Practical Algorithms for 3D Computer Graphics, 2nd Edition

66 2. Basic theory and mathematical results

sin θ = −a

02

cos θ =

p

1 − sin

2

θ

i f |cos θ| < ǫ {

It is not possible to distinguish heading from pitch and the convention that

φ is 0 is assumed, thus:

sin α = −a

21

cos α = a

11

sin φ = 0 cos φ = 1

}

e ls e {

sin α =

a

12

cos θ

cos α =

a

22

cos θ

sin φ =

a

01

cos θ

cos φ =

a

00

cos θ

}

α = AT AN 2(sin α, cos α)

θ = AT AN2(sin θ, cos θ)

φ = AT AN 2(sin φ, cos φ)

Listing 2.5: Conversion from rotational transformation matrix to the equivalent Euler

angles of rotat io n .

line and in Cartesian geometry a straight line is the shortest path between two

points. A similarly simple interpolation for quaternions is desirable. Therefore

for quaternions ...