**LFSD West Coast Challenge 2005
**Long Flights
On Short Days

Original posting 12/31/04

**Z Adjustment Factor Expression**

To calculate a score, we multiply the miles flown times a calculated adjustment factor "z" to devalue longer days. In 2004, we used to use a simple expression that resulted in an elegant curve, but the outcome seemed to favor flights near the equinox, so for 2005 we gave the expression 4 exponent "tool handles" to control the curve shape. Since there is a seasonal lag we use an "if" statement to select separate formulas for waxing and waning days. Each formula contains 2 exponent handles that we use to adjust the shape of the curve. There is interaction between the exponents, but one is used to "stretch" the top of the curve around the solstice and the other is used to control the slope (steepness) through the equinox.

The expression was assembled in an Excel Spreadsheet "Calculator". You can go to the graph page and play with the curve shape by changing the anchored exponent reference cell values. We labeled the cells to make it easier to logically follow the math.

- Spreadsheet Terms
- Exponent "tool handles" in curving sequence:
- "se" is the waxing stretch exponent to flatten the top of the curve early in the year
- "de" is the waxing drop exponent to drop the curve as the days lengthen
- "ce" is the waning climb exponent to climb the curve as the days shorten
- "re" is the waning reach exponent to flatten the top of the curve toward the end of the year

- Days
- "d" is the day of the year with Jan 1 as day 1 and Feb 28 as day 59...
- "dx" ~ d/2. used for waxing days (d<184) calculated by dividing d by 2
- "dy" ~ 183-(d/2). used for waning days (d>183) calculated by dividing d by -2 and adding 183

- Flattening the top of the curve
- "sx" ~ (dx^se)/(90^(se-1)). stretching the early waxing days by raising them to the stretching exponent and then dividing by 90 raised to the (stretching exponent -1)
- "ry" ~ (dy^re)/(90^(re-1)). reaching the late waning days by raising them to the reaching exponent and then dividing by 90 raised to the (reaching exponent -1)

- Taking the Cosine
- "cx" ~ cos sx. taking the cosine of the flattened waxing days
- "cy" ~ cos ry. taking the cosine of the flattened waning days

- Increasing the steepness of the slope of the curves middle z values
- "zx" ~ sx^de. raising the cosine of the flattened waxing days to the dropping exponent
- "zy" ~ sx^ce. raising the cosine of the flattened waning days to the climbing exponent
- "z" ~ use zx. for waxing days and zy for waning days

- Exponent "tool handles" in curving sequence:

2005 Exponents: se=2.0, de=16, re=2.2, ce=26

**Expression:**- To calculate for waxing days: If d<184, then:
- z ~ (cos (((d/2)^se)/(90^(se-1))))^de

- To calculate for waning days: If d>183, then:
- z ~ (cos (((183-(d/2))^re)/(90^(re-1))))^ce

- To calculate for waxing days: If d<184, then:

*Exponent Selection:* The focus was to make the curve work in the prime months where our South
Coast flying has an advantage over other sites. Early November through
early March. Longer flights will be had from SB in April, but flat land
sites have historically posted bigger numbers by mid spring. The
expression for 2005 was frozen on 12/31/04, but in future years we may choose to
tweak the curve down for late winter if Diablo continues to push out further in
the desert flats early in the season.

*Negative Cosine Values:* There are some negative cosine values for days
181 through 185, but the effective z output on those days utilizing the exponent
values for our purpose is zero, so the negative values have no real effect and
we choose to ignore them to make the calculation simpler. We could force
the negative values positive, but that would entail extra clunkiness. The
next time someone creates our universe, if they could make our year 360 days
instead of 365¼ day, life would be a little easier.

*Rounding:* "z" is displayed to 4 decimal places, but is calculated to more
precision within our calculator. Miles are displayed to the nearest mile.
The calculator can display more precision, but we will score to the nearest mile
displayed by the calculator. Miles will be input to the nearest mile.
This seems like a reasonable resolution, especially for the few dinosaurs that
haven't incorporated a gps. The compounding of rounding errors could
permit a shorter flight to win, but it's not a perfect world and its only a game
for fun. Reaching pilots will do well over numerous seasons.