Axonometric Projection: Exploring Depth Perception in Computer Vision

By Fouad Sabry

What is Axonometric Projection

Axonometric projection  is a type of orthographic projection used for creating a pictorial drawing of an object, where the object is rotated around one or more of its axes to reveal multiple sides.

How you will benefit

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

Chapter 1: Axonometric projection

Chapter 2: Isometric projection

Chapter 3: Orthographic projection

Chapter 4: Descriptive geometry

Chapter 5: Oblique projection

Chapter 6: Parallel projection

Chapter 7: Multiview orthographic projection

Chapter 8: Architectural drawing

Chapter 9: Axonometry

Chapter 10: Technical drawing

(II) Answering the public top questions about axonometric projection.

(III) Real world examples for the usage of axonometric projection 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 Axonometric Projection.

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

Book preview

Chapter 1: Axonometric projection

Axonometric projection is a sort of orthographic projection used to create a depiction of an item by rotating it about one or more of its axes to display numerous sides.

Axonometry signifies measurement along the axes. In German literature, axonometry is based on Pohlke's theory, so that axonometric projection could comprise all parallel projection types, including not only orthographic projection (and multiview projection), but also oblique projection. However, outside of German literature, the term axonometric is sometimes only used to differentiate between orthographic views where the principal axes of an object are not orthogonal to the projection plane and orthographic views where the principal axes of an object are orthogonal to the projection plane. (In multiview projection, these are referred to as secondary and primary views, respectively.) Occasionally, the phrase orthographic projection is also reserved just for the major views, which is quite confusing.

Therefore, in German literature, axonometric projection may be considered synonymous with parallel projection; yet, in English literature, axonometric projection may be considered synonymous with a auxiliary view (as opposed to a primary view) in a multiview orthographic projection..

In an axonometric projection, the scale of an object is independent of its location (i.e., an object in the foreground has the same scale as an object in the background); consequently, such images appear distorted, as human vision and photography use perspective projection, in which the perceived scale of an object depends on its distance and location from the viewer. This distortion, which is the direct effect of the presence or absence of foreshortening, is most pronounced if the object is predominantly rectangular. Despite this drawback, axonometric projection can be beneficial for illustrative purposes, particularly because it permits simultaneous transmission of precise measurements.

Depending on the precise angle by which the view deviates from orthogonal, there are three varieties of axonometric projection: isometric projection, dimetric projection, and trimetric projection. In axonometric drawings, as in other forms of diagrams, one axis of space is typically depicted as vertical.

In the isometric view, The most prevalent type of axonometric projection used in engineering drawings, The direction of vision is such that all three axes of space appear to be proportionally compressed, and there is a common angle of 120° between them.

As the foreshortening-induced distortion is uniform, The proportions between lengths are maintained, and the axes have the same scale; This facilitates taking direct measurements from the drawing.

Another advantage is that 120° angles are easily constructed using only a compass and straightedge.

In dimetric projection, the viewing direction is such that two of the three axes of space seem equally compressed, with the attendant scale and presentation angles set by the viewing angle; the scale of the third direction is determined individually. Dimetric drawings typically contain approximations of dimensions.

In trimetric projection, the viewing direction is such that the three axes of space appear unequally compressed. The scale along each of the three axes and the angles between them are determined independently based on the viewing angle. In trimetric drawings, dimensioning approximations are common, although trimetric perspective is rarely employed in technical drawings.

Axonometry was developed in China.

Model of an optical grinding engine (1822), drawn in 30° isometric perspective

illustration of a dimetric perspective drawing from a U.S. patent (1874)

Illustration of a trimetric projection depicting the Bank of China Tower in Hong Kong.

Example of isometric projection in Chinese art from a 15th century CE illustrated version of the Romance of the Three Kingdoms.

Along the River During the Qingming Festival, in its original form, is attributed to Zhang Zeduan (1085–1145). Observe that the image is inconsistent because it alternates between axonometric and perspective projection in different portions of the image.

Similarly to other methods of parallel projection, objects created using axonometric projection do not appear larger or smaller as they approach or recede from the spectator. Despite being useful for architectural designs in which measurements must be extracted directly from the image, the outcome is a perception of distortion because, unlike perspective projection, this is not how human vision or photography generally functions. As seen in the figure to the right, it can also easily lead to circumstances in which depth and height are difficult to estimate.

This visual ambiguity has been utilized in optical art and drawings of impossible objects. Although not exactly axonometric, Waterfall (1961) by M. C. Escher is a well-known artwork in which a channel of water appears to flow independently along a downward path, only to inexplicably fall again as it returns to its source. Thus, the water appears to violate the law of energy conservation.

{End Chapter 1}

Chapter 2: Isometric projection

In technical and engineering drawings, isometric projection is used to create a two-dimensional image of a three-dimensional object. This is an axonometric projection, where the angle between any two axes is 120 degrees and all three appear to be shortened by the same amount.

Isometric, from the Greek for equal measure, is a projection in which the scale remains constant along all axes (unlike some other forms of graphical projection).

Selecting a viewpoint in which the projections of the x and y axes form right angles provides an isometric perspective, y, both the x and z axes are equivalent, or 120°.

For example, using a cube, One does this by fixing one's gaze intently on that person's face.

Next, the cube is rotated ±45° about the vertical axis, followed by a rotation of approximately 35.264° (precisely arcsin ¹⁄√

3

or arctan ¹⁄√

2

, It is perpendicular to the x-axis and has something to do with the Magic angle.

As can be seen in the image, the perimeter