Mastering Interval and Subinterval Drawing in LaTeX: A Comprehensive Guide

When dealing with mathematical and scientific documentation in LaTeX, it is often necessary to draw intervals and subintervals to represent ranges or sets. This tutorial will guide you through the process of drawing intervals and subintervals using LaTeX and the tkz-euclide package, which is highly regarded for its versatility in geometric drawings.

Understanding Intervals in Math

In mathematics, an interval is a set of real numbers that lie between two endpoints. These endpoints can either be inclusive or exclusive, denoted by parentheses or square brackets. An interval can be either open, closed, or half-open. Subintervals of an interval are parts of the original interval.

Installing and Setting Up the tkz-euclide Package

To draw intervals and subintervals in LaTeX, you first need to have the tkz-euclide package installed. Make sure your LaTeX distribution is up to date and install the package if necessary. Once installed, you can include it in your document using the following line:

usepackage{tkz-euclide}
usetkzobj{all}

Drawing Closed and Open Intervals

Let's start with basic examples of drawing closed and open intervals. A closed interval is represented using square brackets, while an open interval uses parentheses. The tkz-euclide package helps in drawing the interval on a line or number axis.

Example 1: Closed Interval

To draw the closed interval [0, 1], you can use the following code:

begin{tikzpicture}
tkzDefPoint(0,0){A}
tkzDefPoint(1,0){B}
tkzDrawSegment[ultra thick,red](A,B)
tkzDrawPoint[black](A)
tkzDrawPoint[black](B)
tkzLabelPoints[left](A)
tkzLabelPoints[right](B)
end{tikzpicture}

Explanation

Here’s a breakdown of the code:

tkzDefPoint(0,0){A} and tkzDefPoint(1,0){B} define the start and end points of the interval. tkzDrawSegment[ultra thick,red](A,B) draws the line segment between points A and B. tkzDrawPoint[black](A) and tkzDrawPoint[black](B) draw black points at the endpoints. tkzLabelPoints[left](A) and tkzLabelPoints[right](B) label the points with their respective coordinates.

Drawing Half-Open Intervals

A half-open interval is either half-closed at the left and open at the right or vice versa. Drawing such intervals follows a similar approach but with slight modifications.

Example 2: Half-Open Interval [0, 1)

Here is the code to draw the half-open interval [0, 1):

begin{tikzpicture}
tkzDefPoint(0,0){A}
tkzDefPoint(1,0){B}
tkzDrawSegment[ultra thick,red](A,B)
tkzDrawPoint[black](A)
tkzDrawCircle[black](B,b)
tkzLabelPoints[left](A)
tkzLabelPoints[right](B)
end{tikzpicture}

Explanation

tkzDrawCircle[black](B,b) draws a small circle at point B to indicate an open endpoint.

Creating Subintervals

A subinterval is a smaller interval within a larger interval. Drawing subintervals within existing intervals requires careful placement of points and lines.

Example 3: Subinterval (0.25, 0.75) within [0, 1]

To draw a subinterval (0.25, 0.75) within the interval [0, 1], use the following code:

begin{tikzpicture}
tkzDefPoint(0,0){A}
tkzDefPoint(1,0){B}
tkzDefPoint(0.25,0){C}
tkzDefPoint(0.75,0){D}
tkzDrawSegment[ultra thick,red](A,B)
tkzDrawSegment[ultra thick,blue](C,D)
tkzDrawPoint[black](A)
tkzDrawPoint[black](B)
tkzDrawPoint[black](C)
tkzDrawPoint[black](D)
tkzLabelPoints[left](A)
tkzLabelPoints[right](B)
tkzLabelPoints[above](C,D)
end{tikzpicture}

Explanation

tkzDefPoint(0.25,0){C} and tkzDefPoint(0.75,0){D} define the start and end points of the subinterval. tkzDrawSegment[ultra thick,blue](C,D) draws the subinterval using a different color (blue) for distinction. tkzLabelPoints[above](C,D) labels the points within the subinterval.

Conclusion

Mastering the art of drawing intervals and subintervals in LaTeX is a valuable skill for anyone working on mathematical, statistical, or scientific documents. With the tkz-euclide package, you can create clear and precise visual representations of complex mathematical concepts. Whether you are creating interval notations for a research paper or a mathematical analysis, these techniques ensure your work is both understandable and visually appealing.

Further Resources

tkz-euclide documentation on CTAN Collection of TikZ examples for Euclidean geometry

Feel free to explore and experiment with different configurations to craft your perfect mathematical illustrations in LaTeX.