Rotations (and other transformations) Rotation as rotation matrix. Storage. Apply to vector matrix vector multiply (15 flops)

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1 Cornell University CS 569: Interactive Computer Graphics Rotations (and other transformations) Lecture Steve Marschner 1 Rotation as rotation matrix 9 floats orthogonal and unit length columns and rows inverse is transpose matrix vector multiply (15 flops) matrix matrix multiply (45 flops) simple; efficient to apply; easy to compose 3x redundant; slow to construct and compose; interpolation ill-behaved 2008 Steve Marschner 3

2 Rotation as Euler angles 3 floats three rotations ouch! simple; compact; efficient to apply gimbal lock; hard to construct; hard to compose; interpolation very ill-behaved 2008 Steve Marschner 5 Rodrigues rotation formula Rotation as axis & angle R(a, θ)x = (cos θ)x + (sin θ)(a x) + (1 cos θ)(a x)a R(a, θ) = (cos θ)i + (sin θ)ã + (1 cos θ)aa T [Leonard McMillan] unit vector axis + angle (4 floats), or axis scaled by angle (3 floats) Rodrigues formula (32 flops + cos/sin/sqrt for setup) ouch! simple; reasonably efficient to apply; construction simple composition not obvious 2008 Steve Marschner Steve Marschner 8

3 Quaternions for Rotation A quaternion is an extension of complex numbers Review: complex numbers 2008 Steve Marschner 10 ONB in quaternions ONB in quaternions Each of i, j and k are square root of 1 A quaternion is an extension of complex numbers: 4D space Cross-multiplication is like cross product 2008 Steve Marschner Steve Marschner 12

4 Quaternion Properties Quaternion for Rotation Associative Linear combination of 1, i, j, k Not commutative Unit quaternion Multiplication 2008 Steve Marschner Steve Marschner 14 Quaternion for Rotation Rotation Using Quaternion Rotate about axis a by angle! A point in space is a quaternion with 0 scalar [Leonard McMillan] Composing rotations q1 and q2 are two rotations irst, q1 then q Steve Marschner Steve Marschner 16

5 Matrix for quaternion Quaternion spline interpolation [Ramamoorthi & Barr SIGGRAP 97] 2008 Steve Marschner Steve Marschner 18 Rotation as quaternion ierarchical Transforms coeffs of 1, i, j, k (4 floats) unit vector quaternion rotation formula quaternion multiplication reasonably efficient to apply; construction simple; well-behaved interpolation Articulated body Every object has local frame of reference Example local coordinate system at center of box Tr difficult to understand at first 2008 Steve Marschner Steve Marschner 20

6 Tree of Transforms DAG/Instancing Trunk Tr Trunk Tr Nodes are model components Edges are transformations Cube Cyl Cyl 2008 Steve Marschner Steve Marschner 22 Trackball Pan/Zoom/Orbit are not enough Want to inspect an object Want to rotate about some axis and angle But only have 2 degrees of freedom Trackball There is a ball in front of image plane Grab the ball to move camera eye (0,0,0) O Image plane

7 ow does it work? Algorithm Assume circle in screen (x-o x ) 2 + (y-o y ) 2 + (z-o z ) 2 = R 2 Assume mouse moves from P0 to P1 Get 3D points P0 and P1 from equation Axis a = (P0-O) x (P1-O) Angle theta = k P1-P0 Rotate by theta around a P0 (next frame) = P1 Trackball Terms Trackball Terms image plane z = z c image plane (x-o x ) 2 + (y-o y ) 2 + (z-o z ) 2 = R 2 D = (x 0, y 0, z 0 ) D Z-O z = sqrt(r 2 -(x-o x ) 2 + (y-o y ) 2 ) = (x 1, y 1, z 1 ) O is arbitrary O (x-o x ) 2 + (y-o y ) 2 + (z-o z ) 2 = R 2 y res O O = (x res /2, y res /2, z c ) R D = - R axis rotation = ( -O)"( -O) angle rotation = k D x res