Game Development Reference
In-Depth Information
Each one of these quaternions is of unit length. 2
Now you can multiply these quaternions to obtain a single one that represents the ro‐
tation, or orientation, defined by the three Euler angles:
q = q yaw q pitch q roll
Performing this multiplication yields:
q = [{cos(φ/2) cos(τ /2) cos(ψ /2) + sin(φ/2) sin(τ /2) sin(ψ /2)},
{sin(φ/2) cos(τ /2) cos(ψ /2) − cos(φ/2) sin(τ /2) sin(ψ /2)} i +
{cos(φ/2) sin(τ /2) cos(ψ /2) + sin(φ/2) cos(τ /2) sin(ψ /2)} j +
{cos(φ/2) cos(τ /2) sin(ψ /2) − sin(φ/2) sin(τ /2) cos(ψ /2)} k ]
Here's the code that takes three Euler angles and returns a quaternion:
inline Quaternion MakeQFromEulerAngles(float x, float y, float z)
{
Quaternion q;
double roll = DegreesToRadians(x);
double pitch = DegreesToRadians(y);
double yaw = DegreesToRadians(z);
double cyaw, cpitch, croll, syaw, spitch, sroll;
double cyawcpitch, syawspitch, cyawspitch, syawcpitch;
cyaw = cos(0.5f * yaw);
cpitch = cos(0.5f * pitch);
croll = cos(0.5f * roll);
syaw = sin(0.5f * yaw);
spitch = sin(0.5f * pitch);
sroll = sin(0.5f * roll);
cyawcpitch = cyaw*cpitch;
syawspitch = syaw*spitch;
cyawspitch = cyaw*spitch;
syawcpitch = syaw*cpitch;
q.n = (float) (cyawcpitch * croll + syawspitch * sroll);
q.v.x = (float) (cyawcpitch * sroll - syawspitch * croll);
q.v.y = (float) (cyawspitch * croll + syawcpitch * sroll);
q.v.z = (float) (syawcpitch * croll - cyawspitch * sroll);
return q;
}
2. You can verify this by recalling the trigonometric relation cos 2 θ + sin 2 θ = 1.
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