of a circle with varying eccentricity levels.