I just had a fun project at work that arrived just in time to leaven my growing boredom.
Bunch of framed awards to be hung on a wall. All different sizes, and we realized a grid layout would look bad. Fortunately we have visual artists and graphic designers on payroll, and one of them spent a morning laying out a workable pattern on the floor.
So our department is given the job of translating that to the actual wall. First step was obvious; after taking photographs and jotting down a quick reference sketch, I plotted cartesian coordinates for every single frame (they were laid out in a corner of the room, meaning I had two walls to use as baseline coordinates).
Of course they needed to fall eight feet above the floor, and start two and a half inches from one edge of the surface they were going on. There was a steel framing member at some arbitrary height above that line, so I measured the difference. Also, the given edge was the far edge, meaning I had to plot from the opposite side of the wall to get the last frame to arrive at exactly 2.5" short. So that gave me two offsets, which I could add to my coordinates to arrive at the actual point.
Since the point chosen was the upper right corner of each frame, two additional numbers were needed, the distance from the upper edge to the hook from which it would hang, and half the width (because the hook needed to be at the center of the frame but the reference was taken from one edge).
And then a new requirement was added. My plotting data showed there would be an approximately 14" gap on the far side of the wall (the original zero edge). This was not acceptable; we had to expand the horizontal coverage. I did this by dividing the difference by the average number of spaces between frames, and came up with a value of close to one and one half inch; a fourth offset that increased in steps across the pattern.
I was working on a legal pad, and for each frame, now, I had to arithmetically sum four different offsets (plus and minus) to each original set of cartesian coordinates.
And as a last problem, the width of the wall was too much to hold a tape to (also, I began layout from the top and my ability to place a tape was progressively blocked by the previously hung material). Thus, many of the final coordinates were arrived at in stepping-stone fashion; 71" to the bottom of a previous frame, then the difference between that and the target of (107" reference + 14.5" offset + 3(1.5") additional horizontal spacing - (11.25"/2) for center.
And, yeah. The last rows needed tweaking to compensate for all the compounding effects of applying an arbitrary and arbitrarily quantized expansion value to a tightly-ordered pattern. Plus there were some architectural elements in the way. But this is where the real art is; knowing not just a mechanical method to translate and plot this kind of data, but understanding the artistic intent well enough to be able to achieve even when problems develop.
So in short, for the majority of the objects I plotted them geometrically -- then I eyeballed them and adjusted if it was necessary. And this is a good basic habit; measure twice if you like, but eyeball once as a sanity check before you cut.
The last show I painted -- I think it may have been the last musical I worked -- was similar sorts of drafting and art. The original design -- New York cityscape and skyline in perspective -- was provided via watercolor renders.
These original images were not square, not scaled accurately to the space, and the perspective was laid in by eye. This latter was the biggest headache; we had to try to achieve the loose look of the original render but we were working in a harder-edged, less forgiving medium. Which meant gross differences in vanishing point stood out, and verticals needed to be made plumb.
The basic method was the same; create an arbitrary reference datum, and plot the cartesian coordinates of each corner or other necessary reference. Then go to the physical set (the wings, drops, and cyclorama) and measure those points. Lastly, strike a line between plotted points, reconstructing the edges of the original artwork.
This is something theater people do all the time. Second nature, basically. Every set wall that is installed is plotted the same way, usually from plaster line and centerline as datum.
Trouble is, a hard cyc or a proscenium is big. And it can't be laid flat on the floor of the paint shop for easier access. When you want to plot two points sixty feet apart, with one of them thirty feet off the ground and the other twelve, it is quite time consuming to get the tape measures there.
So we derive again. The cyc was constructed from 4x8 sheets, meaning we could could add four foot multiples until we came to a fraction, at which point we measure that. Or we could plot from a previously plotted point; because even stretching a tape seven feet is difficult when you are at the top of an extension ladder.
But there's where all the adjustments start to come in. By the time two or three lines have been plotted, there's a number of competing aspects to juggle. The actual plotted value. What looks good in the air. What it looked like in the original render. What is now possible, considering that the lines your line is intersecting with have also moved.
And all of this in a hurry, of course. Whilst balancing tape and chalk and rendering and scrawled estimates on the top of a ladder, stretching out on tip-toes to try to place a point or position a ruling stick or the end of a snap-line.
We didn't always get it right, not that time.
There is a similar exercise in machining. A lathe or mill is designed to move extremely smoothly along calibrated tracks. Which is to say; you can tell to within ten-thousandths of an inch where you are...in respect to a starting point. The fun part of the exercise is establishing that starting point.
A typical method on the lathe is to "touch off"; to bring a tool to where it is just barely pressing against the working face. Now you know where the face is; you can chose to zero the DRO or other guides to that point. For the mill, a more complicated method is to use an edge finder. In this case, the tool indicates when the outer diameter has touched the selected surface, which means the center of the mill is located half that diameter away.
That is of course only where it starts. Often as not you can't reach the reference face you really want with any of these methods, so you have to construct a path to it using offsets. For the Aliens M40 grenades, for instance, the grenade is inside the stock; zero is set arbitrarily at the first flat face achieved, and all the rest of the grenade is projected from that point.
And, yes, drawing in perspective is still more of the same. Heck, even rendering in 3d is the same; in that later case, you set up the setting, the on-set lighting elements, the poses...then you tweak, changing poses to make them read better from the chosen camera angle, cheating objects to remove tangents or improve visibility, and every now and then, actually changing scale or position to make what looks right to the eye (instead of what is mathematically right in the world of ray tracing).
If you do these tweaks from camera, you can find yourself very surprised by what the actual 3d scene looks like when you step back and look at it from another view!
In fine art and the "less than fine" of comic book art the grid method is paramount. As it is for theater people for subjects that are other than monumental and rectilinear. You put a grid over the reference, and one (appropriately scaled) over the blank paper (or the piece of plywood you mean to run through a bandsaw to make some bit of theatrical scenery or prop).
(These days, of course, printers being being so cheap, for small props it makes sense for a lot of people to print out a reference drawing at scale and paste it on to the wood).
Then you go to the grid and eyeball; three squares down and near the middle, ten squares over and just over the line.
Somewhat similar to plotting the contents of a page in perspective; you make a grid, then you eyeball the lines you draw to adhere to what that grid is advising.
The trick is, as always, that the estimates points, the plotted points, the eyeballed curves will never quite be right. The real art is coming back over the (lightly-pencilled!) transfer and applying straight-edges where appropriate, making the curves flow appropriately, and otherwise adjusting until it looks right.