We were working on the shihonage lead the other night and found a point that needed clarifying which really helped the students and you may find helpful as well.
Ude Mawashi Undo is an elliptical motion of the wrist (representing the likely point of connection with uke) traveling in a plane that is as closely parallel to nage's chest as possible. The path of the wrist should never cross the center line if the body or travel such that it goes behind the body (plane of the ellipse should be parallel).
Couple of notes... First the hips define the plane of the body. Good posture dictates the torso is aligned with the hips and the hips are carried on the legs. For the purposes of discussion, imagine that the line connecting the feet lies entirely on the plane of the body (your feet aim your hips).
If Ude Mwashi is in the plane of the body, defined by your hips, which are aimed by your feet, then if you want to aim Ude Mawashi, you need to move your feet. Turning at your waste or your shoulder breaks things. So, if you want to have shihonage lead pass right by uke's one-point, but not into uke or away from uke, you ideally want to mirror uke's body plane with yours. (by the way, you do want to do this, at least for irimi, ok, maybe a couple degrees skew into uke, especially if they were rude enough to use a knife. Manners are learned behaviors after all)
Ok, second point, the ellipse...Yes, circles are special types of ellipses. I can almost guarantee which one of our students reading this would point that out. Circles, however pretty and perfect, are kind of boring. You just go round and round and acceleration never changes (in magnitude, you know who you are). Enough of that. Assume for a second you did move your wrist in a circular path, it would have to cross the center line of your body (your sternum). Now the right hand is squarely (didn't see special rectangles coming, did you?) in the air space of the left hand and vice-versa. If you are doing both hands, they either have to crash into each other and explode or one or both of them have to leave the plane of the body to pass safely by the other. Now, it has to be one of those two options as we know you would never ever mess with the timing. I likely lost too many people when they went to remind themselves who Euclid was and got sucked into the web and are now watching cat videos on youtube, so we will not delve into violating 4th dimensional constraints.
So, we have established it must be an ellipse and that you don't care enough about cat videos to have jumped ship. Now, how eccentric is the ellipse? Ahh, we have branched into something for experimentation. The eccentricity of the elliptical path (how squished is the used-to-be-circle) is a variable. As you make the ellipse longer & shorter you will change the effect on uke's one-point. A really high, sharp ellipse will drop uke into a "hole" in front of you. A less sharp, fatter ellipse will drop them in a "hole" closer to their body.
Ok, get up and do Ude Mawashi, then go grab an uke and play. Yokomen attacks are great for this. Pay attention to the sharpness of the ellipse and what it does to uke. Did I mention we weren't playing with time yet? We have been describing the path of your hand/wrist, not the speed! The effect will work the same if you are working at a slow or fast pace. Have fun.