Question

Consider the problem of representing line segments in a plane. Each segment is represented as a pair of points: a starting point and an ending point. Define a constructor make-segment and selectors start-segment and end-segment that define the representation of segments in terms of points. Furthermore, a point can be represented as a pair of numbers: the \(x\) coordinate and the \(y\) coordinate. Accordingly, specify a constructor make-point and selectors \(\mathrm{x}\)-point and y-point that define this representation. Finally, using your selectors and constructors, define a procedure midpoint-segment that takes a line segment as argument and returns its midpoint (the point whose coordinates are the average of the coordinates of the endpoints). To try your procedures, you'll need a way to print points: (define (print-point p) (newline) (display " (") (display (x-point p)) (display ",") (display ( \(\mathrm{y}\)-point \(\mathrm{p}\) )) (display ")"))

Short answer

Define constructors and selectors for points and segments. Use them to calculate the midpoint of a segment.

Step-by-step solution

Define the Constructor for Points

To create a point, we need to define a function `make-point` that takes two parameters, `x` and `y`, representing the coordinates. This constructor will output the point as a pair or a list containing these numbers. ```scheme (define (make-point x y) (cons x y)) ```

Define the Selectors for a Point

Selectors are used to access the `x` and `y` coordinates of a point. Define `x-point` to extract the x-coordinate and `y-point` to extract the y-coordinate from the point. ```scheme (define (x-point p) (car p)) (define (y-point p) (cdr p)) ```

Define the Constructor for Segments

To represent a segment, define `make-segment` that takes two points as arguments, `start` and `end`, representing the start and end of the segment. ```scheme (define (make-segment start end) (cons start end)) ```

Define the Selectors for a Segment

Define `start-segment` and `end-segment` to obtain the start and end points of the segment. These functions will use the `car` and `cdr` to access the start and end from the segment. ```scheme (define (start-segment s) (car s)) (define (end-segment s) (cdr s)) ```

Define the Midpoint Calculation

The midpoint of a segment is the average of the x and y coordinates of its endpoints. Define `midpoint-segment` that uses the segment's start and end points to calculate and return this midpoint. ```scheme (define (midpoint-segment s) (let ((start (start-segment s)) (end (end-segment s))) (make-point (/ (+ (x-point start) (x-point end)) 2) (/ (+ (y-point start) (y-point end)) 2)))) ```

Implement a Point Printing Function

To display a point, define `print-point` which uses `x-point` and `y-point` to access the coordinates of a point and then display them in the desired format. ```scheme (define (print-point p) (newline) (display " (") (display (x-point p)) (display ",") (display (y-point p)) (display ")")) ```

Key concepts

Constructors

A constructor is a special function used in programming to create instances of a more complex data structure from basic components. In the context of coordinate geometry, constructors are essential for creating points and segments, which are foundational aspects of geometric representation in a plane. For the given exercise, we define two constructors: `make-point` and `make-segment`.

The `make-point` constructor creates a point by taking two numbers as arguments, corresponding to the \(x\) and \(y\) coordinates. It combines these coordinates into a simple pair, often using a list or a similar data structure. Here, this is achieved by using a `cons` function, which combines the \(x\) and \(y\) values into a single entity, representing a point.

• Syntax: `(define (make-point x y) (cons x y))`

`make-segment` works similarly but operates on a higher level by constructing a line segment from two points - a start point and an end point. This function creates a pair that encapsulates the two points, effectively forming the segment.

• Syntax: `(define (make-segment start end) (cons start end))`

Using constructors, we can build the fundamental geometric structures necessary for more complex operations, such as calculating midpoints or intersections.

Selectors

Selectors are functions designed to retrieve specific data from a composite data structure. In our coordinate geometry task, selectors help us access individual components of points and segments.

For a point, selectors allow us to retrieve the \(x\) and \(y\) coordinates. With `x-point` and `y-point`, we can extract the first and second elements of the point's representation, respectively. This is made possible through the `car` and `cdr` functions, which access the first and second elements of a pair:

• `(define (x-point p) (car p))` - retrieves the x-coordinate
• `(define (y-point p) (cdr p))` - retrieves the y-coordinate

When it comes to segments, selectors like `start-segment` and `end-segment` are used to access the start and end points of a segment. They similarly rely on `car` and `cdr` to efficiently extract the points:

• `(define (start-segment s) (car s))` - retrieves the starting point of the segment
• `(define (end-segment s) (cdr s))` - retrieves the ending point of the segment

These selectors are crucial for manipulating complex structures, enabling further computations like identifying midpoints or calculating segment lengths.

Coordinate Geometry

Coordinate geometry is a branch of mathematics that deals with representing and analyzing geometric figures using coordinates, typically in a Cartesian plane. Each point is specified by an ordered pair \( (x, y) \) that represents its position.

In the exercise at hand, we create constructs and use functions to work with geometric entities like points and line segments. One primary application is calculating the midpoint of a segment, which involves averaging the coordinates of the endpoints. This can be derived using:

• Midpoint formula: \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\)

Using our `midpoint-segment` function, we utilize previously defined constructors and selectors to pinpoint this location. Given a segment, we retrieve its start and end points, calculate the mid-coordinates, and formulate a new point at these averages:

• `(define (midpoint-segment s)...`

This concept forms the basis for various geometric calculations and operations, allowing us to solve real-world problems through mathematical models.