3-Dimensional Launch Conditions from Impact Conditions

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.

DegreesRadiansFirst TermSecond term
x2/2!
Third term
x4/4!
5.091.004.000002
10.171.015.00004
20.351.061.0006
30.141.137.003
40.241.247.009

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 - Φ2

2
   =    (1 - A2

2
) (1 - L2

2
)

Expanding, we get:

1 - Φ2

2
   =    1 -  A2

2
 -  L2

2
 + L2 A2

4

Which is easily solved as:

Φ2
  =   A2
 +  L2
 - { L2 A2

2
 }

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.



Last modified  March 25, 2017