# are oblique projections one specific subdivision of trimetric projections?

So I've reaserched a while and come with this broad definitions

a projection is the representation of a 3D object in 2D by the use of "imaginary proyectors"(cameras of some sort).

it has 2 branches,

-perspective proyections : they focus on "focal points" of the drawings, and the general rule that distant objects are smaller than closer objects.

-parallel proyections: they focus on showing the images by "beam lines" that are parallel each other.

Inside parallel proyections are 2 subdivisions

--orthographic: they represent the object by frontal images of the object

--axonometric: focus in distorting the angle of the axes for the figure representation. it's broken in 2 kinds orthogonal and oblique

---orthogonal axonometry focus in the forshortening of the angles of the drawing system, it sub-divides in 3 parts

----isometric: it makes 3 angles equal ($120^\circ$ each)

----dimetric: it makes 2 angles equal

----trimetric: the 3 angles are different

---oblique axonometry focus also in the forshorthening of the angles, but with the limitation that one of them is always $90^\circ$

so by this definition is it logical to conclude that oblique axonometry is a special form of the trimetric axonometry?

If any of this definition is wrong , or could be expressed better, please tell me, also biblographical references would also be appreciated. thanks!

(also sorry for if I am not understood, English is my second language)

Unfortunately I don't have 50 rep to leave my answer as a comment, so I'll do it here. It's probably more appropriate to do it here anyways.

I would wager to say that some, not all, of the Oblique projections are subsets of the Dimetric projection, and that at least one is a subset of the Trimetric projection.

## Premise

Descriptively, per Wikipedia's projection grid graphic, a Dimetric projection (2-angle congruency) contains a third ray that bisects the explementary angle defined by two other rays. This establishes a basis and defines the 3D space, albeit in a 2D viewing plane.

In the Dimetric example therein, the primary angle is $$150°$$, the explementary angle is $$210°$$, and the two bisected angles created from the explementary angle by the third ray are equally $$105°$$.

For both the Military and Cavalier examples, the primary angle is $$90°$$ (one of the basis planes is parallel to the viewing plane, with defining rays diagonal (Military) or parallel/perpendicular (Cavalier)), the explementary angles are both $$270°$$, and the bisected angles are both $$135°$$.

The Top-Down example might not appear to be Dimetric at first glance if you start by inspecting the bottommost $$90°$$ angle as the primary. However, it is indeed Dimetric if you consider the rightmost $$180°$$ angle as the primary. The explementary angle in this case is equal to the primary angle, $$180°$$, and when bisected results in two congruent, right, $$90°$$ angles.

Wikipedia also lists Cabinet projection as a Diagonal projection in their 3D Projection entry. The requirement given for this projection is that one of the faces of the displayed object, not necessarily a basis plane, must be parallel to the viewing plane. Besides that, it appears that the ray angles are all different, meaning that Cabinet projection can be considered a subset of Trimetric projection (0-angle congruency).

Hence, because the Oblique Cabinet projection resembles a Trimetric projection, and that the Oblique Military, Cavalier, and Top-Down projections resemble a Dimetric projection, I don't think we can claim that all Oblique projections are Trimetric (or Dimetric for that matter).

## Inflection

I think the crucial choice in distinguishing conventional Oblique projections from Axonometric ones is that for an Oblique Dimetric projection at least one of the angles between the basis vectors is an integer multiple of $$90°$$ / $$π/4$$: at least $$90°$$ / $$π/4$$ and at most $$2 \cdot 90°$$ or $$180°$$ / $$2 \cdot π/4$$ or $$\pi/2$$. Axonometric Dimetric projections can have a primary angle of any value (seemingly between $$\pi/4$$ and $$\pi/2$$), where the explement is still bisected to form 2 remaining congruent angles.

For Military and Cavalier projections, the primary angle choice is right, or $$π/4$$. For Top-Down, the choice is straight, or $$π/2$$.

We could theoretically conceive of other projections where the primary angle choice is acute, $$<π/4$$, or reflex, $$>π/2$$. It doesn't appear that many sources online engage with these forms of projections, probably because they screw up shapes so much as to make them appear inconsistent with reality and from an M. C. Escher-type work, or because they screw up mathematics.

This discussion doesn't even touch on scaling of the basis vectors aside from orientation/direction, so I'm sure you could add further classifications: one class for three congruent* basis vectors, one class for two congruent basis vectors and 1 of different length, and one class for three basis vectors of different lengths.

*: In the sense of congruent line segments as described in Hilbert's first Congruence axiom. I can't seem to find a term that describes vectors of equal magnitude. Parallel and antiparallel seem to describe vectors in the same or opposite direction of each other, while collinear describes vectors lying along the same line in space, necessarily being parallel or antiparallel. Congruence in other contexts may mean equality of all components of a larger abstract object, referring to magnitudes and angles, like how equilateral refers to equal magnitudes of segments constituting a shape, but not of the segment magnitudes themselves. Equi-magnitudinal could serve as the substitute, like how some sources offer co-directional as a substitute for anti/parallel, but this likely warrants another post on English or Math Stack Exchange.