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Local and World Coordinate Systems

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  1. Coordinate Systems and an introduction to matrices Jeff Chastine

  2. The Local Coordinate System • Sometimes called “Object Space” • It’s the coordinate system the model was made in Jeff Chastine

  3. The Local Coordinate System • Sometimes called “Object Space” • It’s the coordinate system the model was made in (0, 0, 0) Jeff Chastine

  4. The World SPACE • The coordinate system of the virtual environment (619, 10, 628) Jeff Chastine

  5. (619, 10, 628) Jeff Chastine

  6. Question • How did get the monster positioned correctly in the world? • Let’s come back to that… Jeff Chastine

  7. Camera Space • It’s all relative to the camera… Jeff Chastine

  8. Camera Space • It’s all relative to the camera… and the camera never moves! (0, 0, -10) Jeff Chastine

  9. The Big Picture • How to we get from space to space? ? ? Jeff Chastine

  10. The Big Picture • How to we get from space to space? • For every model • Have a (M)odel matrix! • Transforms from object to world space ? M Jeff Chastine

  11. The Big Picture • How to we get from space to space? • To put in camera space • Have a (V)iew matrix • Usually need only one of these V M Jeff Chastine

  12. The Big Picture • How to we get from space to space? • The ModelView matrix • Sometimes these are combined into one matrix • Usually keep them separate for convenience V M MV Jeff Chastine

  13. Matrix - What? • A mathematical structure that can: • Translate (a.k.a. move) • Rotate • Scale • Usually a 4x4 array of values • Idea: multiply each point by a matrix to get the new point • Your graphics card eats matrices for breakfast The Identity Matrix Jeff Chastine

  14. Back to The Big Picture • If you multiply a matrix by a matrix, you get a matrix! • How might we make the model matrix? M Jeff Chastine

  15. Back to The Big Picture Translation matrix T Rotation matrix R1 Rotation matrix R2 Scale matrix S • If you multiply a matrix by a matrix, you get a matrix! • How might we make the model matrix? M Jeff Chastine

  16. Back to The Big Picture Translation matrix T Rotation matrix R1 Rotation matrix R2 Scale matrix S • If you multiply a matrix by a matrix, you get a matrix! • How might we make the model matrix? M T * R1 * R2 * S = M Jeff Chastine

  17. Matrix Order • Multiply left to right • Results are drastically different (an angry vertex) Jeff Chastine

  18. Matrix Order • Multiply left to right • Results are drastically different • Order of operations • Rotate 45° Jeff Chastine

  19. Matrix Order • Multiply left to right • Results are drastically different • Order of operations • Rotate 45° • Translate 10 units Jeff Chastine

  20. Matrix Order • Multiply left to right • Results are drastically different • Order of operations • Rotate 45° • Translate 10 units before after Jeff Chastine

  21. Matrix Order • Multiply left to right • Results are drastically different • Order of operations Jeff Chastine

  22. Matrix Order • Multiply left to right • Results are drastically different • Order of operations • Translate 10 units Jeff Chastine

  23. Matrix Order • Multiply left to right • Results are drastically different • Order of operations • Translate 10 units • Rotate 45° Jeff Chastine

  24. Matrix Order • Multiply left to right • Results are drastically different • Order of operations • Translate 10 units • Rotate 45° after before Jeff Chastine

  25. Back to The Big Picture Translation matrix T Rotation matrix R1 Rotation matrix R2 Scale matrix S • If you multiply a matrix by a matrix, you get a matrix! • How might we make the model matrix? M T * R1 * R2 * S = M Backwards Jeff Chastine

  26. Back to The Big Picture Translation matrix T Rotation matrix R1 Rotation matrix R2 Scale matrix S • If you multiply a matrix by a matrix, you get a matrix! • How might we make the model matrix? M S * R1 * R2 * T = M Jeff Chastine

  27. The (P)rojection Matrix • Projects from 3D into 2D • Two kinds: • Orthographic: depth doesn’t matter, parallel remains parallel • Perspective: Used to give depth to the scene (a vanishing point) • End result: Normalized Device Coordinates (NDCs between -1.0 and +1.0) Jeff Chastine

  28. Orthographic vs. Perspective Jeff Chastine

  29. An Old Vertex Shader in vec4 vPosition; // The vertex in NDC void main () { gl_Position = vPosition; } Originally we passed using NDCs (-1 to +1) Jeff Chastine

  30. A Better Vertex Shader in vec4 vPosition; // The vertex in the local coordinate system uniform mat4 mM; // The matrix for the pose of the model uniform mat4 mV; // The matrix for the pose of the camera uniform mat4 mP; // The projection matrix (perspective) void main () { gl_Position = mP*mV*mM*vPosition; } New position in NDC Original (local) position Jeff Chastine

  31. SMILE – It’s the END! Jeff Chastine

  32. How about more than one object? • Hierarchical Transformations • Composing transformations • Coordinate systems/frames

  33. Basic idea: 1) Move fixed point to origin 2) Rotate 3) Move the fixed point back Remember, postmultiplication applies transforms in reverse Result: M = T RT –1 What does this look like graphically? Composing Transformations: Rotation about a Fixed Point

  34. Rotate around a fixed pointT-1

  35. Rotate around a fixed pointR Ө

  36. Rotate around a fixed pointR Ө

  37. Rotate around a fixed pointT Ө

  38. Rotation about z axis by 30 degrees with a fixed point of (1.0, 2.0, 3.0) Remember that last transform specified in the program is the first applied OpenGL/glm Example model *= glm::translate(1.0, 2.0, 3.0)* glm::rotate(30.0, 0.0, 0.0, 1.0)* glm::translate(-1.0, -2.0, -3.0); cube.render(view*model, &shader); ...

  39. For example, a robot arm Transformation Hierarchies

  40. Transformation Hierarchies • Let’s examine:

  41. Transformation Hierarchies • What is a better way?

  42. Transformation Hierarchies • What is a better way?

  43. World Coordinates Transformation Hierarchies Transformation: Upper Arm -> World • We can have transformations be in relation to each other • How do we do this in openGL and glm? Upper Arm Coordinates Transformation: Lower -> Upper Lower Arm Coordinates Transformation: Hand-> Lower Hand Coordinates

  44. World Coordinates Transformation Hierarchies Transformation: Upper Arm -> World • Activity: how you would have an object B orbiting object A, and object A is constantly translating. Upper Arm Coordinates Transformation: Lower -> Upper Lower Arm Coordinates Transformation: Hand-> Lower Hand Coordinates

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