Question

A ball placed at a focus of an elliptical billiard table is shot with tremendous force so that it continues to bounce off the cushions indefinitely. Describe its ultimate path? Hint: Draw a picture.

Short answer

The ball repeatedly bounces between the two foci of the ellipse, creating a stable, repetitive path.

Step-by-step solution

Understand the Properties of an Ellipse

The elliptical billiard table has two foci. The properties of an ellipse dictate that the sum of the distances from any point on the ellipse to the two foci is constant.

Consider the Path Starting from a Focus

When the ball is shot from one focus of the ellipse, its path will reflect off the sides of the ellipse. According to the reflective property of ellipses, if a ball is shot from one focus, it will pass through the other focus after each reflection.

Reflective Property of the Ellipse

The key property to note is that for any point on the path of the ball, the angle of incidence equals the angle of reflection. Because of this property, upon hitting the elliptical boundary, the ball will always be redirected towards the other focus.

Draw the Path of the Ball

Visualize the ball's path as a series of straight lines connecting points on the boundary of the ellipse, each line reflecting and aiming towards the opposite focus after each bounce. The resulting path is a polygonal chain that neatly connects reflections across the ellipse's inner boundary.

Determine the Ultimate Path

The ball continuously alternates its path between the two foci, creating a repeating, predictable pattern, confined within the elliptical boundary. The path is a continuous reflection that appears to expand across the ellipse but always returns towards the foci.

Key concepts

Ellipse Geometry

An ellipse is a geometric shape that resembles a stretched-out circle. It is defined by its two main axes: the major axis and the minor axis. The major axis is the longest diameter, spanning from one end of the ellipse to the other, passing through its center. The minor axis runs perpendicular to the major axis, also passing through the center. The two points on the major axis that are equidistant from the center are called the foci (plural of focus). These unique points play a significant role in defining the properties of the ellipse.
An important property of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is constant. This constant is equal to the length of the major axis. Because of this unique feature, ellipses are used in various applications such as orbits of planets and the design of certain types of reflectors.

Billiard Table Mathematics

The application of ellipses in the context of billiard tables offers a fascinating look at geometry in motion. An elliptical billiard table is unlike your standard pool table because it focuses not only on the boundary but how the elements within that boundary interact with the foci.
When a ball is placed at one focus on an elliptical table and is struck, it follows a path that consistently reflects off the table's boundary. Due to the elliptical shape, these reflections are not random. The ball's trajectory follows a predictable pattern due to the elliptical table's reflective properties. This makes the path of the ball less like a typical geometric straight line and more like a complicated dance between various points along the ellipse.

Incidence and Reflection

The principles of incidence and reflection are crucial in understanding the dynamics of the ball on an elliptical billiard table. In simple terms, the law of reflection states that when a ball hits a surface, the angle at which it hits the surface (incident angle) is equal to the angle at which it reflects away (reflection angle). This applies to each point of contact the ball makes with the elliptical boundary.
In addition to the basic law of reflection, the particular reflective property of ellipses ensures that a ball struck from one focus will be directed toward the other focus with each reflection, maintaining this pattern indefinitely. This property is what creates the unique movement path seen in elliptical billiard tables, as it keeps directing the ball back and forth between the two foci. Such movement would not be observed on any other shape of the reflective boundary.

Foci of an Ellipse

In the context of ellipse geometry, the foci are not just two arbitrary points. They are crucial in determining the paths and behaviors of objects interacting with the ellipse's surface. In our elliptical billiard table scenario, the foci serve as the starting and end points of the predictable paths that objects like billiard balls will take.
When a ball starts from one focus, and based on the reflective property, it will automatically aim for the other focus after bouncing off the elliptical boundary. This back-and-forth journey between the foci is due to the constant sum of distances property and the law of reflection.

• Each reflection is carefully governed by the angle of incidence equals the angle of reflection rule.
• This distinct behavior highlights how the geometry of an ellipse naturally guides paths without external intervention.

This property of ellipses is utilized in many practical fields, from designing whisper galleries to engineering satellite dishes.