Axonometric Projections

Only orthographic projection cannot give a true impression of ​​the general three-dimensional shape of the object. This limitation can be overcome with axonometric projections. "Axonometric" means "measuring along axes". Axonometric projection shows an image of an object as viewed from a skew direction in order to reveal more than one side in the same picture. Whereas the term orthographic is sometimes reserved specifically for depictions of objects where the axis or plane of the object is parallel with the projection plane, in axonometric projection the plane or axis of the object is always drawn not parallel to the projection plane.

With axonometric projections the scale of distant features is the same as for near features, such pictures will look distorted, as it is not how our eyes or photography work. This distortion is especially evident if the object to view is mostly composed of rectangular features. Despite this limitation, axonometric projection can be useful for purposes of illustration.

The three types of axonometric projections are isometric projection, dimetric projection, and trimetric projection, depending on the exact angle at which the view deviates from the orthogonal. Typically in axonometric drawing, one axis of space is shown as the vertical.

In isometric projection, the most commonly-used form of axonometric projection in engineering drawing, the direction of viewing is so that the three axes of space appear equally foreshortened, of which the displayed angles among them and also the scale of foreshortening are universally known. However in creating a final, isometric instrument drawing, in most cases a full-size scale, i.e., without using a foreshortening factor, is employed to good effect because the resultant distortion is difficult to perceive. Another advantage is that, in engineering drawings, 60° angles are more easily constructed using only a compass and straightedge.

In dimetric projection, the direction of viewing is so that two of the three axes of space appear equally foreshortened, of which the attendant scale and angles of presentation are determined according to the angle of viewing; the scale of the third direction (vertical) is determined separately.

In trimetric projection, the direction of viewing is so that all of the three axes of space appear unequally foreshortened. The scale along each of the three axes and the angles among them are determined separately as dictated by the angle of viewing. Trimetric perspective is seldom used and is found in only a few video games.

Approximations are common in dimetric and trimetric drawings. Axonometric projection from three-dimensional space onto the plane z = n can be obtained as follows:

(2.31)

Note that this transformation is the transfer of images in the z direction on the value of n, due to the transformation:

(2.32)

which follows the projection of infinity on the plane z = 0, give the transformation

(2.33)

As stated earlier, axonometric projections are based by any arbitrary rotation of the coordinate axes, committed in any order, followed by projection onto the plane z = 0. The overall matrix of the trimetric projection is:

(2.34)

where φ is the angle of rotation around the axis y and θ is angle of rotation around the axis x.

In general, the trimetric projection coefficients distortion for each of the projected main axes (x, y, z) is not equal to each other. The imposition of restrictions on the coefficients reduces range of trimetric projections. However, for any given trimetric projection distortion factor is calculated by applying a general transformation matrix to the unit vectors along the principal axes. In particular:

(2.35)

where [ U ] is a matrix of unit vectors along untransformed axes x, y and z respectively, and [ T ] - the total matrix of trimetric projection. Then the distortion coefficients along the principal projected axes are:

(2.36)

Fig.2.2. Example of the trimetric projection

Dimetric projection is a trimetric projection with two identical distortion coefficients and any value of third distortion coefficient. The unit vectors in the main axes x, y and z are converted to

(2.37)

Now the square of the exposed transformation unit vector length along the axis of x (the square of the distortion) is

(2.38)

Similarly, the squares of the distortion coefficients along the axes y and z are

(2.39)

(2.40)

Equating the distortion coefficients for the x and y axis (can be used by any of the three axes) gives one equation with two unknown rotation angles φ and θ:

(2.41)

Using the identities cos²φ = 1 - sin²φ and cos²θ = 1 - sin²θ, we can write

(2.42)

The second relation between φ and θ we obtain by fixing the distortion coefficient along the axis z. Combining equation (2.40) and (2.42) and using the equality cos²φ = 1 - sin²φ, we get

(2.43)

or

(2.44)

Putting u = sin²θ, we find solutions

(2.45)

The decision sin²θ = 1 must be discard, since its substitution in (2.42) gives an infinite result. Consequently, the

(2.46)

Substitution in equation (2.42) leads to

(2.47)

This shows that the range of distortion factor is 0 <= f <= 1.

Fig.2.3. Example of the dimetric projection

Dimetric projection allows measurements with the same scaling factor of two transformed principal axes. Measure along the third axis requires a different scale factor. This can lead to confusion and errors, if you want the exact scaling dimensions of the projected object. Isometric view solves this problem.

In an isometric projection all three distortion coefficients are equal. Now equate the coefficients of the squares of the distortions along the axes y and z and we obtain:

(2.48)

From the previous equality and (2.42) follow:

(2.49)

Then

(2.50)

and φ = ±45⁰. Distortion coefficient for an isometric projection is

(2.51)

In fact an isometric projection is a particular case of dimetric projection with f_z = 0,8165.

Fig.2.4. Example of the isometric projection

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