Rotary engraving reference
Tapered cup arc projection
A chuck rotary turns degrees, not millimeters. Your software converts millimeters to rotation through the one circumference you enter. So a logo on a narrower band of a tapered or stepped tumbler needs a larger graphic to sweep the right angle. Set up your cup below and read off the graphic size.
Graphic to draw
0mm
Logo band Ø / circ.
0mm
Rotation swept
0°
Scale factor
0×
Cup geometry
Distances run from the mouth (top) down to the base.
Reverse check — you found a frame that works
If a fudged box already framed the logo, back-solve the diameter of the band it lives on.
Why this happens
The controller only knows one circumference — the number you typed in. It rotates 360° per driven circumference of graphic height. Two features the same number of degrees apart cover different physical distances depending on their radius. On a smaller-diameter band the same degrees cover less arc, so a true-size graphic under-rotates and the frame comes up short.
Why it also shifts
Same cause. Every position along the rotary axis is converted with that same driven number, so on the narrower band the whole layout compresses toward the origin by the ratio (logo circ. ÷ driven circ.). If your origin is left-justified, the art scoots left. The size error and the offset are one bug wearing two hats.
The clean fix
The fudge box works, but it patches calibrating to the wrong surface. Enter the circumference of the band the logo actually sits on and a true-size graphic frames and positions correctly. The rule for tapered or stepped cups: calibrate to the band you are engraving, not the widest point. A chuck rotary can only be dialed in on one band at a time.