Quaternion and Virtual Trackball

1
Quaternion and Virtual Trackball

• CSE 781 Introduction to 3D Image Generation
• Han-Wei Shen
• Winter 2007

2
Euler Rotation Problems

• Gimbal Lock lose one degree of freedom
• Problem happens when the axes of rotation line up
  on top of each other. For example

Step one Rotate(0, 0,0,1) Step2 Rotate(90,
0,1,0) Step 3 Rotate(??, 0,0,1)

This is same as
rotation x !! ?
3
Euler Rotation Problems

• Rotations with Euler angles to change from one
  orientation to another are not unique. Example
  (x,y,z) rotation to achieve the following

R
R
x
y
y
OR
z
z
x
z
x
R
z
y
R
z
y
Rotate(180, 1,0,0)
Rotate(180, 0,1,0) then Rotate(180,0,0,1) Eule
r angles (0,0,0) -gt (180,0,0)
Euler angles (0,0,0) -gt (0,180,180)
4
Quaternion

• Invented in 1843 as an extension to the complex
  numbers
• Used by computer graphics since 1985
• Quaternion
• Provide an alternative method to specify rotation
  
• Can avoid the gimbal lock problem
• Allow unique, smooth and continuous rotation
  interpolations

5
Mathematical Background

• A quaternion is a 4-tuple of real number, which
  can be seen as a vector and a scalar
• Q qx, qy, qz, qw qv qw, where
  
• qw is the real part and
• qv iqx jqy kqz (qx, qy, qz)
  is the
• imaginary part
• ii jj kk -1
• jk -kj i ki-ikj ij-ji k
• All the regular vector operations (dot product,
  cross product, scalar product, addition, etc)
  applied to the imaginary part qv

6
Basic Operations

• Multiplication QR (qv x rv rwqv qwrv,
• qwrw -
  qv.rv)
• Addition QR (qvrv, qwrw)
• Conjugate Q (-qv, qw)
• Norm (magnitude) QQ QQ
  qxqxqyqyqzqzqwqw
• Identity i (0,1)
• Inverse Q (1/ Norm(Q)) Q
• Some more rules can be found in the reference
  book (real time rendering) pp46

Imaginary
real
-1
7
Polar Representation

• Remember a 2D unit complex number
• cosq i sinq e
• A unit quaternion Q may be written as
• Q (sinf uq , cosf) cosf sinf uq, where
  uq is a unit 3-tuple vector
• We can also write this unit quaternion as
• Q e

iq
uqf
8
Quaternion Rotation

• A rotation can be represented by a unit
  quaternion Q (sinfuq, cosf)
• Given a point p (x,y,z) -gt we first convert it
  to a quaternion p ixjykz 0 (pv, 0)
• Then, QpQ is in fact a rotation of p around uq
  by an angle 2f !!

-1
9
Rotation Concatenation

• Concatenation is easy just multiply all the
  quaternions Q1, Q2, Q3, . Together
• There is a one-to-one mapping between a
  quaternion rotation and 4x4 rotation matrix.

(Q3 (Q2 ( Q1 P Q1 ) Q2 ) Q3 )
(Q3Q2Q1) P (Q1Q2Q3 )
-1 -1 -1
-1 -1 -1
10
Quaternion to Rotation Matrix

• Given a quaternion w xi yj kz, it can be
  translated to the rotation matrix R
• 1-2y2-2z2 2xy2wz 2xz-2wy
• R 2xy-2wz 1-2x2-2z2 2yz2wx
• 2xz2wy 2yz-2wx 1-2x2-2y2
• Also you can convert a matrix to quaternion (see
  the reference book for detail)

11
Interpolation of Rotation

• Should avoid sudden change of orientation and
  also should maintain a constant angular speed
• Each rotation can be represented as a point on
  the surface of a 4D unit sphere
• Need to perform smooth interpolation along this
  4D sphere

R
How to interpolate A and B to get R?
A
B
12
Interpolation Rotation

• Spherical Linear Interpolation (slerp)
• Given two unit quaternion (i.e., two
  rotations), we can create a smooth interpolation
  using slerp
• slerp(Q1, Q2, t)
• sin (f(1-t))
  sin(ft)
• sinf
  sinf
• where 0lttlt1
• To compute f, we can use this property
• cosf Q1xQ2xQ1yQ2yQ1zQ2zQ1wQ2w

Q1 Q2
13
3D Rotations with Euler Angles

• A simple but non-intuitive method specify
  separate x, y, z axis rotation angles based on
  the mouses horizontal, vertical, and diagonal
  movements

cos(q) -sin(q) 0 0 sin(q) cos(q) 0
0 0 0 1 0 0 0
0 1
cos(q) 0 sin(q) 0 0 1 0
0 -sin(q) 0 cos(q) 0 0 0
0 1
1 0 0 0 0 cos(q)
-sin(q) 0 0 sin(q) cos(q) 0 0
0 0 1
14
Euler Rotation Problems

• Gimbal Lock lose one degree of freedom
• Problem happens when the axes of rotation line up
  on top of each other. For example

z
Step one Rotate(0, 1,0,0) Step2 Rotate(90,
0,1,0) Step 3 Rotate(??, 0,0,1)

