Introduction

Relevance of the Topic

Spatial Geometry is one of the fundamental pillars of Mathematics, essential for our understanding of the space around us. Within this vast field of study, 'Deformations in Projections' holds a special place, bringing to light the complexity and flexibility of the space that surrounds us. Through this study, we will learn how the images of geometric figures can change depending on the observer's perspective, which has numerous practical applications in art, photography, cartography, engineering, among others.

Contextualization

In the Mathematics curriculum of the 2nd year of High School, Spatial Geometry is introduced after Plane Geometry. Understanding geometry in three-dimensional spaces is a natural step to deepen our understanding of the world we live in. After mastering the study of spatial figures, attention turns to how these figures can be projected onto flat surfaces, which is essential in cartography and graphic design, for example. At this point, understanding the deformations that occur in such projections becomes integral to understanding how shapes can appear different depending on how we observe them.

Therefore, understanding Deformations in Projections is a crucial step in the student's journey to becoming an abstract and critical thinker. This topic is a key piece that ties together various areas of study, including Mathematics, Arts, and Sciences. Understanding the importance of this topic not only enhances our mathematical knowledge but also enriches our appreciation of other disciplines.

Theoretical Development

Components

• Perspective Projection: It is a graphical representation technique in which the viewer's and object's points of view are taken into account. Perspective projection does not preserve the true shape or angle of objects, due to the converging viewpoint axis. This is the basis for the perceptible deformations in paintings, photographs, and other types of drawings.

• Parallel Projection: In contrast to perspective projection, in parallel projection, the light rays projected from each point of the object are parallel to each other. This projection is used in maps and engineering projects, where it is crucial to preserve the original shapes and angles of the figures.

• Isometric Projection: A special form of parallel projection in which the observer views the object from a specific direction (at 30⁰, 45⁰, or 60⁰, for example). Parallel lines on the object appear parallel in the projection, resulting in a deformation in shape, but not in size or angles.

Key Terms

• Deformation: Alteration in the shape, size, or texture of an object. It can occur in projections, especially in perspectives, where the distance from an object to the observer affects its appearance.

• Isometry: A geometric transformation that preserves the shape, but not necessarily the sizes and angles. It is a fundamental concept in geometry, especially in isometric projection.

• Perspective: A drawing technique that represents a three-dimensional scene on a two-dimensional plane, incorporating the distance and angle of the observer's view.

Examples and Cases

• Renaissance paintings: Many masterpieces of the Renaissance heavily use perspective projection, so that the figures and landscapes depicted on the canvas appear to have real depth. However, if we look at the same scene in real life, the perception of depth and the proportion of the figures can be drastically different.

• Maps: From the globe to the world map, all maps are distortions of reality in some way. The projection used to transform a spherical surface (Earth) into a flat surface (the map) produces inevitable deformations. Each type of projection uses different strategies to minimize certain types of distortion.

• Isometric illustrations of constructions: Often used in instruction manuals, computer games, or architectural projects, these illustrations are drawn in an isometric projection. Although they provide a detailed view of the design, they do not completely represent the true appearance of the structure.

These examples clearly illustrate the intersection between Mathematics and Arts, emphasizing the importance of understanding Deformations in Projections.

Detailed Summary

Key Points

• Type of Projection: It is essential to understand the difference between perspective, parallel, and isometric projections, as each of them produces specific distortions and deformations in geometric figures.

• Deformation in Perspective Projection: The deformations in this projection result from the fact that the distance between the object and the observer affects the object's appearance. The farther away, the smaller the object appears.

• Deformation in Parallel Projection: In this projection, deformations mainly occur due to how the projection deals with angles. Parallel lines on the object appear to meet at infinity, resulting in a distortion in the perception of lengths.

• Isometric Deformation: In this projection, there is deformation in shape, but not in sizes and angles. Parallel lines on the object are still parallel in the projection.

• Key Term: Deformation: It is fundamental to understand the concept of deformation, as it is intrinsically linked to projections and the alterations they can cause in figures.

• Key Term: Isometry: This is an important geometric transformation at the core of the phenomenon of deformations in projections, as it is a transformation that preserves the shape.

Conclusions

• Deformations in projections are an intrinsic aspect of representing three-dimensional figures on two-dimensional surfaces.

• Each type of projection has its own characteristics and produces its own distortions and deformations.

• Understanding deformations in projections is essential for various practical applications, such as interpreting paintings and photographs, graphic design, engineering, among others.

Exercises

1. Perspective Projection vs. Parallel Projection: Draw a simple object, such as a square, and project it onto a flat surface using both perspective and parallel projection. Observe and describe the resulting deformations.

2. Deformation in Maps: Compare two different types of maps that attempt to represent the spherical surface of the Earth (for example, the globe and a world map). Identify the deformations present in each type of projection used in the making of the maps.

3. Recognizing Deformations: Given a set of illustrations, some in perspective projection and others in isometric projection, identify which of them likely underwent deformations in shape and in which the sizes and angles were preserved. Justify your answers.