3D Game Engine Design 1 3D Game Engine Design Ch D MAP LAB

3D Game Engine Design 2 Quaternion  Quaternion Algebra  Def >  A Quaternion is defined by q = w + xi + yj + zk where w, x, y, z are in R and  Def>  Let then

3D Game Engine Design 3 Quaternion(cont’d)

3D Game Engine Design 4 Quaternion(cont’d)  Thm> Let then and let then quaternion multiplication can be defined using vector dot product and cross product  Thm> A quaternion q may be also viewed as a 4D vector( w, x, y, z).then the dot product of two quaternion is

3D Game Engine Design 5 Quaternion(cont’d)  Def>  A unit quaternion is a quaternion q for which  Thm>  The inverse of unit quaternion and the product of unit quaternion are themselves unit quaternion.  Unit quaternion can be represented by

3D Game Engine Design 6 Quaternion(cont’d) has length 1. However observe that the quaternion product  similar to unit-length complex numbers  Euler’s identity for complex numbers generalizes to quaternion.  Thm>  The power of unit quaternion  The logarithm of unit quaternion

3D Game Engine Design 7 Quaternion(cont’d)  Caution : standard identities are not allowed because of non-commutative of quaternion.  That is,

3D Game Engine Design 8 Quaternion(cont’d)  Relationship to quaternion to rotation  Thm>  Unit quaternion represents the rotation of the 3D vector by an angle about the 3D axis. And rotated vector is represented by.  Proof> it is enough to show that  R(v) is 3D vector  R(v) is length preserving function.  R(v) is linear transformation.  R(v) doesn’t have reflection component.  Rotation axis is really u. ( 1)  Rotation angle (2)  We will prove (1),(2)

3D Game Engine Design 9 Quaternion(cont’d)  (1)  (2) u v w

3D Game Engine Design 10 Quaternion(cont’d)

3D Game Engine Design 11 Quaternion(cont’d)  Conversion between various rotation representations Angle-axis rotation Quaternion rotationRotation matrix

3D Game Engine Design 12 Quaternion(cont’d)  Conversion between angle-axis and rotation matrix  Thm> angle-axis to rotation matrix

3D Game Engine Design 13 Quaternion(cont’d)  Thm> Rotation matrix to angle-axis =0 or 180(??)

3D Game Engine Design 14 Quaternion(cont’d)  Conversion between quaternion and angle-axis  Thm> angle-axis to quaternion  Thm> quaternion to angle-axis

3D Game Engine Design 15 Quaternion(cont’d)  Conversion between quaternion and rotation matrix  Thm> quaternion to rotation matrix  Thm> rotation matrix to quaternion