Sunday, June 27, 2010

Path is a Circular Arc for Fixed Angle between Links

It was known to Euclid that the lines drawn from the ends of the diameter to a point on a circle had a 90° included angle. This can be generalized but was only assumed in the second solution to the minimum links problem. The proof is as follows. If one assumes that the link from A to the connection point, X, is a vector with magnitude a and angle α with respect to the x axis and that from X to B is of magnitude b and angle β then B is the product of a matrix, M, and a vector containing a and b as components. The included angle between the segments is assumed to be δ and if α is known then so is β.

One can find the inverse of M if α, β and δ are known.

Since β is known in terms of α and δ, a and b can be found from another matrix operating on the unit vector defined by α. From trigonometry we also know that x and y are proportional to a.

From the first row of the matrix equation for a and b one sees that the square of a is a linear combination of x and y but by the Pythagorean Theorem it is also the sum of the squares of x and y which enables us to show that the equation for the x and y is that of a circle.

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