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by David W. Henderson and Daina Taimina
The four bar linkage is a mechanism that lies in a plane (see C01) (or spherical surface, C03) and consists of four bars connected by joints that allow rotation only in the plane (or sphere) of the mechanism. We will add this red color to indicate the slight changes that are necessary when considering spherical linkages instead of planar linkages.
In normal practice one of the links is fixed so that it does not move. In the linkage to the right we assume that the link OC is fixed and investigate the possibilities of motion for the other three links. We suggest that the reader play with the simulation by changing the length of links h and b and experiencing the motions that are possible.
We call the link OA the input crank and link CB the output crank. Similarly, we call the angle θ the input angle and angle the output angle.
On a sphere we measure the lengths of the edges by the angle (in radian measure) that the side subtends from the center of the sphere. We assume here that all links have lengths less than = 180 degrees (that is, we assume that links have lengths less than half a great circle) This is not a serious restriction because a link of length l > can always be replaced by a link of length  l that together with the original link completes a great circle. However, below we sometimes will need to consider what happens when two links are collinear (for example, when AOC are in a line) then the sum (in the example, a+g) of the lengths of the two links may be l > and then the distance between the end points (in the example, A and C) will not be l but will rather be the length of the shorter great circle arc, l . [We suggest that you try this out by drawing great circle arcs on an orange or tennis ball.] For this reason, when we write l_{S} we will mean the distance between the end points.
The reader may experiment with
the above crankcrank simulation
and discover that the input crank will be able to swing opposite C (when
the input angle is π =
180 degrees) only if the distance between C and A is allowed to
be a+g_{S}
and this will only be allowed by the mechanism if a+g_{S} < b+h. See also movie of a crankcrank
linkage. There will be a maximum input angle if a+g_{S}
> b+h. (If a+g_{S}
= b+h then the link folds, that is, it has a configuration in which
all four links lie in the same straight line.) Experience this in the rockercrank simulation
and the movie.
By applying the Law of Cosines to this configuration (when ABC is a straight line) we see that this maximum angle satisfies:

. 

Or using the spherical Law of Cosines:
.
Similarly, there will be a minimum input angle whenever a+bh > g (if a<g) or g+bh > a (if a>g) which are equivalent to bh > ga. The minimum angle is actualized when either BAC is straight or ACB is straight, as in the figure.
In both of these cases we can use the Law of Cosines to conclude:

. 

.
Thus we have four types of input cranks:
The analysis of the output crank is exactly symmetric to the above with the lengths a and b interchanged. In particular:
If b+g_{S} > a+h, then there is a maximum output angle _{max} and, by applying the Law of Cosines to this configuration (when BAO is a straight line) we see that this maximum angle satisfies:

. 

If ah > gb, there is a minimum output angle _{min} and, by applying the Law of Cosinges (when either ABO is straight or BOA is straight) we see that this minimum angle satisfies:
.
Thus we have four types of output cranks:
Putting these together we get eight types of 4bar linkages:
You can check that the other eight combinations (a 0 or π rocker combined with a crank or rocker) are not possible. This can be done either analytically (by showing that the resulting inequalities have no solutions), or geometrically (by noting that a 0 or π rocker has a motion that is symmetric by mirror symmetry across OC whereas the crank and rocker do not have this symmetry).
All the other 4bar linkages are in the case when one or more of the inequalities become equalities, in each of these cases the linkage can be folded. That is, the linkage has a configuration in which all the links line up with OC. Some 4bar linkages can be folded in more than one way; for example, the linkage with a=h=b=g can be folded in three different ways (try it!).