Step 1: Design It!
I designed the knife sharpener, first using pencil and paper, then in CAD using Autodesk's free 123D. This design is made so that it can be used horizontally or vertically. When it is used horizontally, the mug lays partially on its side, but the stand holds it 20 degrees from horizontal. This means that the knife stays vertical while you sharpen it. When it is used vertically, the mug sits up side down on its rim and the knife is angled at 20 degrees. Both work well so it's a matter of user preference. I used calipers to measure lots of mugs and found that most common mugs have a diameter of 3.2" plus or minus about 0.05". I sized my design to fit the larger of this range and I used a small amount of electrical tape to take up the slack for slightly smaller mugs.
Step 2: Print It Out!
Step 3: Test It
After breaking off all the support material, I tried it out on one of my knives. It held onto the mug snugly, but had a few problems. The slots for the knife were too skinny so the knife couldn't slide back and forth smoothly. Also, in the horizontal position the base was a bit too small to keep everything stable. I decided to fix those parts in the design and also move to a higher resolution printer.
Step 4: Version 2
I modified the model with some updates and printed using the Objet Connex 500, a much higher resolution printer. The results were great! Try printing your own and let me know how it goes! Feel free to modify, improve and add on to it. It's also on Thingiverse!