## Best Turn of a Paraglider in a Thermal

This code tries to compare and simulate two paragliders turning in a thermal.

The 2 figures at the end of each line indicate the speed of the wing tips.
How much toggle is applied.
And the code tries to find the best speed combination to make optimal use of the thermal

Some background info

The thermal is a math degree 3 parabola .

The turn radius of the glider is determined by the 2 toggle speeds for the inner and outer wing.

vi , inner wing speed

vo, outer wing speed

speeds can go from stallspeed to trim speed

projected wing span is s

inner speed vi is at radius - (s / 2)
outer speed vo  is at radius + (s /2)

center speed v = ( vi + vo ) /2

vi  / (r-s/2) = vo / (r+s/2)

vo*r - vo*s/2 = vi*r + vi*s/2

r*(vo-vi) = (vi+vo)*s/2

radius r = (s/2) * (vi+vo) / (vo-vi)

The centrifugal acceleratione while turning is

z  = v*v / r

while the usual weight stays pointing down with g = 9.81 m/ss

You bank angle is tan a = z/g

The sink of the glider is determined by using the center speed v
and applying it on the polar definition of the glider.
But this sink would apply for straight and level flight.

The adjusted sink = sink / cos a , with a the bank angle

The total lift or sink of the glider in the thermal is then defined as

total = lift by thermal + adjusted sink

with lift by thermal a positibe value
and adjusted sink a negative value in m/s

The code iterates through vi and vo from stall speed to trim speed

The polar is defined from minimum sink to maximum speed
Which is not 100 percent correct for this approach here.
The part of the polar from stall speed to minimum sink is missing.
If the center speed is below  minimum sink speed I set it to minimum sink.