3.3 Interpolation of rotations represented by quaternions

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1 3 S 3.3 Interpolation of rotations represented by quaternions : set of unit quaternions ; S :set of unit 3D vectors Interpolation will be performed on 3 S (direct linear interpolation produces nonlinear motion) 1

2 Interpolation of rotations represented by quaternions How? use spherical linear interpolation (slerp) q q Since and represent the the same rotation, we use great W the circular arc with shorter path. This can be determined by de testing the inner product:

3 Slerp is desirable because slerp produces the shortest path between the two orientations on that unit sphere in 4D; this is equivalent to finding the "minimum torque" rotation in 3D space, which you can think of as the smoothest transition between two orientations 3 it travels this path at a constant speed, which basically means you have full control over the nature of the transition

4 Interpolation of rotations represented by quaternions Two approaches: 3 (1) View S as a group slerp( q, q; u) q1 1 1 u q q, 0 u Given a vector v vˆ R with v ˆ S, the exponential can be defined as 4

5 Interpolation of rotations represented by quaternions 3 Given a vector v vˆ R with v ˆ S, the exponential can be defined as exp is one-to-one when. Hence, can define log as the inverse of exp. Consequently, can define 5

6 Interpolation of rotations represented by quaternions 3 Given a vector v vˆ R with v ˆ S, the exponential can be defined as exp is one-to-one when. Hence, can define log as the inverse of exp. Consequently, can define 6

7 7 Interpolation of rotations represented by quaternions ˆ. ) log( so ) ˆsin, (cos ˆ) exp( since / sin, / sin, / sin ˆ ) ( sin, cos 1., ),,, ( 1/ 3 v q v q v then z y x v and z y x w Define z y x w have we S z y x w q If

8 Interpolation of rotations represented by quaternions (ii) From 4D geometry sin((1 t) ) sin( t) slerp( p0, p1,; t) p0 p1 sin sin (*) where q 0 q1 cos Show that slerp(q0, q1; t) is a unit quaternion 8

9 Before we show that slerp(q0, q1; t) is a unit quaternion, let s see the geometric meaning of this definition first. First, the formula is symmetric. The symmetry can be seen in the fact that Slerp(p 0, p 1 ; t) = Slerp(p 1, p 0 ; 1 t). In the limit as 0, this formula reduces to the corresponding symmetric formula for linear interpolation, slerp( p0, p1,; t) (1 t) p0 tp1 A Slerp path is, in fact, the spherical geometry equivalent of a path along a line segment in the plane; a great circle is a spherical geodesic. 9

10 Next, we show the correctness (or, derivation) of (*). The interpolation between p0 and p1 in 4D space can be written as p( t) c p 0 ( t) p0 c1 ( t) 1 c ( ) c 1 ( t) where and are realvalued functions for 0<= t <= 1 with c 0) 1, c (0) 0, c (1) 0 t 0( 1 0 0, and c1 (1) 1. As t uniformly varies between 0 and 1, the values p(t) are required to uniformly vary along the circular arc from p0 to p1. That is, the angle between p(t) and p0 is 10

11 cos( t) and the angle between p(t) and p1 is cos(( 1t) ). Taking the inner product of p(t) with p0, we get cos( t) c ( t) cos( ) c1 ( 0 t Taking the inner product of p(t) with p1, we get cos(( 1t) ) cos( ) c ( t) c1 ( Solving (1) and (), we get c c 0 1 sin((1 t) ) ( t) sin sin( t ) ( t) sin (1) Thus, the correctness of (*) has been proven. 11 ) 0 t ) ()

12 Interpolation of rotations represented by quaternions Show that slerp(q1, q; u) is indeed a unit quaternion 1

13 13 Interpolation of rotations represented by quaternions Since q1 and q are unit quaternions, we have sin ) ( sin ) ( sin ) )sin( ) sin((1 sin ) ) ((1 sin ) ;, ( z y x w u z z y y x x w w u u z y x w u z y x w u q q Slerp

14 Interpolation of rotations represented by quaternions So, 14

15 Interpolation of rotations represented by quaternions Show that q1 q q1 q Slerp ( q1, q;1/ ) q q cos( / ) Question: Does the nd approach generate the same curve as the 1st approach? 1 15

16 Interpolation of rotations represented by quaternions Show that Proof: q1 q q1 q Slerp ( q1, q;1/ ) q q cos( / ) Slerp( q Hence, 1 q, 1 q 1 ;1/ ) q q1 q cos( / ) cos( / ) 1 16

17 How to interpolate between a series of orientations? Problem with slerping between points: first order discontinuity Solution: use cubicbezier interpolation 3 on S 17

18 (i) Shoemake s approach 18 where {

19 What do they mean? 19 Let D =Double(p,q). Then q is the midpoint of the Circular arc from p to D, i.e., q p p Hence, D D D p D cos ( p q) q p D ( p q) p