This is same as
rotation x !! ?
15
3D Rotations with Trackball

• Imagine the objects are rotated along with a
  imaginary hemi-sphere

16
Virtual Trackball

• Allow the user to define 3D rotation using mouse
  click in 2D windows
• Work similarly like the hardware trackball devices

17
Virtual Trackball

• Superimpose a hemi-sphere onto the viewport
• This hemi-sphere is projected to a circle
  inscribed to the viewport
• The mouse position is projected orthographically
  to this hemi-sphere

z
y
(x,y,0)
x
18
Virtual Trackball

• Keep track the previous mouse position and the
  current position
• Calculate their projection positions p1 and p2 to
  the virtual hemi-sphere
• We then rotate the sphere from p1 to p2 by
  finding the proper rotation axis and angle
• This rotation ( in eye space!) is then applied to
  the object (call the rotation before you define
  the camera with gluLookAt())
• You should also remember to accumulate the
  current rotation to the previous modelview matrix

19
Virtual Trackball

• The axis of rotation is given by the normal to
  the plane determined by the origin, p1 , and p2
• The angle between p1
• and p2 is given by

n p1 ? p1
sin q
20
Virtual Trackball

• How to calculate p1 and p2?
• Assuming the mouse position is (x,y), then the
  sphere point P also has x and y coordinates equal
  to x and y
• Assume the radius of the hemi-sphere is 1. So the
  z coordinate of P is
• Note normalize viewport y extend
• to -1 to 1
• If a point is outside the circle, project
• it to the nearest point on the circle
• (set z to 0 and renormalize (x,y))

21
Virtual Trackball
Visualization of the algorithm
22
Example

• Example from Ed Angels OpenGL Primer
• In this example, the virtual trackball is used to
  rotate a color cube
• The code for the colorcube function is omitted
• I will not cover the following code, but I am
  sure you will find it useful

23
Initialization

• define bool int / if system does not support
• bool type /
• define false 0
• define true 1
• define M_PI 3.14159 / if not in math.h /
• int winWidth, winHeight
• float angle 0.0, axis3, trans3
• bool trackingMouse false
• bool redrawContinue false
• bool trackballMove false
• float lastPos3 0.0, 0.0, 0.0
• int curx, cury
• int startX, startY

24
The Projection Step

• voidtrackball_ptov(int x, int y, int width, int
  height, float v3)
• float d, a
• / project x,y onto a hemisphere centered
  within width, height , note z is up here/
• v0 (2.0x - width) / width
• v1 (height - 2.0Fy) / height
• d sqrt(v0v0 v1v1)
• v2 cos((M_PI/2.0) ((d lt 1.0) ? d
  1.0))
• a 1.0 / sqrt(v0v0 v1v1
  v2v2)
• v0 a v1 a v2 a

25
glutMotionFunc (1)

• Void mouseMotion(int x, int y)
• float curPos3,
• dx, dy, dz
• / compute position on hemisphere /
• trackball_ptov(x, y, winWidth, winHeight,
  curPos)
• if(trackingMouse)
• / compute the change in position
• on the hemisphere /
• dx curPos0 - lastPos0
• dy curPos1 - lastPos1
• dz curPos2 - lastPos2

26
glutMotionFunc (2)

• if (dx dy dz)
• / compute theta and cross product /
• angle 90.0 sqrt(dxdx dydy dzdz)
• axis0 lastPos1curPos2
• lastPos2curPos1
• axis1 lastPos2curPos0
• lastPos0curPos2
• axis2 lastPos0curPos1
• lastPos1curPos0
• / update position /
• lastPos0 curPos0
• lastPos1 curPos1
• lastPos2 curPos2
• glutPostRedisplay()

27
Idle and Display Callbacks

• void spinCube()
• if (redrawContinue) glutPostRedisplay()
• void display()
• glClear(GL_COLOR_BUFFER_BITGL_DEPTH_BUFFER_B
  IT)
• if (trackballMove)
• glRotatef(angle, axis0, axis1, axis2)
• colorcube()
• glutSwapBuffers()

28
Mouse Callback

• void mouseButton(int button, int state, int x,
  int y)
• if(buttonGLUT_RIGHT_BUTTON) exit(0)
• / holding down left button
• allows user to rotate cube /
• if(buttonGLUT_LEFT_BUTTON) switch(state)
• case GLUT_DOWN
• ywinHeight-y
• startMotion( x,y)
• break
• case GLUT_UP
• stopMotion( x,y)
• break

29
Start Function

• void startMotion(int x, int y)
• trackingMouse true
• redrawContinue false
• startX x
• startY y
• curx x
• cury y
• trackball_ptov(x, y, winWidth, winHeight,
  lastPos)
• trackballMovetrue

30
Stop Function

• void stopMotion(int x, int y)
• trackingMouse false
• / check if position has changed /
• if (startX ! x startY ! y)
• redrawContinue true
• else
• angle 0.0
• redrawContinue false
• trackballMove false