際際滷

際際滷Share a Scribd company logo

Computer Graphics - Lecture 02 transformation

鐃
Anton Gerdelan

際際滷s from when I was teaching CS4052 Computer Graphics at Trinity College Dublin in Ireland. These slides aren't used any more so they may as well be available to the public! There are some mistakes in the slides, I'll try to comment below these. This is the third lecture - on using linear algebra for transformations.

1 of 26
Downloaded 34 times
Transformation Computer Graphics 4052 6 October Dr Anton Gerdelan gerdela@scss.tcd.ie
Simplest Example Easiest way to scale our triangle? Easiest way to move our triangle? Demo time
First Maths Revision Q. Define a 3d point 3d vector 3d unit vector Q. What do we use each for in graphics?
Modify from Main Programme? How can we update/change the amount that we move or scale? Hint: use a shader keyword Demo time
Rotation Q. How can we rotate our triangle? How can we rotate a 3d point? Rotation is around the origin (the 0,0,0 point)
Transformation Matrices Translation / rotation / scale are called affine transforms Multiply 3x3 matrix with 3d vector to apply it
Matrix * Vector Q. What is the result in the vector? i.e. what does this matrix do?
Row-Major vs. Column-Major We will use column-major notation Multiplication order for column-major is right-to-left It is possible to use row-major instead (most DX apps do) Reversed in row-major Column
3X3 Rotation Around Z-Axis cos(theta) -sin(theta) 0.0 sin(theta) cos(theta) 0.0 0.0 0.0 1.0 Theta will be in radians in C Right-hand rule for rotation direction Q. Which way will my triangle turn on screen?
Rotation Demo Define matrix as an array of floats (how many for a 3x3?) Array memory is in column order for OpenGL Update shader uniform for matrix inside main loop glUniform...() family of functions glUniformMatrix3fv() takes an array of 9 floats
TRANSFORMATION PART II
4x4 Homogenous Matrices You can use 3d matrices, but we tend to use 4d matrices in graphics, and 4d vectors/points. These are not 4d hyper-geometry - it's a sneaky exploit. Q. Any idea why we might like 4d matrices?
Matrix * Vector Rules A 3x3 matrix (mat3) can only multiply with a 3d vector (vec3) A 4x4 matrix (mat4) can only multiply with a 4d (vec4) Q. If we have a 4x4 matrix and a 3d point, how do we make our 3d point into a 4d point?
4d Vectors XYZ and W vec4 in GLSL For POINTS set the 4th component to 1.0 For VECTORS set the 4th component to 0.0 Q. Any idea why? vec4 (1.0, 5.0, -20.0, 0.0); vec4 (0.0, -1.0, 0.0, 1.0); This is a dirty trick!
Now, What Was the Point in Going 4d? We can combine many matrices together by multiplication mat4 M = R * T * S; vec4 result = M * vec4 (vp.xyz, 1.0); Send fewer matrix uniforms updated over the bus Create a transformation pipeline (more on that soon)
4X4 Homogenous Matrix Sx 0.0 0.0 Tx 0.0 Sy 0.0 Ty 0.0 0.0 Sz Tz 0.0 0.0 0.0 1.0 rotation and scaling translation x, y, z Putting translation in the final column lets us do our sneaky trick...
Matrix * Vector Q. Can you figure it out? What the 0 or 1 does at the end of a vec4?
Matrix * Vector Q. Can you figure it out? What the 0 or 1 does at the end of a vec4?
Matrix Multiplication a b c d e f g h AB = = (ae + bg) (af + bh) (ce + dg) (cf + dh) Each cell in result is = Sum of: A(our row, first col) * B(first row, our col) + ... ... ... + A(our row, last col) * B(last row, our col) A B
Identity Matrix 1.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 Main diagonal
Transpose Matrix Swaps between column-major and row-major layout Flip values over the main diagonal Q. Can y'all compute the transpose of this? 1.0 0.5 0.0 0.0 0.0 2.0 0.0 0.0 -5.0 0.0 1.0 -1.0 0.0 0.0 0.0 1.0
Inverse Matrix Reverses any matrix transformation Or transform relative to another object Quite complicated to compute Work out determinant, then multiply with cofactors 1.0 0.0 0.0 2.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 Q. Guess?
Most Important Homework 1.Find out how to build & use the following for 4x4 matrices on paper: Identity Scaling Translation Rotation around X axis, Y axis, and Z axis Matrix * Matrix 2.Spot the difference between row and column-major layouts Hint: column-major has the translation in part in a column 3.Know which order multiplication goes in R-to-L or L-to-R?
Guidelines Get a 3d maths library for C/C++ or make your own Christophe Riccio's GLM http://glm.g-truc.net/ I made a simple one (Blackboard) Make a cheat-sheet (or grab mine off Blackboard) Know how the maths work for all of these operations If unusure textbooks and online sources! This stuff definitely comes up in job interview tests, especially certain famous companies starting with 'H'
Next Extending the transformation pipeline to add a virtual camera viewing position and angle using perspective
Notes I deliberately skipped some things Vector addition Unit vectors and normalisation Dot product of 2 vectors Cross product of 2 vectors I plan to introduce this where we actually use it (lighting) ...and because I ramble too much

More Related Content

Computer Graphics - Lecture 02 transformation

  • 1. Transformation Computer Graphics 4052 6 October Dr Anton Gerdelan gerdela@scss.tcd.ie
  • 2. Simplest Example Easiest way to scale our triangle? Easiest way to move our triangle? Demo time
  • 3. First Maths Revision Q. Define a 3d point 3d vector 3d unit vector Q. What do we use each for in graphics?
  • 4. Modify from Main Programme? How can we update/change the amount that we move or scale? Hint: use a shader keyword Demo time
  • 5. Rotation Q. How can we rotate our triangle? How can we rotate a 3d point? Rotation is around the origin (the 0,0,0 point)
  • 6. Transformation Matrices Translation / rotation / scale are called affine transforms Multiply 3x3 matrix with 3d vector to apply it
  • 7. Matrix * Vector Q. What is the result in the vector? i.e. what does this matrix do?
  • 8. Row-Major vs. Column-Major We will use column-major notation Multiplication order for column-major is right-to-left It is possible to use row-major instead (most DX apps do) Reversed in row-major Column
  • 9. 3X3 Rotation Around Z-Axis cos(theta) -sin(theta) 0.0 sin(theta) cos(theta) 0.0 0.0 0.0 1.0 Theta will be in radians in C Right-hand rule for rotation direction Q. Which way will my triangle turn on screen?
  • 10. Rotation Demo Define matrix as an array of floats (how many for a 3x3?) Array memory is in column order for OpenGL Update shader uniform for matrix inside main loop glUniform...() family of functions glUniformMatrix3fv() takes an array of 9 floats
  • 12. 4x4 Homogenous Matrices You can use 3d matrices, but we tend to use 4d matrices in graphics, and 4d vectors/points. These are not 4d hyper-geometry - it's a sneaky exploit. Q. Any idea why we might like 4d matrices?
  • 13. