How can you determine the position of a circular rod that is supported by
two other circular rods? You need to use a little trigonometry.
This is the same three-rod diagram that you saw in the previous section, but with two
triangles overlaid on the picture. Remember that the centers of the two
supporting rods are known and fixed, and that the radii of all the rods are
known. We're trying to determine the location of the point (Xtop, Ytop).
Let's consider the triangles:
- The first triangle connects the center points of the three rods. It
contains the angle labeled A and has sides a, b, and c.
- The other triangle is a right triangle that contains the angle
B and has sides c, d, and e. Notice that e and d are parallel
to the X and Y axes.
What is the distance between the center of the left supporting rod and the
center of the top rod? In other words, what is the length of side a?
Click here for the answer
If we could determine the length of sides d and e, we would be able to
solve for the length of side c by using the Pythagorean Theorem. But coming
up with these two lengths is easy because we know exactly where the centers of
the left and right rods are. Thus,
Now, given that you know the length of a, can you come up with equations that
give the values of Xtop and Ytop in terms of a and the angle A+B?
Click here for the answer
All that remains is to determine the sine and cosine of the angle A+B.
These two trig identities will prove useful:
Now all we need to do is to find the sine and cosine of A and B. B is
easy, since it is involved in a right triangle:
A is a bit more difficult, since it involves using the law of cosines:
Whew! Now if we put all this together, we can obtain the position of the
supported rod.
Eric N. Eide
Hamlet Project
Department of Computer Science
University of Utah