Contextualization

Mathematics has a diverse range of applications in various areas of everyday life, and one of them is cartography, the science of graphical representation of the Earth's surface. In particular, mathematical projection, which is a fundamental tool in map creation, raises some interesting questions related to geometry and the concept of deformation.

Cartographic projections are mathematical techniques for representing the characteristics of the Earth's surface (sphere) on a flat surface (map). However, this transformation inevitably causes deformations, as it is not possible to perfectly map a sphere onto a plane without distortions. And that's where we focus our project - investigating deformations in projections.

There are several cartographic projections, each with its own peculiarities and implications regarding deformations. Let's focus on two well-known projections: the cylindrical projection and the conical projection. What deformations do these projections cause? How do they affect angles and areas? How can we measure these deformations? These are questions we will explore.

When looking at world maps that we routinely see, it can be easy to forget that they are representing a three-dimensional object on a flat surface - and this representation brings several challenges. By understanding deformations in projections, we can better understand the nature of maps and the mathematics behind them.

This knowledge is essential not only for cartographers but also in professions such as pilots, engineers, geographers, and even politicians, who use maps to make decisions that affect millions of people. This understanding helps to avoid mistakes that can occur when interpreting an inappropriate map.

For example, if we use a map with a cylindrical projection, which exaggerates areas near the poles, we may overestimate the size of Greenland compared to Africa, for instance. Thus, understanding deformations in projections is essential to avoid misconceptions of reality.

Below, we suggest some resources in Portuguese that will help you delve into the topic and will be the basis for our project:

1. What are the deformations that occur in cartographic projections?
2. Types of cartographic projections and their characteristics
3. Deformations of cartographic projections
4. Cartographic Projections: what are they, types, characteristics

Practical Activity

Activity Title: Mapping Deformations

Project Objective:

The main objective of this project is to investigate the deformations caused by cylindrical and conical projections, allowing students the opportunity to intuitively understand the challenges and implications of these projections in the graphical representation of our world.

Detailed Project Description:

In this project, the group of students (3 to 5 members) will be responsible for creating their own cylindrical and conical projections of a sphere (representing the Earth), measuring the deformations of angles and areas caused by these projections, and finally presenting their findings in a detailed written report.

Required Materials:

• Large balloon or soft ball that can be painted
• Tracing paper
• Pencils and erasers
• Ruler and compass
• Gouache paint or markers
• Tape measure or ruler
• Camera (optional)

Detailed Step-by-Step for Activity Execution:

1. Creation of the Sphere: Inflate the balloon/ball representing the Earth. You can paint or draw the Earth's surface on the balloon, highlighting some important features such as continents, equator lines, meridians, parallels, etc.

2. Cylindrical Projection: Wrap the balloon vertically with the tracing paper, representing the cylinder. With the help of markers, transfer the contours of the continents, equator lines, and meridians to the tracing paper. Remove the paper and flatten it to create the cylindrical projection on a plane.

3. Conical Projection: Similarly, wrap the inclined balloon with another piece of tracing paper, representing the cone. Transfer the contours and flatten the paper to create the conical projection.

4. Measurement of Deformations: Choose some notable locations (e.g., cities, lakes, etc.) in your projections and measure the distances between them on the sphere and in the projections to identify area deformations. Additionally, measure the angles between the meridian and parallel lines on the sphere and in the projections to identify angle deformations.

5. Analysis and Discussion: Compare the deformations observed in the two projections and relate them to the theoretical properties of cylindrical and conical projections. Discuss the implications of these deformations in the daily use of maps.

6. Report Elaboration: Based on your observations and discussions, create a comprehensive report detailing all stages of the project, its results, and conclusions. Remember, the report should include the following topics: Introduction, Development, Conclusions, and Bibliography used.

Project Deliverables:

The deliverables of this project involve both a practical component and a written component. The practical component is the creation of the projections themselves, along with the collection and analysis of the corresponding measurements. It is recommended to document this process with photographs or video.

The written component is the final report. This report should start with an introduction to the topic of deformations in projections, their relevance and real-world application, and the project's objective. Then, in the development section, the student should introduce the theory behind the project, describe the activity in detail, indicate the methodology used, and present and discuss the results obtained.

The conclusion of the report should then summarize the main points of the project, the lessons learned, and the conclusions drawn. The final bibliography should list the resources used to help complete the project.

Students should plan their time so that the project, including the elaboration of the report, is completed within one month.