Appendix

Appendix 1 - Approximate calculation for obliqueness

In the body of the article, we derived the relationship between total obliqueness of impact Φ and its components loft L and face angle A as:

cos(Φ)  =  cos(A) cos(L)

It was asserted that the relationship could be approximated as a Pythagorean addition, with A and L being the sides of a right triangle and Φ being the hypotenuse. Let us derive that here.

The cosine function can be expanded as a MacLaurin series:

cos(x)  =  1  -  x2/2!  +  x4/4!  - ...

Remember that x is in radians, not degrees. When we look at x in degrees, the significance of the terms of the series falls off very fast, noted in the table below.

 Degrees Radians First Term Second termx2/2! Third termx4/4! 5 .09 1 .004 .000002 10 .17 1 .015 .00004 20 .35 1 .061 .0006 30 .14 1 .137 .003 40 .24 1 .247 .009

The contribution of the third term is almost invisible, even for an angle as high as 40°. So let's just use the first two terms in our expansion. This allows us to rewrite the expression for Φ as:

 1 - Φ22 =    (1 - A22 ) (1 - L22 )

Expanding, we get:

 1 - Φ22 =    1 - A22 - L22 + L2 A24

Which is easily solved as:

 Φ2 = A2 + L2 - { L2 A22 }

The last term (in brackets) turns out to be very small for reasonable values of L and A. Remember that an A (face angle) of 10° is large almost to the point of pathological. What happens if we throw out the last term and use the rest for our approximation?

 Φ2 = A2 + L2

This is the same form as the Pythagorean Theorem, finding the hypotenuse of a right triangle from the lengths of its sides. For angles of L less than 40° and A less than 10°, the error from Pythagorean addition is less than 1/5 of a degree. That is very acceptable for most purposes.