20 (i) Approach II 0 Then, use qi-1, ai-1, bi and qi as the control points of segment i

21 Generation of the segment Ci(u) Use De Casteljau algorithm 1

22 Path Following Issues: orientation handling, path smoothing, path along a surface Orientation Handling: - define a local coordinate system (u, v, w) for the camera (as it travels along a path) Frenet Frame: w u P( s) P( s) P ( s) v wu

23 Solution for (i) when P ( s) 0 for a segment 3 if P ( s) 0 for a < s < b then how should a local coordinate system (uc, vc, wc ) for P(c), a < s < b, be defined? w w w c a interpolate b v a and v b to get v c

24 Solution for (ii) when P(s) is not continuous at a point 4

25 Main problem with using the Frenet frame as the local coordinate frame to define the orientation of the camera or object following the path is: the resulting motions are too extreme and not natural looking Using the w-axis (tangent vector) as the view direction of a camera can be undesirable. Why? 5 Often, the tangent vector does not appear to correspond to the direction of where it s going even though it is in the instantaneous (analytic) sense.

26 The more natural orientation, for someone riding in a car or riding a bike, would be to look further ahead along the curve rather than to look tangential to the curve. Solution for (ii) the resulting motions are too extreme and not natural looking 6 Define view vector as COI - POS

27 Smoothing a Path - to remove the jerkiness of a path whose points are generated by a digitizing process 7 - usually, local methods are used

28 Smoothing a Path (i) Smoothing with linear interpolation of adjacent points 8 repeated applications would flatten out the curve

29 (ii) Smoothing with cubic interpolation of adjacent points Compute C(1/) ; 9

30 At left end: Compute C(1/3) 30

31 (iii) Smoothing with convolution kernels - new point is generated by applying a smoothing kernel to the data points viewed as a step function 31

32 (iii) Smoothing with convolution kernels - new point is generated by applying a smoothing kernel to the data points viewed as a step function 3

33 Determining a Path along a Surface: (i) along a polygonal surface mesh a. plane intersection b. greedy algorithm c. shortest path 33

34 Determining a Path along a Surface: (i) along a polygonal surface mesh a. plane intersection 34 Determine a plane that contains the start point and the destination point and is generally perpendicular to the surface

35 Determining a Path along a Surface: b. greedy algorithm 35 For each edge emanating from the current vertex, calculate the projection of the edge onto the straight line between the current vertex and the destination vertex. Divide this distance by the length of the edge to get the cosine of the angle between the edge and the straight line. The edge with the largest cosine is the edge most in the direction of the straight line; choose this edge to add to the path.

36 Determining a Path along a Surface: (i) along a polygonal surface mesh c. shortest path 36 Unfold the faces of the mesh to be on a plane. The shortest path is a straight line between the start vertex and the destination vertex that lies within the unfolded mesh.

37 Determining a Path along a Surface: (i) along a polygonal surface mesh c. shortest path Jingdon Chen, Yijie Han, Shortest Path on a Polyhedron, Part I: Computing Shortest Path, International Journal of Computational Geometry and Applications, 6, (199),

38 Determining a Path along a Surface: (i) along a polygonal surface mesh c. shortest path Basic idea: 1. Triangulate all the faces. Triangulate the face that contains S so that S becomes a vertex 3. Unfolding, using the following approach 38

39 Determining a Path along a Surface: (i) along a polygonal surface mesh c. shortest path Unfolding strategy: 1. Avoid situation such as the one shown on the right. Use geodesic path 39

40 Determining a Path along a Surface: (ii) along a high-order parametric surface d s 40 construct a line in parameter space and transfer to the surface. (Question: how to find d and s?)

41 Path Finding: finding a collision-free path in a given environment A topic usually addressed in the robotics literature. Complexity of the problem increases when the environment is not stationary, and the problem becomes more complex if the obstacles movement is not predictable. No good solutions to this problem have been found yet, even though some (computation intensive) greedy 41 algorithms have been proposed.

42 Path Finding: finding a collision-free path in a given environment Recently, also considered in graphics, such as walk through a plaza or a room with a lot of people moving around. 4

43 End of Interpolation III 43

44 Interpolation of rotations represented by quaternions If q is a unit quaternion, then q and q represent the same rotation. Why? Need to prove that q[0, P] q 1 ( q)[0, P]( q) 1 44 Since -q is also a unit quaternion, we have Hence, 1 q q ( q ) ( q 1 ) ( q)[0, P]( q) 1 ( q)[0, P]( ( q 1 )) q[0, P] q 1

45 Interpolation of rotations represented by quaternions Actually, more game programmers are using a third approach to interpolate between quaternions: 1. linearly interpolate two quaternions from t = 0 to t = 1. normalize the result 45

46 For C A t( B A), 0 t 1 we have CD OD t sin OE ED B (cos, sin) 1-t C OE 1 t t ED FG t cos O F E G D A (1, 0) 46 Hence, t sin tan 1 ( ) 1t t cos

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