3D: scale, shear and rotate settings

Question

Hi.  I am relatively new to Illustrator.
I am trying to create a simple 3D map, but not with the default iso settings.  It's actually a remake of a map made many years ago on a squar grid using a rise of 1 and a run of 2 (all done by hand on graph paper).  This creates an angle that is 26.5651°.
For iso, [scale: H=100%, V=86.602%] [shear:  hoizontal axis -30°]  [rotate: -30°] and so on, varying for the left, right and top.  I have this working fine for iso.
My problem arose when I went to the 26.5651 angle.  What settings should I use for the three sides (left, right, top)?
Is there any way to get the extrude and bevel tool to do work with my settings?
Thanks

Answer 1

I am trying to create a simple 3D map, but not with the default iso settings....made many years ago on a squar grid using a rise of 1 and a run of 2...
You are confusing actual orthographic projection from an actual 3D model (3D Effect), with an archaic 2D "cheat" that is really not a correctly proportioned orthographic projection at all.
This creates an angle that is 26.5651°.
Yes, but more significantly it is simply the projected angle that results from vertically scaling a true-length 45° line by an arbitrary 50% (the 1:2 ratio). In orthographic projection, that amount of foreshortening would mean that you are viewing the top plane of your coordinate system as if it is tilted upward toward you 30°[arcsine (.5 )= 30°], not the isometric angle of 35°16".
For iso, [scale: H=100%, V=86.602%] [shear: hoizontal axis -30°] [rotate: -30°]
Contrary to the misconception perpetuated by many of those who use the 1:2 cheat, that is not isometric. It's just an arbitrary 2D oblique. By definition, in isometric (read "same measure") drawing, all three axes are of the same length. Using the 1:2 ratio, the length of your left and right axes (the hypoteneuse of a 1x2 rectangle) measures 2.2361. If your vertical axis measures .86602, then obviously your three axes are not of equal length, and the drawing is not isometric. The corresponding lengths/angles of the axes in axonometric (isometric, dimetric, trimetric) drawing are not arbitrary. For any given set of axis angles, there is but one correct set of forshortened lengths.
So from where are you getting the 86.602% figure? I'll tell you: That's the cosine of 30°. Now, if the 1:2 ratio did, in fact, result in a vertical axis of that measure, then the 1:2 cheat would result in a proper dimetric (not isometric) projection. That is, if the top of a cube is tilted upward from your line of sight by 30°, then the front of that cube is necessarily tilted away from perpendicular to your line of sight by 30°. The top would be foreshortened by a factor of .5, and the front would be foreshortened by a factor of .866.
But that is not the vertical measure that the 1:2 cheat gives you. Xara Designer Pro, for example, has the fake "isometric" 1:2 grid you are talking about built in as a standard feature:
Note that the vertical axis measure of the 1:2 ratio is not .866. It's .7071. (Otherwise, it wouldn't fit the square grid which is the whole point of the 1:2 ratio cheat.)
Inkscale, on the other hand, provides an actual isometric grid feature:
Note that all three axes are of the same length, and are separated by 120°.
Here's a properly proportioned dimetric cube, using your desired 1:2 aspect-ratio 26.57° axis angles (manually drawn in Illustrator, but could be drawn in any of the three programs mentioned):
Note that the vertical axis is .866 (cosine of 30°).
Which of the three boxes above look like a properly proportined cube to you? Note how the first screenshot looks short and squat. It's simply out of proportion. That's because it's not an orthographic projection of a cube. The other two are.
Isometric:
Left and right axes angled 30° from horizontal. (or more correctly, 120° from the vertical axis).
Unit cube is rotated 45° and tilted upward toward you 35°16".
All three axes foreshortened length: .8165
[One instance of] dimetric:
Left and right axes angled your desired 26.57° from horizontal
Unit cube is rotated 45° and tilted upward toward you 30°
Left and right axes foreshortened length: .7906
Vertical axis foreshortened length: .866
1:2 cheat:
Left and right axes angled your desired 26.57° from horizontal
Unit cube is rotated 45° and tilted upward toward you 30°
Left and right axes foreshortened length: .7906
Vertical axis foreshortened length: .71 (incorrect value for orthographic projection)
So the fact is, the 1:2 fake "isometric" cheat yields neither an isometric nor a dimetric drawing. It just yields a distorted 2D oblique. But again, 3D Effect is an actual orthographic projection generated by 3D geometry. Thus my initial comment that you are confusing a mere geometrically incorrect 2D cheat with real orthographic projection. You're trying to apply a set of arbitrary geometric values to a system that is not arbitrary in its geometric calculations.
My problem arose when I went to the 26.5651 angle....Is there any way to get the extrude and bevel tool to do work with my settings?
So can you get 3D Effect to replicate the disproportionate scaling of your 1:2 cheat? Sure. You just have to use settings which cause 3D Effect to make the same proportional error, thereby extruding a square to a squat box, rather than to a cube. (Albeit you will get into other proportional errors when you map artwork to the extruded sides.)
Like everything else in Illustrator, the extrusion depth setting in 3D Effect uses points as its basic unit of measure. So to extrude a 1" square into a cube, you use an extrusion depth of 72 points (1").
Because of the cumbersome way the rotation fields are implemented in the 3D Effect modal dialog, it's much simpler to perform the initial 45° rotation of the base art before applying the effect. That way, you only have to mess with one of the rotation fields (i.e.; numerically specify a simple rotation, rather than a compound rotation).
Now do the same thing again, but just distort the extrusion depth by the same proportion of a correct dimetric vertical axis length to that of the 1:2 cheat:
The routine is not limited to cubes. Draw the top face of any shape(s), rotate 45°, and apply the same 3D Effect settings. This should work for your map, to match things previously drawn using the 1:2 cheat:
JET