Isometric Projection: Exploring Spatial Perception in Computer Vision

Chapter 1: 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 of the resulting 2D representation of a cube is a regular hexagon, with all the black lines being the same length and the areas of all the cube's faces being the same.

You can get the look without the math by placing a sheet of isometric graph paper underneath your regular drawing paper.

The same applies here, A 3D scene can be viewed from an isometric perspective.

To begin, the camera must be level with the ground and parallel to the coordinate axes, it is first rotated horizontally (around the vertical axis) by ±45°, then 35.264° around the horizontal axis.

Isometric projection can also be imagined as a view inside a cube, seen from one of the upper corners and moving to the opposite wall, lower corner.

The x-axis makes a right-down diagonal, The y-axis dips down and to the left on a diagonal, and vertically upward along the z-axis.

Height on the image also acts as a depth indicator.

Lines drawn along the axes are at 120° to one another.

A camera with these characteristics would require an object-space telecentric lens, just as it would for any axonometric or orthographic projection, to ensure that projected lengths remain constant as the observer moves away from the camera.

The term isometric is frequently used incorrectly to describe axonometric projections. However, there are in fact three distinct axonometric projections: oblique, dimetric, and isometric.

Isometric projection requires two views, one from above and one from below, Considering how contradictory the second's worth may seem, it warrants some clarification.

Let’s first imagine a cube with sides of length 2, and centered on the point where the axes meet, meaning that every one of its faces has an axis intersection exactly 1 unit from the origin.

We can calculate the length of the line from its center to the middle of any edge as √

2

using Pythagoras' theorem .

By rotating the cube by 45° on the x-axis, to emphasize (1, 1, As a result, (1) becomes into (1), 0, √

2

) as depicted in the diagram.

The second rotation aims to bring the same point on the positive z-axis and so needs to perform a rotation of value equal to the arctangent of ¹⁄√

2

which is approximately 35.264°.

An isometric perspective can be attained from one of eight different angles, Depending on the direction of the observer's octant.

The isometric transform from a point ax,y,z in 3D space to a point bx,y in 2D space looking into the first octant can be written mathematically with rotation matrices as:

where α = arcsin(tan 30°) ≈ 35.264° and β = 45°.

As was mentioned up top,, this is a rotation around the vertical (here y) axis by β, followed by a rotation around the horizontal (here x) axis by α.

The next step is a xy-plane orthographic projection:

By rotating to the opposite sides and then inverting the view orientation, we get the other seven choices.

The concept of isometry had been around for centuries in a basic empirical form until it was defined by Professor William Farish (1759-1837).

Isometric projection, like all parallel projections, makes objects look the same size whether they are close to or far from the spectator. While this is useful for architectural designs where precise measurements must be collected, it creates an illusion of distortion because human eyesight and photography do not ordinarily function in this way. As demonstrated to the right or above, it can also lead to circumstances where judging depth and altitude is challenging. The Penrose Stairs are one example of a seemingly contradictory or impossible shape that can result from this.

Isometric graphics, also known as parallel projection graphics, are commonly used in video games and pixel art, instead of looking straight down or to the side, the viewpoint is skewed to show details in the environment that would otherwise be hidden, Creating an illusion of depth in three dimensions.

in spite of the label, However, not all isometric computer graphics actually use an isometric viewpoint, the x, y, and z axes are not necessarily oriented 120° to each other.

Instead, Several perspectives are considered, Typically, a dimetric projection with a 2:1 pixel ratio is used.

The terms ³⁄4 perspective, ³⁄4 view, 2.5D, pseudo 3D and pseudo 2D are also common synonyms, although there may be some subtle differences in interpretation depending on the situation.

With the rise of more capable 3D graphics technologies and the shift toward games focusing on action and distinctive characters, isometric projection has become increasingly rare.

{End Chapter 1}

Chapter 2: Orthographic projection

Orthographic projection (also orthogonal projection and analemma) results in affine transformation of each picture plane on the viewing surface. In an oblique projection, the projection lines are not orthogonal to the projection plane.

In multiview projection, orthographic can refer to a technique in which the principal axes or planes of the subject are parallel with the projection plane to create the primary views. If the major planes or axes of an object in an orthographic projection are not parallel to the projection plane, the depiction is axonometric or an auxiliary view. (Axonometric projection and parallel projection are synonymous.) Plans, elevations, and sections are subtypes of primary views; isometric, dimetric, and trimetric projections are subtypes of auxiliary views.

A telecentric lens that gives an orthographic projection is an object-space lens.

The following matrix defines a straightforward orthographic projection onto the z = 0 plane:

For each point v = (vx, vy, vz), point converted Pv would be

Frequently, it is more advantageous to employ homogeneous coordinates. For homogeneous coordinates, the above transformation can be expressed as

For each homogeneous vector v = (vx, vy, vz, 1), The vector Pv after transformation would be

In computer graphics, one of the most frequently used matrices for orthographic projection is specified by the 6-tuple (left, right, bottom, top, near, far), which specifies the clipping planes. These planes create a box with the smallest corner at (left, bottom, -near) and the largest corner at (right, top, -far) (right, top, -far).

The box is then scaled to the unit cube, which is defined as having its minimum corner at (1,1,1) and its greatest corner at (1,1,1). (1,1,1).

The following matrix represents the orthographic transformation::

This can be expressed as a scaling S followed by a translation T according to the form

The inversion of the projection matrix P−1, It can be employed as the unprojection matrix:

Isometric projection, dimetric projection, and trimetric projection are three subtypes of orthographic projection, depending on the exact angle at which the view deviates from the orthogonal. 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.

Multiview projection produces up to six images of an object, known as primary views, with each projection plane parallel to one of the object's coordinate axes. The relative positioning of the views is determined by one of two schemes: first-angle or third-angle projection. The appearances of views are projected onto planes that form a six-sided box around the object in each case. Although it is possible to draw six different sides, three views of a drawing provide sufficient information to create a three-dimensional object. These perspectives