Game Development Reference
In-Depth Information
The cosine and sine of the heading angle is equal to the ratio of the x- and y-velocity
components to the horizontal velocity component.
v
v
v
v
y
cos
θ
=
x
sin
θ
=
(10.33)
h
h
The bank angle relative to a fixed point on the ground is the same as the bank angle relative
to the center of gravity. Its definition remains the same as was defined in Figure 10-21.
The lift, thrust, and drag forces acting on an airplane are evaluated parallel and normal to
the velocity vector of the airplane. The directions parallel and normal to the velocity vector are
defined by the climb, heading, and bank angles. In Chapter 2, the concept of a two-dimensional
rotation matrix was introduced as a way to perform a coordinate system rotation about a
coordinate axis. With airplanes there are three angles (climb, heading, bank) representing
three coordinate system rotations. Therefore, to transform the lift, thrust, and drag forces
into x-, y-, and z-direction components requires multiplying three two-dimensional rotation
matrices together.
⎡
⎤
⎡
⎤ ⎡
⎤ ⎡
⎤ ⎡
⎤
F
1
0
0
cos
φ
0
−
sin
φ
cos
θ
−
sin
θ
0
F
−
F
⎢
x
⎥
⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
T
D
⎥
⎢
⎥
⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥
F
F
=
0
cos
ψ
−
sin
ψ
0
1
0
sin
θ
cos
θ
0
0
(10.34)
⎢
⎥
⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥
y
⎢
⎥
⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥
0
sin
ψ
cos
ψ
sin
φ
0
cos
φ
0
0
1
F
⎢
⎥
⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥
⎣
⎦
⎣
⎦ ⎣
⎦ ⎣
⎦ ⎣
⎦
z
L
In Equation (10.34), the climb angle is
f
, the heading angle is
q
, and the bank angle is
y
.
When the three rotation matrix multiplications are performed, the result is three equations for
the x-, y, and z-components of force on the airplane.
(
)
(
)
F
=
cos
θφ
cos
F
−
F
+
sin
θψ θφψ
sin
−
cos
sin
cos
F
(10.35a)
x
T
D
L
(
)
(
)
F
=
sin
θφ
cos
F
−
F
+ −
cos
θψ θφψ
sin
−
sin
sin
cos
F
(10.35b)
y
T
D
L
(
)
F
=
sin
φ
F
−
F
+
cos
φ ψ
cos
F
(10.35c)
z
T
D
L
The expressions shown in Equation (10.35) are the force equations that will be used when
we create the Flight Simulator.
Takeoff
For an airplane to fly, it must first gain enough lift to take off from the ground. While the airplane is
traveling down a flat runway, the climb angle is zero, and the lift and gravity forces are vertical.
The velocity required for takeoff is a function of the weight of the airplane, the lift coefficient,
the air density, and the reference area of the wing.
2
W
v
=
(10.36)
CA
r
L
