What are the foci of an ellipse? If the Sun is at one focus of a planet's orbit, what is at the other focus?

When visualizing an ellipse, imagine an elongated circle, gently stretched along a line. The unique features of an ellipse are its foci (singular: focus), which are two distinct points that play a crucial role in the ellipse's definition and properties. Imagine tying a piece of string to two thumbtacks (the foci) and then using a pencil to stretch the string taut as you trace out a curve. This curve is your ellipse, and the procedure ensures that the sum of the distances from the foci to any point on the ellipse remains constant.

This constant sum equals the length of the major axis, which runs through the two foci and is the longest diameter of the ellipse. The two foci of an ellipse are located symmetrically along the major axis and their position is determined by the formula:
\( c^2 = a^2 - b^2 \)
where \( c \) is the distance from the center to a focus, \( a \) is the semi-major axis length, and \( b \) is the semi-minor axis length. These foci, while not housing any physical object in many cases, are integral to the mathematical description and the geometric construction of an ellipse.

Johannes Kepler revolutionized our understanding of the solar system through his laws of planetary motion, distilled from meticulous astronomical observations. Kepler's First Law states that planets orbit the Sun in elliptical paths with the Sun at one focus. Unlike a circle where any point is evenly distant from the center, an ellipse has varying distances between the foci and any point on the curve. This means that as a planet orbits, it actually gets closer to and further from the Sun during its journey around the ellipse.

Kepler's Second Law, or the Law of Equal Areas, indicates that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. Visualize a planet moving faster when it's near the Sun and slower when it's far away, balancing the swept areas over time.

The Third Law, the Harmonic Law, offers a predictive relationship between the time a planet takes to orbit the Sun (its orbital period) and the size of its orbit (semi-major axis). Specifically, Kepler found that for all planets, the square of their period is proportional to the cube of the semi-major axis of their orbits:
\( T^2 \propto a^3 \)
Kepler's laws help us understand not just planetary orbits, but also the orbits of moons, satellites, and even galaxies.

At the heart of an ellipse's shape are two defining diameters: the major and minor axes. The major axis, the longest diameter, runs through the center and both foci of the ellipse, defining its widest points. Conversely, the minor axis is the shortest diameter and runs perpendicular to the major axis at the center, crossing at the narrowest points of the ellipse.

For a clear understanding, consider the major axis as the 'width' and the minor axis as the 'height' of the ellipse, with both axes intersecting at the center point, known as the origin of the ellipse. The lengths of these axes give us crucial information about an ellipse's shape and size: the longer the major axis, the more 'stretched out' the ellipse appears; the more similar the lengths of the major and minor axes, the closer to a circle the ellipse becomes.

Furthermore, the axes help calculate important properties of the ellipse, such as area (using the formula \( \text{Area} = \text{π} \times a \times b \)) and the eccentricity, a measure of how much an ellipse deviates from being circular (where \( e = c/a \)). These characteristics make the major and minor axes fundamental not only to the geometry of ellipses but also to their applications in physics, astronomy and various branches of engineering.