Assuming cartesian, affine, orthogonal coordinate systems wherein axes are perpendicular towards each other and usually parallel to the viewport edges, the planar 2D coordinate system (for specifying positions, linear paths and areal shapes) of
- horizontal X-axis (abscissa) for designating left and right positions constituting distances like widths, breadths or (without a negative direction) lengths
- vertical Y-axis (ordinate) for designating top, up or upper and bottom, down or lower positions constituting altitudes like (positive) heights and elevations or (negative) depths
- either the plan, ground, base or floor, i.e. what appears flat in top-view,
- or the canvas, face or facade, i.e. what is flat in front-view and
- rarely the side, wall or fence
of a spatial 3D coordinate system (for also specifying volumetric bodies). The positive directions conventionally follow the right hand rule with the thumb pointing towards X+, the index finger towards Y+ and the middle finger towards Z+.
In the 1st case, especially used with maps, the Z-axis becomes the vertical dimension, which is often positive only, and either the X-axis or the XY-plane is considered horizontal. In the former case, the Y-axis has no conventional designation matching the horizontal ∶ vertical pair, and it is only ever used when both of these axis are oriented the same as the drawing canvas, which explains somewhat why there is no third term. In the latter case, no designation is necessary.
Front-Back or Back-Front Z-Axis
Only in the 2nd case, there are formal alternative names available: the third axis is called an applicate in environments where the others are known as abscissa and ordinate. This means, an applicate always follows the virtual line of sight of a spectator ranging – according to the right hand rule – either from negative back to positive front, e.g. in most paper diagrams with the origin in the lower left, Y+ pointing up and X+ pointing right, or from negative (or zero) proximity to positive distance, e.g. on screens and all other planar media inherently organized in horizontal lines or rows and vertical columns running from top to bottom. One could derive neologisms from these terms, e.g. *abscissal ∶ ordinal ∶ *applical or *applicational.
Layout systems often use the 2nd convention, but only support a 2.5D space with multiple planes in distinct, indexed and possibly named layers that are stacked above one another in a canonical order with a thickness of zero. The equivalent to horizontal-vertical position for the pseudo-dimensional axis would be level with the derived neologism *levial.
It often makes sense to establish a relative, local coordinate system for any (moving) object in 3D space. Its trajectory (from–towards) or (if stale) bearing of its assumed line of sight, e.g. along a nose, determines the longitudinal axis. Its lateral axis is perpendicular to this, e.g. alongside wings, and together they span a plane that is often (approximately) parallel to the ground and which the third, normal axis is perpendicular to. This triple longitudinal ∶ lateral ∶ normal basically matches <wanted term> ∶ horizontal ∶ vertical, thus longitudinal could be argued for as a reasonable candidate for the third axis in absolute, global coordinate systems as well.
Looking at a related language, there are native terms for horizontal and vertical in German: waagerecht and senkrecht, respectively. They either originate in or at least align with the desired states of beam balance scales (Balkenwaage) and plummets (Senkblei, Senklot). Accordingly, Ænglish cognates would perhaps be *scaleright and *plumbright. There is no term for the third axis or orientation in German either, but analogous construction could yield something like *eyeright, *sightright, *shotright, *wayright or *pathright.
Since X-Y-Z axes are frequently colored red, green and blue, respectively (cf. RGB color space) in 3D applications, and vert means green in French as well as in English heraldic tinctures, one could form neologisms to match vertical (Y), i.e.: *rougical or *gulical (X, synonyms for horizontal) and *azurical (perhaps rather *azurial, *azureal, *azural) or *bleucal (Z).
Fortunately, most people do not have to deal with cylindrical, polar, equatorial, horizontal or other spherical and yet more exotic coordinate systems on a regular basis. Those require terms like pole, horizon and equator, longitude and latitude, ascension and declension, azimuth, celestial, ecliptic and galactic, zenith and nadir.