사용자:Kanie/연습장
위키백과 ― 우리 모두의 백과사전.
- What is the quaternion q1 that represents the rotation of 180 degree about the x-axis?
- What is the quaternion q2 that represents the rotation of 180 degree about the z-axis?
- What rotation is represented by composite quaternion q = q1q2?
- rotation of 180 degree about the y-axis
- Let
be a point and let
be a quaternion whose scalar part is zero and whose vector part is equal to
. Show that if
is a unit quaternion, the product qXq − 1 is a purely imaginary quaternion and the vector part of qXq − 1 satisfies:
-
- Show that q and -q represent same rotation using the result of Exercise 4.
- Therefore q and − q represents the same rotation.
- Compare the number of additions and multiplications needed to perform the following operations:
- Compose two rotation matrices.
- given n × n matrices A and B,
- this requires at least (n-1) additions and n multiplications per single element
- 3 × 3 matrix multiplication requires (2 additions + 3 multiplication) * 9 elements = 18 additions + 27 multiplications)
- given n × n matrices A and B,
- Compose two quaternions
- this requires 12 additions + 16 multiplications
- Apply a rotation matrix to a vector
- (2 additions + 3 multiplications) * 3 elements = 6 additions + 9 multiplications
- Apply a quaternion to a vector (as in Exercise 4)
: 3 additions + 4 multiplications
: 2 additions + 3 multiplications
- total : (4 additions + 5 multiplications) * 3 + 5 additions + 7 multiplications
- = 17 additions + 22 multiplications
- (if times 2 is counted as multiplication, then 6 more multiplications) : 17 additions + 28 multiplications
- Compose two rotation matrices.
- Show that a rigid body rotating at angular velocity
can be represented by the quaternion differential equations
Hint: Recall that the angular velocityindicates that the body is instantaneously rotating about the ω axis with magnitude
. Suppose that a body were to rotate with a constant angular velocity
. Then the rotation of the body after a period of time
is represented by the quaternion
At times(for small
), the orientation of the body is (to within the first order)
computeby differentiating the above equation