3-Dimensional
Launch Conditions from Impact Conditions
AppendixAppendix 1 - Approximate calculation for obliquenessIn 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 term
x2/2! | Third term
x4/4! | | 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 - | 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?
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
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