Transcript CS 445/645 Fall 2001 Splines Last Class • Hermite and Bézier Splines – Understand how these curves are defined • The G matrix – Understand how.

CS 445/645
Fall 2001
Splines
Last Class
• Hermite and Bézier Splines
– Understand how these curves are defined
• The G matrix
– Understand how the curve definitions lead to
the coefficient matrices
• The M matrix
– Blending functions
– Convex Hulls
Assignment 4, part 2
• One-week extension…
– Program due December 5th at midnight
• Quaternions not required
– Will be extra credit
• Reason for extension
– We think you can have more fun with this
assignment if you play with a couple new ideas
Quaternions
• Remember why we don’t like Euler angles
– Gimbal lock
– Arbitrary multiplication order
– Interpolation
Gimbal Lock
• Occurs when two axes are
aligned
• Second and third rotations
have effect of
transforming earlier
rotations
– ex: Rot x, Rot y, Rot z
• If Rot y = 90 degrees, Rot z
== -Rot x
Multiplication Order
• (qx, qy, qz) = RzRyRx
– Rotate qx degrees about x-axis
– Rotate qy degrees about y-axis
– Rotate qz degrees about z-axis
• Axis order is not defined
– (y, z, x), (x, z, y), (z, y, x)…
are all legal
– Pick one
Interpolation
Local Rotations
• Multiple applications of small Euler angles
cause numerical errors and drift
– Consider 720 rotations about y of 1 degree
 cos(1)
 0

 sin(1)

 0
0
1
0
0
0 cos(1) 0

0
0
1
0 sin(1)
720
 .9999817

0

 .017452374

0

0 .017452374 0
1
0
0
0 .9999817 0

0
0
1
720
– Should end up back where you started
 .9999817

0

 .017452374

0

0 .017452374 0
1
0
0
0 .9999817 0

0
0
1
720
1 1
0  ?  0 
   
0   0 
   
0   0 
Local Rotations
• Not only is the Euler computation
expensive
• But it won’t provide the correct result
• Matrices may not be orthonormal
Quaternion
•
•
•
•
Doesn’t suffer from Gimbal Lock
Multiplication order is clear
Interpolation is sensible
Compositions are cheap
Implementing Quaternions
•
•
•
•
Instead of storing xrot and yrot
Store quaternion
Initially quaternion is set to (0, 0, 0, 1)
If user moves mouse, multiply initial quaternion
by a second quaternion corresponding to the
rotation the user caused
• When drawing scene convert current quaternion to
rotation matrix and multiply on stack
User Interaction
• In handleMouseMovement()
– You can keep track of the change in x and y
between when the user clicked the mouse and
released
– Imagine the user had a sticky finger and
touched the surface of a sphere encircling your
model and then spun the sphere with their
finger
– How would the sphere have rotated?
User Interaction
• Compute points of surface of sphere where
user first clicked and released
• Use these points to compute
• Amount of rotation (in degrees)
• Cross product of vectors between sphere center and
these two points
• These will be your quaternion parameters
User Interaction
• Compute amount of rotation (cos (theta / 2) is
scalar of quaternion)
• Multiply rotation by scalar so rotations match
mouse translation
• Axis of rotation (use cross product)
• Zero-out the z-component of axis
• Normalize axis of rotation
• Multiply axis of rotation by sin (theta/2)
• You might have to normalize again
• Multiply previous quaternion by this new
quaternion to compose rotations
Cube Mapping
• Consider projection of triangle onto 2 faces
of cube
Other things
• glTexSubImage2D
• glUnproject
• glDepthFunc (GL_LEQUAL)
Making Movies
• Next Class