Revisiting the Mathematical Model for Determining Coordinates of Points in a Trimetric Projection

In a recent article in this journal, Nikolić et al. presented a mathematical model for computing the coordinates in the trimetric projection of a 3D object, in terms of the projections of unit areas on the three dihedral planes. We revisit this model and analytically formalize its geometric principles, noting that such projections amount to the direction cosines of the unit normal to the viewplane. Thus, we reinterpret their proposal as providing three direction numbers that define a scaled version of this unit normal. The model also derives formulas relating the trimetric parameters (i.e., trimetric angles and foreshortening ratios). We observe that these relationships, found in classical literature, admit more compact expressions through simpler derivations. Also, we compile and reexamine various methods for selecting the trimetric projection, making them more accessible. In particular, the turn and tilt rotations of interactive user interfaces provide an intuitive way to choose the direction cosines. Ultimately, any method defines the mathematical model through a transformation matrix that maps 3D world coordinates to viewplane coordinates in the viewing pipeline.