This instructable discusses how to manipulate artwork in Illustrator for use with a vinyl cutter for glass etching. It also applies to any 2D artwork you might want to wrap around a conical glass. You will need a calculator with trigonometric functions, or you can use this calculation page I made.
The last step shows numbers you’ll want to use, but you can also scroll through for derivation (especially helpful in case I got something wrong).
Step 1: What Do We Want?
Put more pointedly: what are the critical dimensions of the cup, and what are the critical dimensions in illustrator? And, how can we go from the first set of measurements to the second?
Step 2: Converting Glass Parameters to Illustrator Parameters
We have three parameters for the conical glasses - top and bottom diameters (d2, d1 respectively), and the distance along the glass between these measurements (h). Please only measure for conical sections.
We want the parameters necessary to make an arc - inside and outside radii (r1 and r2 for inverted cone glasses), and the angle this arc subtends (theta).
We have three unknowns and we can write three independent and simultaneous equations, and because we are very good at math, we can massage these equations into forms that let us directly calculate r1, r2, and theta.
In illustrator, the window for an arc mesh lets you choose between -100% and 100% Bend, where 100% represents a full 180 degree arc. We can multiply the theta by 100/pi to get Bend %. If you did your calculations using degrees, multiply by 5/9.
We still need to set up the rectangle of artwork. Even if your artwork does not wrap around the glass, you need your artwork to be inside a rectangle that would wrap your entire glass. Then, once everything has been distorted, you can choose how much to cut out for your mask. Because you measured the height along the glass, that height should be the height of your rectangle (which becomes distance between the two arcs in Illustrator). The width of your rectangle should become the smaller circumference when arc-ed.
In Illustrator, the smaller arc shares endpoints with edge of the rectangle (lower edge for a positive bend radius, upper edge for a negative bend radius.
Please note that this image is for a conical glass whose top is larger than the bottom. The *smaller radius* should be used for this calculation (r1 for a glass those top radius is smaller than the bottom radius). We can draw a close up image of the arc image to understand r1 relates to the width of the artwork rectangle. Then, thanks to the law of sines, we can calculate the width.
Now we have all parameters necessary for Illustrator! Last image is a summary of these values.
Step 3: A Note About Text
1. You can create text in your rectangular design, then distort it with the arc mesh with all other components.
2. You can create baselines for the text, distort those, then write using the 'type on a path' tool
The choice is entirely yours, and the difference is often small. Here are two considerations for which method to choose:
a. If your text is very tall, then method 1 will distort it a lot. If your text is short, then method 1 won't look much different from method 2.
b. Method 2 does not work well for blocks of text, which would need separate baselines for each line in the block
if you have other thoughts on handling text, please write them in the comments!
Step 4: Summary
To distort images for wrapping around a conical glass...
a. Measure your glass
b. Calculate necessary variables shown in pictures (or use the calculation website I made)
c. Set up your artwork rectangle according to these dimensions
d. Once your artwork is finished, use Object-->Envelope Distort --> Make with Warp and choose 'Arc' from the dropdown menu. Use the previously calculated bend percentage.
Note: If you want to use Effect --> Warp --> Arc, you must group objects together beforehand or it will warp them individually. Envelope Distort and Warp are not the same thing, though they will give you similar artwork.
e. You will probably still need to use Object --> Expand Appearance to get the lines how you want if you are using a vinyl cutter.
Please note that the masks may still need a little finagling, especially if they need to be very precise. Illustrator only allows for integer bend percentages (which is more than a degree per percentage), and most measurements are not perfect. You can see from the last photos that despite caliper prowess, the values were not perfectly right. For most small masks, you don't need to alter your artwork. For anything wrapping around the glass, or especially a larger design, this is a good method for getting the artwork to wrap well.
Best of luck with your project! Please leave comments below!