 # Common 3-dimensional coordinate system for assessment of directional changes

We congratulate Ruellas et al for their innovative concept of a common coordinate system for cone-beam computed tomography orientation, which has direct implications on cross-sectional and longitudinal studies. However, one core point of this article’s methodology demands further discussion.

This article concludes that 3D distances are not affected by head orientation, and the amount of directional change in each plane of 3D space is strongly influenced by head orientation. This proves the fact that scalars are invariant under coordinate transformation, but vectors are not. In this case, the 3D distances are Euclidean distances between 2 landmarks, which is a scalar (no directional property), and the x-, y- and z-axes components of 3D distances are vectors (with directional properties).

The results of this study showed insignificant changes in 3D distances between before and after orientations of mandibular width (Dist RCo-LCo) (B.or − A.or = 0.01 mm) and mandibular length (Dist RCo-Pog) (B.or − A.or = 0.25 mm). Similarly, between before and after orientations, maxillary changes (T2-T1) and mandibular changes (T2-T1) were 0.07 and 0.03 mm, respectively.

It is pertinent to mention here that the Euclidean distances in 3 dimensions are defined by the following equation.

3 D Euclidean distance ( D P Q ) = ( x 1 − x 2 ) 2 + ( y 1 − y 2 ) 2 + ( z 1 − z 2 ) 2

where P ( x 1 , y 1 , z 1 ) and Q ( x 2 , y 2 , z 2 ) are the coordinates of 2 landmarks.

The x (x 2 − x 1 ), y (y 2 − y 1 ), and z (z 2 − z 1 ) components of 3D distance between any 2 fixed landmarks will change due to translational and rotational coordinate system transformation. However, the 3D Euclidean distances resulting from equation 1 are invariant.

3 D Euclidean distance = D P Q = D P Q ′ = ( x 1 ′ − x 2 ′ ) 2 + ( y 1 ′ − y 2 ′ ) 2 + ( z 1 ′ − z 2 ′ ) 2