Transformations for Virtual Worlds. (thanks, Greg Welch)

Transcription

1 Transformations for Virtual Worlds (thanks, Greg Welch)

2 Coordinate System Transformations What is a transformation? The instantaneous relationship between a pair of coordinate systems. Defines the relative position, orientation and scale of two coordinate systems Notation: T A_B is the transformation from coordinate system B to coordinate system A

3 Simple 2D Example P P A = T A_B P B (4,5) P A P B (1,3) (4,5) = (3,2) (1,3) A T A_B (3,2) B T A_B A_B converts points in B to points in A T A_B measures the position of B s origin in A The vector runs from A to B

4 Properties of Coordinate System Transformations Used to convert the coordinates of a point specified in one coordinate system to another. P A = T A_B P B Can be inverted Inverse of T A_B = T B_A

5 Properties of Coordinate System Transformations Can be composed to compute the relationship between several coordinate systems T A_C = T A_B T B_C Note: Nice property of subscript cancellation Example: T Shoulder_Hand T A_C T Shoulder_Hand = T Shoulder_Elbow T Elbow_Hand

6 Transformation Representations Can be represented by a 4x4 Transformation Matrix Alternately can use the VQS notation Represent transformation as a Vector (translation), Quaternion (rotation), and a (uniform) Scaling factor

7 Transformation Matrices TRANSLATION T X T Y T Z SCALE S X S Y S Z ROTATION R 11 R 12 R 13 0 R 21 R 22 R 23 0 R 31 R 32 R

8 VQS Notation TRANSLATION: V = (T X, T Y, T Z ) ROTATION: Q = (Q X, Q Y, Q Z, Q W ) SCALE: S = S UNIFORM

9 Transformations: Why Quaternions? Allow simple interpolation More compact Angle and axis of rotation easy to extract More efficient (composing and inverting) More tractable mathematically than matrices or Euler angles

10 VQS Transform from P to P p = [v, q, s] p = s (q * p * q -1 ) + v where p is treated as a quaternion with zero scalar component, and the result has a zero scalar component so can be treated as a vector.

11 Coordinate System Graphs Graphical representation of the coordinate systems in a virtual world and their relationship Nodes represent coordinate systems Edges represent transformations

12 Coordinate System Graphs world tracker room objects head left eye left virtual image left screen hand right eye right virtual image right screen

13 Coordinate System Graphs: How do I use them? Can be used to determine the transformations involved in converting between coordinate systems. Example: Finding world coord of head space point T World_Head P World P World = T World_Head P Head T World_Head = T World_Room T Room_Tracker T Tracker_Head P World P World = T World_Room T Room_Tracker T Tracker_Head P Head

14 Coordinate System Graphs and Virtual World Interactions Can be used to determine the transformations involved in any virtual world interaction Specifying Actions With Invariants Based on frame-to-frame invariants A relation between a set of transformations in the current frame and a set from the previous frame

15 Specifying Virtual World Interactions Coordinate system hierarchy and frame-to-frame invariants can be used to specify many forms of virtual world interaction: Grabbing Flying Scaling

16 Specifying Actions With Invariants Example: Grabbing a Virtual Object Objective: Keep an object fixed to the user s hand Frame-to-frame invariant: T n+1 Object_Hand = T Tn n Object_Hand Result: Updated object position T n+1 World_Object

17 Specifying Actions With Invariants Use coordinate system graph to expand: T Object_Hand T Object_Hand = T Object_World T World_Room T Room_Tracker T Tracker_Hand Substitute: T 2 Object_World T 2 Object_World T 2 World_Room T 2 Room_Tracker T 2 Tracker_Hand = T 1 Object_World T1 T1 T 1 World_Room T 1 Room_Tracker T T1 1 Tracker_Hand

18 Specifying Actions With Invariants Solve: T 2 T1 T1 Object_World = T 1 Object_World T 1 World_Room T 1 Room_Tracker T 1 Tracker_Hand T 2 Hand_Tracker T 2 Tracker_Room T 2 Object_World T 2 Hand_Tracker T 2 Room_World World_Room Invert T 2 Object_World to obtain T 2 World_Object

19 Coordinate System Graph world room object tracker hand

20 Where do I learn more? Computer graphics texts (e.g. Foley, vandam, Feiner, and Hughes) on reserve at MSE for Kumar s Computer Graphics class Paper by Robinett and Holloway READ IT! available on page about today s talk Paper on quaternions by Shoemake and Chou

21 References Foley, J., A. van Dam, S. Feiner, J. Hughes (1990). Computer Graphics: Principles and Practice (2nd ed.). Addison-Wesley Publishing Co., Reading MA. Robinett, W., R. Holloway (1992). Implementation of flying, scaling, and grabbing in virtual worlds, ACM Symposium on Interactive 3D Graphics, Cambridge MA, March Shoemake, K. (1985). Animating rotations using quaternion curves, Computer Graphics: Proc. of SIGGRAPH 85. Chou, J. (1992). Quaternion Kinematic and Dynamic Differential Equations, IEEE Transactions on Robotics and Automation,, Vol. 8, No. 1, Feb