Closed Form Solution of Absolute Orientation using Unit Quaternions

Horn, Berthold KP. "Closed-form solution of absolute orientation using unit quaternions." JOSA A 4.4 (1987): 629-642.

http://people.csail.mit.edu/bkph/papers/Absolute_Orientation.pdf
See Sim3Solver of ORB-SLAM2 project.

AxisAngle to Quaternion

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

where:

the axis is normalised so: ax*ax + ay*ay + az*az = 1
the quaternion is also normalised so cos(angle/2)2 + ax*ax * sin(angle/2)2 + ay*ay * sin(angle/2)2+ az*az * sin(angle/2)2 = 1

// assumes axis is already normalised
public void set(AxisAngle4d a1) {
  double s = Math.sin(a1.angle/2);
  x = a1.x * s;
  y = a1.y * s;
  z = a1.z * s;
  w = Math.cos(a1.angle/2);
}

http://www.euclideanspace.com/maths/geometry/rotations/conversions/angleToQuaternion/

CMAttitudeReferenceFrame

Enum constants for indicating the reference frames from which all attitude samples are referenced.


typedef enum {
     CMAttitudeReferenceFrameXArbitraryZVertical = 1 << 0,
     CMAttitudeReferenceFrameXArbitraryCorrectedZVertical = 1 << 1,
     CMAttitudeReferenceFrameXMagneticNorthZVertical = 1 << 2,
     CMAttitudeReferenceFrameXTrueNorthZVertical = 1 << 3
} CMAttitudeReferenceFrame


CMAttitudeReferenceFrameXArbitraryZVertical
: Describes a reference frame in which the Z axis is vertical and the X axis points in an arbitrary direction in the horizontal plane.

CMAttitudeReferenceFrameXArbitraryCorrectedZVertica
: Describes the same reference frame as CMAttitudeReferenceFrameXArbitraryZVertical except that the magnetometer, when available and calibrated, is used to improve long-term yaw accuracy. Using this constant instead of CMAttitudeReferenceFrameXArbitraryZVertical results in increased CPU usage.

CMAttitudeReferenceFrameXMagneticNorthZVertical
: Describes a reference frame in which the Z axis is vertical and the X axis points toward magnetic north. Note that using this reference frame may require device movement to calibrate the magnetometer.

CMAttitudeReferenceFrameXTrueNorthZVertical
: Describes a reference frame in which the Z axis is vertical and the X axis points toward true north. Note that using this reference frame may require device movement to calibrate the magnetometer. It also requires the location to be available in order to calculate the difference between magnetic and true north.

all available in iOS 5.0 and later.


source : Apple dev page.

Z-Y-X Euler Angles

Z-Y-X Euler Angles

Z-Y-X Euler Angles (or X-Y-Z fixed angles)
Rx'y'z' (a,b,c) =

[ cos(a)*cos(b), cos(a)*sin(b)*sin(c) - cos(c)*sin(a), sin(a)*sin(c) + cos(a)*cos(c)*sin(b)]
[ cos(b)*sin(a), cos(a)*cos(c) + sin(a)*sin(b)*sin(c), cos(c)*sin(a)*sin(b) - cos(a)*sin(c)]
[       -sin(b),                        cos(b)*sin(c),                        cos(b)*cos(c)]



X-Y-Z Euler Angles
Rz'y'x' (a,b,c) =

[                        cos(b)*cos(c),                       -cos(b)*sin(c),         sin(b)]
[ cos(a)*sin(c) + cos(c)*sin(a)*sin(b), cos(a)*cos(c) - sin(a)*sin(b)*sin(c), -cos(b)*sin(a)]
[ sin(a)*sin(c) - cos(a)*cos(c)*sin(b), cos(c)*sin(a) + cos(a)*sin(b)*sin(c),  cos(a)*cos(b)]

Rz'y'x' (c,b,a) =
[                        cos(a)*cos(b),                       -cos(b)*sin(a),         sin(b)]
[ cos(c)*sin(a) + cos(a)*sin(b)*sin(c), cos(a)*cos(c) - sin(a)*sin(b)*sin(c), -cos(b)*sin(c)]
[ sin(a)*sin(c) - cos(a)*cos(c)*sin(b), cos(a)*sin(c) + cos(c)*sin(a)*sin(b),  cos(b)*cos(c)]

ARCBALL

회전축과 회전각도를 결정하여 회전 Quaternion 을 구해냄.

마우스 입력으로 3차원 회전축과 각도를 어떻게 결정하는지 알면 끝.

현재점과 anchor 점을 결정할 때 2D 입력을 spherical mapping을 사용하여 3D Vector로 mapping.

본 논문 : 

ARCBALL: a user interface for specifying three-dimensional orientation using a mouse
http://dl.acm.org/citation.cfm?id=155312

Eigenvalues of rotation matrix : 0~180 degrees

Eigenvalue의 의미를 보여주는 그래프. y축은 imaginary axis.

추가 1)

When an n×n rotation matrix, Q, does not include −1 as an eigenvalue, so that none of the planar rotations of which it is composed are 180° rotations, then Q+I is an invertible matrix. Most rotation matrices fit this description.

http://en.wikipedia.org/wiki/Rotation_matrix