Screw Theory - Algebra of Screws

Algebra of Screws

Let a screw be an ordered pair

where S and V are three dimensional real vectors. The sum and difference of these ordered pairs are computed componentwise. Screws are often called dual vectors.

Now, introduce the ordered pair of real numbers â=(a, b) called dual scalars. Let the addition and subtraction of these numbers be componentwise, and define multiplication as

The multiplication of a screw S=(S, V) by the dual scalar â=(a, b) is computed componentwise to be,

Finally, introduce the dot and cross products of screws by the formulas:

and

The dot and cross products of screws satisfy the identities of vector algebra, and allow computations that directly parallel computations in the algebra of vectors.

Let the dual scalar ẑ=(φ, d) define a dual angle, then the infinite series definitions of sine and cosine yield the relations

In general, the function of a dual variable is defined to be f(ẑ)=(f(φ), df′(φ)), where f′(φ) is the derivative of f(φ).

These definitions allow the following results:

• Unit screws are Plücker coordinates of a line and satisfy the relation

• Let ẑ=(φ, d) be the dual angle, where φ is the angle between the axes of S and T around their common normal, and d is the distance between these axes along the common normal, then

• Let N be the unit screw that defines the common normal to the axes of S and T, and ẑ=(φ, d) is the dual angle between these axes, then