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>> Rotation and rolling. So an important type of cyclic motion is a rolling. And rolling
combines rotation and motion, so motion, object moving from one place to another. Something
like a Ferris wheel simply has rotation in place. Now one concept which I think is helpful
to picture is that rolling is very similar to tipping over so when we have something
tipping over, we have points which it's tipping around and the farther we are from that point
on the object, the faster the motion. It's the same thing with the rolling, but with
rolling that point where the tipping motion takes place keeps shifting but momentarily
it is stationary as the rest of the object turns around it. So another way to picture
rolling, which is closely related is that rolling is a combination that is equal parts
sliding motion and spinning rotation. So if we had just pure sliding then we would have
all points of the ball moving together at the same speed. On the other hand if the ball
was spinning in place then the center would be stationary, the top would say be moving
clockwise and the bottom counter-clockwise. Now with rolling it's a combination of equal
parts sliding and spinning so the bottom is momentarily stationary, the middle is moving
at a certain speed, and the top rim is moving at twice the speed of the middle. We see this
in this example of motion blur when a bicycle moves past you. So here the camera's fixed
and the bicycle is moving and you notice that there is very little motion blur on the lower
part compared to the upper part. On the other hand, if the camera was tracking the motion
of the bike then, we would just be seeing the spinning of the wheel and in that case
the outer rim is travelling faster than points near the center for pure spinning. In that
case, the outer rim would be more blurred than the inner part. Maybe this picture helps
to clarify what the motion is like so the stationary camera actually sees rolling whereas
the moving camera, say tracking the wheel, the motion of the moving camera now we see
the spinning of the wheel instead of the rolling of the wheel. I wanted to emphasize this because
motion often looks different depending on what the camera is doing and how the camera
is moving. Now as an object rolls there is a close connection between the distance that
it travels and the rotation. So a rolling ball will go a distance that is slightly greater
than three diameters when it makes one revolution, slightly more being a pi. So we see that,
in this example, the ball goes around a little more then three diameters and then returns
to its original orientation. Now if the ball is turning and that distance is not synchronized
with the rotation then we don't actually have rolling, this would be a slipping so the ball,
the distance that it travels is disconnected from the amount that it?s turning. So when
that is animated, it looks very different whether it's rolling or slipping. In actual
bowling you see both those situations. Slipping is common when the ball is thrown very hard,
very fast but then when it slows down it tends to roll. Now obviously there's a connection
between the rotation and the speed of the motion so the speed of, say a rolling wheel,
depends on the size of the wheel, and the rate at which it's rotating. That the size
of the wheel would indicate the amplitude and the rate of rotation would be the frequency
of the cyclic motion. This table shows you the speed in miles per hour, for automobile
tires, which are about two feet in diameter, depending on how much they rotate with each
frame. So if the tires makes a half turn in one frame, then the car is travelling about
50 miles an hour. So when cars are animated, it requires that the speed that the car is
moving is consistent with the rotation of the wheels, we see this a few frames from
Madagascar 3, in this point, the car is moving fairly slowly, so just tracking some marks
on the front wheel. You see it's only turning by a small amount but that's consistent with
the speed the car is going and the distance that it travels from frame to frame. In CG
animation, this pairing is usually done automatically, so it's somewhat more of a challenge for say
hand drawn animation. So since we're talking about rotating wheels I should mention this
common visual illusion that you may be seeing in westerns or even for automobile wheels
and that is that sometimes you see a vehicle that is moving in one direction but the rotation
of the wheel seems to be opposite from what you would expect, given the direction of the
motion. So in this case, the wheel seems to be turning in reverse, this is called the
wagon wheel illusion, it's often seen when you see a stage coach in a western. Here's
a video showing that with a propeller, so you see in this case, the propeller starts
turning and the propeller is actually going to be turning faster and faster and faster
but at this point it looks like it's going in reverse, slows down, then it's travelling
and now it's in reverse, you can somewhat see from the motion blur that indeed its going
quite fast. Now what happens here, this wagon wheel illusion is an example of what's called
the Nyquist effect, so let's see how this works, let's say that the...we have these
two images of the wheel on frames 1 and 2, and that the actual rotation is counter-clockwise,
but the illusion is that the wheel is turning clockwise. Now what happens is let's think
about this spoke here near in the center, the actual rotation the spoke from frame one
to frame two moves over to here, however the illusion is that the spoke has actually moved
backwards and we think we're seeing it just to the right of where it was but in reality
what we're thinking is that spoke is actually another spoke which has rotated to that spot.
So it's this confusion when we can't tell one spoke from another, the brain automatically
assumes that the new position is from the closest previous position. So like I say the
brain tracks the movement of the spokes and by perception of motion, it assumes that it
is seeing the nearest location on each frame. Now of course this means that if you want
to avoid this illusion, it limits how fast a wheel can spin. This is actually something
that showed up when this scene in Madagascar 3 was first shown in daily's, it appeared
as if the penguin was rotating backwards as he was moving so it was just a wagon wheel
illusion with the rotation of the penguin, the animator of course recognized that and
it was corrected to avoid that distracting visual effect in the final version of the
film. So in summary, rolling is a synchronization of rotating motion and motion from place to
place, a rolling wheel travels slightly more than three diameters with each turn, if the
rotation is not synchronized with the motion, with the distance that the wheel travels then
the visual will not look as if the wheel is rolling, instead it will appear to be sliding
with free spinning. The speed, at which the wheel is travelling say in miles per hour
is set by the radius of the wheel and the rotation rate. And then finally there is this
wagon wheel illusion where the wheel seems to be turning in reverse, and this happens
when the rotation rate is high and as we saw it's an example of the Nyquist effect. So
rolling is an important type of cyclic motion with rotation. In some of the other tutorials
we'll see other types of rotating motion such as tumbling and spinning. So I'll see you
then.