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Mathematical Description: P01 - Universal Joint

by David W. Henderson and Daina Taimina


The universal joint P01 can be considered to be a spherical 4-bar linkage with , the output crank b variable in the range , and the fixed (grounded) link g being the angle between the input and output shafts which can be adjusted in the range . Note that the links a, h, and g are not actually links, however, the constraints of the mechanism operate as if they were spherical links. See 4-bar linkages for a discussion of the eight main types of spherical 4-bar linkages.


When this mechanism is used in the drive shaft of an automobile, then it is usual that  and in this case the mechanism is called Hooke’s joint. See History of Hooke’s Joint. For the Hooke’s joint, using the notation from 4-bar linkages, we have . Thus (noting that on the sphere, |l|S = |l|, if  |l| , and |l|S =  - |l|, if  |l|  )


From the chart in 4-bar linkages we have that this is a double crank, which means that both the input and output shafts can rotate completely around, as can be seen in the movie above.   {The reader can check that as long as b is within  of  and g is within  of , then this universal joint will still be a double crank. In fact, this is important in its automotive use because as an automobile goes over bumps the angle g between the shafts will change.}


We now look at the relationship between the rotation of the input shaft to the rotation of out shaft. One of the problems with Hooke’s joint is that, though one rotation of the input shaft results in one rotation of the output shaft, the rotations are not in sync during the revolution. We look at the standard case of , which we depict in Figure 1. Since , the corresponding arcs are  of great circles, thus A must the pole for the great circle (dashed in the figure) passing through O and B; and the great circle arcs h and a must intersect this great circle at right angles (since A is the pole). Likewise, B must be the pole for the great circle passing through A and C; and the great circle arcs h and b must also intersect this great circle at right angles. Since the arc h is perpendicular to both of the dashed great circles, then the arc must bisect lune determined by these two great circles. Since h is  great circle in length, then angle of this lune (at P) must be , as marked. In addition, we use  to label the angle at A and  to label the angle at B. Now, since A is the pole for the great circle P-O-B then the arc OB  subtends the angle  at the center of the sphere and so the radian measure of OB is . By the same reasoning the radian measure of the arc AC is .





The angle  measures the rotation of the Input Shaft and the angle  measures the rotation of the output shaft. Note that, when  =0 then  =0 also. As  changes in the positive (CCW) direction,  will be changing in the negative (CW) direction; thus, when  is positive,  will be negative and we will need to use  when denoting the angle in the triangle OCA.

In order to find the relationship between g (in radian measure) and the angles  and , we use properties of spherical triangles that are familiar to anyone who has studied spherical geometry  but unfortunately few people study spherical geometry these days. You can find the formulas we need at the end of this document.


If we apply the “Spherical Pythagorean Theorem” to the right triangle OPC, we obtain:


If we apply the Law of Sines to triangle OCA, then we obtain:


From triangle OCB we obtain:


Putting these together we get:




We put the negative as indicated because g in practice is near  and thus  is positive. Note that when  then  and thus the two shafts turn in unison.


Spherical Trigonometry Formulas

If we have a spherical triangle with sides r, s, t (in radian measure) whose opposite angles are labeled , then the Law of Sines states:


The Law of Cosines states:


In the case that the angle opposite c is a right angle then we get the “Spherical Pythagorean Theorem”:



For more information, see

Henderson, David W., and Taimina, Daina. Experiencing Geometry: Euclidean and Non-Euclidean With History. Upper Saddle River, NJ: Prentice Hall, 2005.

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