Introduction

Mathematics has wider applications than we can imagine. Cartographic projection is one such application. Yes, the art of making maps and representing the earth, a three-dimensional globe, on a two-dimensional plane. More than mere drawings, maps are powerful tools for geographically representing our planet and involve substantial mathematics in their creation. This is the starting point of our project: deformations resulting from projections.

The core issue in this project is the inherent deformation that occurs when we go from a sphere to a plane. As you know, wrapping a soccer ball with paper without creasing it is virtually impossible. Something similar happens when we try to represent something three-dimensional such as the earth on a flat surface like a map.

We will work with the concept of cylindrical and conical projections to understand how the projection process affects angles and areas in cartographic representations. Spatial thinking and the perception that mathematics is essential in map design are valuable skills not only for mathematics but also for a better understanding of the world we live in.

Background

Cylindrical and conical projections are two main types of projections used in cartography. In a cylindrical projection, the earth's sphere is projected onto an imaginary cylinder, which is then opened up to form a plane. In a conical projection, the sphere is projected onto an imaginary cone, which is also opened up to form a plane. However, both processes generate deformations observable in the differences in size, shape, and distance between countries on different types of maps.

These deformations occur due to the difference between the earth's spherical surface and the flat surface of the paper. Therefore, our challenge is to understand why these deformations occur and how we can quantify or minimize them using the mathematical knowledge involved in cylindrical and conical projections.

Suggested resources for study:

1. Book: "Cartography: Representation, Communication, and Visualization of Spatial Data" – Ardemirio de Barros Silva: This book provides an introduction to the study of cartography and its applications, including deformations in projections.
2. Website: Mathematical Multi (http://www.matematicamulti.com.br/): This website is a great source of articles and tutorials on a variety of mathematical topics, including geometry and projections.
3. YouTube: "Matemática Rio with Prof Rafael Procopio" Channel: This channel has a series of videos explaining mathematical concepts in a fun and engaging way, including the geometry of projections.

We will focus on this fascinating world of projections using mathematics as our main tool. Shall we begin?

Learning Activity

Activity Title:

"Modeling and Measuring Deformations in Cartographic Projections"

Project Goal:

Investigate and quantify the deformation of angles and areas that occur in cylindrical and conical projections in cartography. The project also aims to develop socio-emotional skills such as time management, communication, problem-solving, and creative thinking.

Project Description:

Students will be divided into groups of 3 to 5 participants and each group will receive a globe and world maps in two different projections: cylindrical and conical. The task of the groups will be to investigate how deformations occur on maps by making measurements and calculations to quantify these deformations.

Using digital technology (specific software or mobile phone applications), the groups will take measurements of angles and areas on both maps and compare them with the measurements made on the globe.

The groups will also research the main applications and advantages and disadvantages of each type of projection.

Required Materials:

1. Globe
2. World maps in cylindrical and conical projections
3. Ruler, protractor, and compass
4. Specific software or mobile phone applications for measuring angles and areas
5. Material for research: books, internet, etc.

Step by Step:

1. Each group must first study the concept of cylindrical and conical projections and understand how they are made. To do this, use the suggested reference materials and any others you can find.

2. Next, each group must measure the angles and areas of several selected countries on the maps and the globe, recording the results.

3. The groups should then use the software or applications to perform the same measurements on the digital maps and compare them with the previous results.

4. The groups must analyze the results obtained and discuss the deformations found, quantifying the differences, and explaining them based on the concepts studied.

5. The groups must research applications of cylindrical and conical projections, listing advantages and disadvantages of each.

6. Finally, each group must prepare a report containing: introduction (contextualization of the theme, relevance, and objective of the project), development (theory of projections, description of the activity carried out, methodology used, and results obtained), conclusion (reiteration of the main points, lessons learned, and conclusions about the project), and bibliography. This document should reflect all the work done by the groups and will be crucial for the project evaluation.

The project is suggested to have a duration of four weeks, with each student dedicating more than twelve hours to its execution.

Project Deliverables:

At the end of the project, students must deliver:

1. The final written report, containing a detailed description of the entire project, as per step 6 above. This report should contain all calculations and measurements made, as well as comparative graphs or tables to illustrate the deformations found. It should also contain a discussion of the results, explaining the deformations based on the concepts studied. The report should be clear and organized, allowing the reader to fully understand the work done.

2. A digital map for each projection, showing the countries measured and their respective angles and areas, with indications of the deformations found.

3. A short video (5 to 10 minutes) presenting the project, explaining the projections, the measurements made, and the results found. This video will be used to assess students' communication skills and their understanding of the topic.

The evaluation will be based on the accuracy and clarity of the measurements and calculations, the quality of the report and the video, and the students' ability to explain the concepts and results clearly and coherently.

Remember, the main goal is to understand the deformations that occur in the projections and to be able to explain them using mathematics and geography. Good luck!