How to convert direction vector to euler angles?

Question

I'm looking for a way to convert direction vector (X,Y,Z) into Euler angles (heading, pitch, bank). I know that direction vector by itself is not enough to get the bank angle, so there's also another so-called Up vector. Having direction vector (X,Y,Z) and up vector (X,Y,Z) how do I convert that into Euler angles?

Accepted Answer

Let's see if I understand correctly. This is about the orientation of a rigid body in three dimensional space, like an air plane during flight. The nose of that airplane points towards the direction vector

```
D=(XD,YD,ZD) .
```

Towards the roof is the up vector

```
U=(XU,YU,ZU) .
```

Then heading `H` would be the direction vector `D` projected onto the earth surface:

```
H=(XD,YD,0) ,
```

with an associated angle

```
angle_H=atan2(YD,XD) .
```

Pitch P would be the up/down angle of the nose with respect to the horizon, if the direction vector `D` is normalized you get it from

```
ZD=sin(angle_P)
```

resulting in

```
angle_P=asin(ZD) .
```

Finally, for the bank angle we consider the direction of the wings, assuming the wings are perpendicular to the body. If the plane flies straight towards `D`, the wings point perpendicular to `D` and parallel to the earth surface:

```
W0 = ( -YD, XD, 0 )
```

This would be a bank angle of 0. The expected Up Vector would be perpendicular to `W0` and perpendicular to `D`

```
U0 = W0 × D
```

with `×` denoting the cross product. `U` equals `U0` if the bank angle is zero, otherwise the angle between `U` and `U0` is the bank angle `angle_B`, which can be calculated from

```
cos(angle_B) = Dot(U0,U) / abs(U0) / abs(U)
sin(angle_B) = Dot(W0,U) / abs(W0) / abs(U) .
```

Here 'abs' calculates the length of the vector. From that you get the bank angle as

```
angle_B = atan2( Dot(W0,U) / abs(W0), Dot(U0,U) / abs(U0) ) .
```

The normalization factors cancel each other if `U` and `D` are normalized.

How to convert direction vector to euler angles?

Solution

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