Descriptive Geometry: Unlocking the Visual Realm: Exploring Descriptive Geometry in Computer Vision

By Fouad Sabry

What is Descriptive Geometry

Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and in art. The theoretical basis for descriptive geometry is provided by planar geometric projections. The earliest known publication on the technique was "Underweysung der Messung mit dem Zirckel und Richtscheyt", published in Linien, Nuremberg: 1525, by Albrecht Dürer. Italian architect Guarino Guarini was also a pioneer of projective and descriptive geometry, as is clear from his Placita Philosophica (1665), Euclides Adauctus (1671) and Architettura Civile, anticipating the work of Gaspard Monge (1746-1818), who is usually credited with the invention of descriptive geometry. Gaspard Monge is usually considered the "father of descriptive geometry" due to his developments in geometric problem solving. His first discoveries were in 1765 while he was working as a draftsman for military fortifications, although his findings were published later on.

How you will benefit

(I) Insights, and validations about the following topics:

Chapter 1: Descriptive geometry

Chapter 2: Analytic geometry

Chapter 3: Affine transformation

Chapter 4: Orthographic projection

Chapter 5: 3D projection

Chapter 6: Oblique projection

Chapter 7: Vanishing point

Chapter 8: Picture plane

Chapter 9: Line (geometry)

Chapter 10: Parallel projection

(II) Answering the public top questions about descriptive geometry.

(III) Real world examples for the usage of descriptive geometry in many fields.

Who this book is for

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Descriptive Geometry.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: May 5, 2024

Book preview

Chapter 1: Descriptive geometry

Descriptive geometry is the branch of geometry that permits the representation of three-dimensional objects in two dimensions by employing a particular set of processes. These techniques are essential for engineering, architecture, and design, as well as art.

The protocols of Monge make it possible to draw an imaginary object in such a way that it may be modeled in three dimensions. All geometric characteristics of the fictitious item are accounted for in correct size and shape, and they may be visualized from any place in space. Every image is displayed on a two-dimensional surface.

Descriptive geometry employs the image-creating approach of parallel, hypothetical projectors issuing from an imagined object and intersecting a parallel, imaginary projection plane at right angles. The accumulation of intersecting points generates the desired image.

Two pictures of an item are projected in arbitrarily perpendicular directions. Each image view contains three spatial dimensions, two of which are presented as full-scale, mutually perpendicular axes and the third as an invisible (point view) axis receding into the image space (depth). Each of the two adjacent image perspectives contains a full-scale representation of one of space's three dimensions.

Either of these images could serve as the basis for a third projected vision.

The third perspective may initiate a fourth projection, and indefinitely more.

Each of these sequential projections is a circuitous, 90° turn in space in order to view the object from a different direction.

Each successive projection employs a dimension at full scale that appeared as a point-of-view dimension in the preceding image. To attain the full-scale view of this dimension and include it into the new view, one must disregard the prior view and move to the second previous view, in which this dimension appears at full-scale.

Each unique perspective is generated by projecting in one of an endless number of directions, perpendicular to the preceding projection direction.

(Envision the many directions of the spokes of a wagon wheel each perpendicular to the direction of the axle.) The result is one of stepping circuitously about an object in 90° turns and viewing the object from each step.

Each new view added to an orthographic projection layout display appears as a unfolding of the glass box model..

Other than the Orthographic, six conventional primary views (Front; Right Side; Left Side; Top; Bottom; Rear), descriptive geometry tries to give four essential solution views: the true length of a line (i.e, full size, not foreshortened), a line's point view (end view), the real shape of a plane (i.e, according to scale, (or not abbreviated), and the edge view of a plane (i.e, (view of a plane with the line of sight perpendicular to the line of sight associated with the line of sight to produce the plane's true shape).

Frequently, these assist to decide the direction of projection for the next view.

By the 90° circuitous stepping process, Any projection from the point perspective of a line reveals its full length; Its point view is obtained by projecting in a direction parallel to a genuine length line view, Any line's point view projected onto a plane produces the edge view of the plane; A projection perpendicular to the edge view of a plane will reveal its true shape (to scale).

These diverse perspectives can be utilized to tackle engineering difficulties provided by solid-geometry concepts.

Studying descriptive geometry is heuristically beneficial. It promotes visualization, spatial analysis, and the intuitive capacity to detect the optimal viewing direction for presenting a geometric issue to be solved. Exemplificative examples:

Two skewed lines (perhaps pipes) are placed in general positions to discover the location of their shortest link (common perpendicular)

Two skew lines (pipes) at general positions so that the shortest connector can be seen in full scale.

Two skew lines at general positions such that the shortest parallel connector to a given plane can be viewed to scale (say, to determine the position and the dimension of the shortest connector at a constant distance from a radiating surface)

A surface on which a perpendicularly drilled hole can be viewed in its entirety, as if one were gazing through the hole (say, to test for clearances with other drilled holes)

A plane that is equidistant between two skew lines in general positions (for example, to verify safe radiation distance?)

The shortest distance between a given point and a plane (say, to locate the most economical position for bracing)

The intersecting line between two surfaces, including curved surfaces (maybe for the most cost-effective section sizing?)

The actual magnitude of the angle formed by two planes.

A standard for showing computer-modeling views similar to sequential orthographic projections has not yet been accepted. One candidate for this is depicted in the images below. The visuals in the illustrations were made using engineering computer graphics in three dimensions.

Three-dimensional computer modeling generates virtual space behind the tube and may generate any view of a model from any angle within this virtual space. It does so without the requirement for adjacent orthographic views and, as a result, may appear to render Descriptive Geometry's tortuous, step-by-step methodology obsolete. Due to the fact that descriptive geometry is the study of the valid or permissible representation of three or more dimensional space on a flat plane, it is essential for enhancing computer modeling capabilities.

Given the X, Y, and Z coordinates of P, R, S, and U, the X-Y and X-Z projections 1 and 2 are drawn to scale, respectively.

To see one of the lines accurately (length in the projection equals length in 3D space), one must: SU in this instance, projection 3 is drawn with hinge line H2,3 parallel to S2U2.

To gain a conclusion about