Descriptive geometry

The geometry is a field of mathematics broad studying all the properties and measures the space or the plane giving special attention to the problems of the area and diameters of the figures, or the volume of solid bodies. Geometry is divided into several branches or fields, one of them is known as descriptive geometry which works based on dimensions .

Definition

Projective geometry is a discipline whose function study all geometric properties and the relation of spatial type that exist figures from the orthogonal projections occurring on a flat surface . It is then in charge of being able to represent the figures of real space in a two-dimensional plane and for this it uses a series of measures and techniques that help in the process.

Descriptive geometry characteristics

Among the main features are the following:

• It is part of the geometry .
• Represents figures in three-dimensional form in two-dimensional space .
• It can facilitate representation on a two-dimensional paper .
• Exactly describe the shape of bodies.
• It provides fundamentals , principles and ways to solve and to be able to graphically communicate the elements.
• Use two different models; the language of representation and the treatise on geometry.

History

It was created by Gaspar Monge , a mathematician of French nationality who managed to create a form to be able to represent shapes and bodies on a plane regardless of the dimensions, however, its history begins since humanity had its origins, when it was he practiced rock art .

Little by little, the technique was improving and with the invention of stonework , more complex designs began to be created to determine the way in which the stones should be carved for construction, this process took place throughout the Middle Ages.

In the Renaissance , descriptive geometry managed to develop further thanks to important figures such as Leonardo da Vinci and Filippo Brunelleschi who used some mathematical concepts to elaborate their works. In 1975 , Gaspar Monge appeared who made an important publication where he explained how to use descriptive geometry.

Descriptive geometry applications

The descriptive geometry can be applied to establish all the principles used to resolve and communicate graphically the elements that exist in space, they can be straight , dots , solids , volumes , flat surfaces and curves . It is the way in which the bodies that are conceived in space can be represented and shaped, establishing correlative relationships between them, in three dimensions and in a flat form.

It is also applied in the horizontal or iconographic planes where you can draw what is observed if the image were seen from infinity in the upper part. With it, it is then possible to get to know the position and shape of a body in space and with this it is possible to solve a large number of analytical problems .

Elements

The basic elements of descriptive geometry are three, these are mentioned below:

• Point : it is considered as the first element that does not have a geometric definition and is represented by a very small circle next to a letter that identifies it.
• Straight : it is an infinite set of points which extend indefinitely in opposite directions. In order to refer to it, it is first necessary to select the points in it, since these are responsible for determining the line. It is represented by uppercase or lowercase letters.
• Flat : is flat is a flat surface that can be extended indefinitely. It is represented geographically by a four-sided figure and a capital letter.

Beginning

The fundamental principles of descriptive geometry are as follows:

• The sight directions for both adjacent views are mutually perpendicular.
• The points that correspond in adjacent views must be connected by means of visual and dotted lines that make it possible to join the adjacent points.
• The measurements of the parallels are equal to the lines of the sightings taking into account all the views adjacent to the same view.
• A normal view of a line is one that, in the direction of sight , is perpendicular to the line.
• A terminal view of the line is the one that, in the direction of the line of sight, turns out to be parallel to the line so that said line can be represented as a point.
• Parallel lines display as parallel from any view that is orthogonal.
• Two perpendicular lines can be displayed in any normal view of the lines. They cannot appear perpendicular unless vision is normal in one of them.
• For any point that exists in an oblique plane, the three main lines that are part of the plane can be drawn.

Types of planes

There are several types of planes that can be used in descriptive geometry, among them we can find oblique planes , perpendicular to the edge plane , perpendicular to the vertical plane , perpendicular to the profile plane , parallel to the horizontal plane , parallel to the frontal plane .

Types of straight lines

In descriptive geometry we can find oblique lines , frontal line, horizontal line, parallel line to the two projection planes, profile line, vertical perpendicular, perpendicular point line, oblique, oblique to one plane and parallel to the other, perpendicular to a plane that are located in another plane, straight in profile and coinciding with the ground line.

Types of projections

The types of projections that can be found in descriptive geometry are the following:

• Parallel or cylindrical projection : where the center of the projection is at an infinite distance and all projection lines are parallel. It can be divided into orthogonal projection and oblique projection.
• Central or conical projection : the center of the projection is located at a finite distance and all the projection lines turn out to be convergent.

Geometric forms

With regard to geometric shapes in descriptive geometry we can find two important groups:

• Plane geometric shapes : include figures such as polygons, lines, and conic sections.
• Ruled geometric shapes : here we find regular polyhedra, wedge, pyramid, cylinder, cone, sphere, prism, pyramids, ellipsoid, paraboloid, cone, hyperboloid.

Importance

Descriptive geometry turns out to be of great importance in all those areas in which it is necessary to represent elements on flat surfaces , for this reason, it is vital for a large number of sciences that include areas such as engineering , architecture , topography and design . It is a good way of being able to understand the three-dimensional space that surrounds individuals in order to develop a logical thought structure . In addition, it becomes the basis of other disciplines such as the mechanics of deformable and fluid bodies.

Descriptive Geometry Books

Some books in which important information about descriptive geometry can be found are:

• Descriptive geometry : theory and problems of Jorge Nakamura Muroy.
• Descriptive geometry theory and problems of Minor Clyde Hawk.
• Descriptive geometry book by Alejandro Miranda.
• Descriptive Geometry by Gaspard Monge. Nabu Press.
• Descriptive geometry: The flat conception of three – dimensional space by Pérez García, Alberto María.