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O.K. - lets get started. The first thing I did was create the piston, crank, and wheel pieces in Illustrator - each piece on a separate layer. Then I imported this into After Effects as a comp. Then I opened the comp and using the pan-behind tool, I adjusted the anchor points: For the wheel I moved the anchor point to the center; For the crank I moved the anchor point to where the crank attaches to the wheel; For the piston I moved the anchor point to where the piston attaches to the crank.
The concept for the animation is pretty simple - we make the wheel the parent of the crank and the crank the parent of the piston. Then as the wheel rotates, we need to rotate the crank around its new anchor point in the opposite direction by the same amount. This will keep the crank horizontal, which is not exactly what we want. So in addition, we need to add in an adjustment factor that will keep the crank/piston junction level with the center of the wheel. Then we will need to rotate the piston by an amount opposite of the crank adjustment to keep it level. The following pictures will illustrate this:
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Starting Position |
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Wheel rotates, no rotation of crank and piston |
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Crank rotates amount opposite to wheel rotation |
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After crank adjustment |
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After piston adjustment |
Pretty simple, right?
Now we want to set up the parenting. Since the wheel is going to be driving the animation, we will set it up as the parent of the crank, which will be the parent of the piston. This part is interesting, so watch closely! Notice the position values for the crank and the piston before the parenting.
Now look at the position values for the crank and piston after we assign the parents.
What the heck just happened? Well, it turns out that the position of a layer is in "comp space" unless the layer has a parent. In that case the position value is in the "layer space" of the PARENT. That means that the child's position is relative to the upper left-hand corner of the parent layer. Why do we care about this? We're going to be calculating the radius of the wheel and the length of the crank so we need to keep this in mind. If we assumed that positions of the layers were relative to the comp (which is normally the case) our calculations will be completely hosed after we do the parenting.
O.K., so what do we do about this? The expression that we're going to develop for the rotation of the crank needs to know the radius of the wheel and the length of the crank. Normally (with no parenting involved) this would be pretty simple. Since the anchor point of the crank is on the edge of the wheel, the formula for the radius of the wheel would be:
r=length(this_comp.layer("wheel").position,position);
And since the anchor point of the crank is at one end of the crank and the anchor point of the piston is at the other end of the crank, we would expect the formula for the length of the crank to be:
l=length(position,this_comp.layer("piston").position);
But as we now know, these formulas won't work. It turns out that we have several options to solve this problem. Key points to remember here: The wheel has no parent so its position is in comp space. Each anchor point is in that layer's own layer space. Positions of children layers are in the layer space of the parent. So the key to making this work is that when you calculate a distance you just need to make sure that both points are in the same space. It turns out that After Effects provides a couple of useful functions to assist us. These are part of the "layer space transform methods". The two that could be useful to us are "to_comp" (which converts from layer space to comp space) and "from_comp" (which - you guessed it - converts from comp space to layer space).
O.K., on to the radius of the wheel. Since we're doing this calculation in the crank layer, we have available the crank's position (which is in the wheel's layer space), the crank's anchor point (which is in the crank's layer space), the wheel's position (which is in comp space) and the wheel's anchor point (which is in the wheel's layer space) to work with. So any of these formulas should work:
r=length(this_comp.layer("wheel").anchor_point,position);
r=length(from_comp(this_comp.layer("wheel").position),anchor_point);
r=length(this_comp.layer("wheel").position,to_comp(anchor_point));
We'll use the first one. This works because the crank's position and the wheel's anchor point are both already in the wheel's layer space. Get it? Alrighty then!
Now for the length of the crank. Since this formula will be used in the crank layer, we have the crank's position (which is in the wheel's layer space), the crank's anchor point (which is in the crank's layer space), the piston's position (which is in the crank's layer space) and the piston's anchor point (which is in the piston's layer space). Because both elements are already in the crank's layer space, the obvious choice in this case is:
l=length(anchor_point,this_comp.layer("piston").position);
So it turns out we didn't need the "layer space transform methods" after all, but it's nice to know about them. If, for example, we needed to create an expression in the piston layer that calculated the distance to the wheel we would need to use one of these transforms.
O.K. math-o-phobes I apologize, but it's going to get a little bumpy for a while. To make this all work, we need to use a little trigonometry (you all remember that from school, right?) For those of you still with me, we'll slog through this as quickly as possible. The following diagram shows the relationships we need to deal with:
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where:
A = the angle of rotation of the wheel
B = adjustment angle for the crank
r = radius of the wheel
c = length of the crank
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from our trig class we know:
r * sin(A) = d
and
c * sin(B) = d
so
sin(B) = r * sin(A) / c
or,
B = arc sin(r*sin(A) / c)
O.K. - just one more math thingy and we'll be done. For sin and arc sin we're going to be using the JavaScript functions "Math.sin" and "Math.asin". These functions want their angles in radians, not degrees. Since After Effects expresses rotation angles in degrees, not radians, we'll have to do a conversion. Fortunately, this isn't too tough. JavaScript kindly offers us the value of pi as "Math.PI". Since there are 180 degrees in pi radians we can make the following two conversion factors:
rad2deg = 180 / Math.PI; //radians to degrees
deg2rad = Math.PI / 180; //degrees to radians
That's it! Now we have all the pieces necessary to create the expressions for the rotation of the crank and the piston. For the crank we just add the negative of the wheel rotation to the adjustment angle.
rad2deg=180/Math.PI;
deg2rad=Math.PI/180;
r=length(this_comp.layer("wheel").anchor_point,position);
l=length(anchor_point,this_comp.layer("piston").position);
-this_comp.layer("wheel").rotation+
Math.asin(r*Math.sin(this_comp.layer("wheel").rotation*deg2rad)/l)*rad2deg
For the piston's rotation we just use the negative of the adjustment angle.
-1*(this_comp.layer("wheel").rotation+this_comp.layer("crank").rotation)
Here's a screen shot with the complete formulas in place:
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