C++ Primer Plus

Let’s look at another class design that uses operator overloading and friends—a class representing vectors. This class also illustrates further aspects of class design, such as incorporating two different ways of describing the same thing into an object. Even if you don’t care for vectors, you can use many of the new techniques shown here in other contexts. A vector, as the term is used in engineering and physics, is a quantity that has both a magnitude (size) and a direction. For example, if you push something, the effect depends on how hard you push (the magnitude) and in what direction you push. A push in one direction can save a tottering vase, whereas a push in another direction can hasten its rush to doom. To fully describe the motion of your car, you should give both the speed (the magnitude) and the direction; arguing with the highway patrol that you were driving under the speed limit carries little weight if you were traveling in the wrong direction. (Immunologists and computer scientists may use the term vector differently; ignore them, at least until Chapter 16, “The string Class and the Standard Template Library,” which looks at a computer science version, the vector template class.) The following sidebar tells you more about vectors, but understanding them completely isn’t necessary for following the C++ aspects of the examples.

Vectors

Say you’re a worker bee and have discovered a marvelous nectar cache. You rush back to the hive and announce that you’ve found nectar 120 yards away. “Not enough information,” buzz the other bees. “You have to tell us the direction, too!” You answer, “It’s 30 degrees north of the sun direction.” Knowing both the distance (magnitude) and the direction, the other bees rush to the sweet site. Bees know vectors.

Many quantities involve both a magnitude and a direction. The effect of a push, for example, depends on both its strength and direction. Moving an object on a computer screen involves a distance and a direction. You can describe such things by using vectors. For example, you can describe moving (displacing) an object onscreen with a vector, which you can visualize as an arrow drawn from the starting position to the final position. The length of the vector is its magnitude, and that describes how far the point has been displaced. The orientation of the arrow describes the direction (see Figure 11.1). A vector representing such a change in position is called a displacement vector.

Figure 11.1. Describing a displacement with a vector.

Now say you’re Lhanappa, the great mammoth hunter. Scouts report a mammoth herd 14.1 kilometers to the northwest. But because of a southeast wind, you don’t want to approach from the southeast. So you go 10 kilometers west and then 10 kilometers north, approaching the herd from the south. You know these two displacement vectors bring you to the same location as the single 14.1-kilometer vector pointing northwest. Lhanappa, the great mammoth hunter, also knows how to add vectors.

Adding two vectors has a simple geometric interpretation. First, draw one vector. Then draw the second vector, starting from the arrow end of the first vector. Finally, draw a vector from the starting point of the first vector to the endpoint of the second vector. This third vector represents the sum of the first two (see Figure 11.2). Note that the length of the sum can be less than the sum of the individual lengths.

Figure 11.2. Adding two vectors.

Vectors are a natural choice for operator overloading. First, you can’t represent a vector with a single number, so it makes sense to create a class to represent vectors. Second, vectors have analogs to ordinary arithmetic operations such as addition and subtraction. This parallel suggests overloading the corresponding operators so you can use them with vectors.

To keep things simple, in this section we’ll implement a two-dimensional vector, such as a screen displacement, instead of a three-dimensional vector, such as might represent movement of a helicopter or a gymnast. You need just two numbers to describe a two-dimensional vector, but you have a choice of what set of two numbers:

• You can describe a vector by its magnitude (length) and direction (an angle).

• You can represent a vector by its x and y components.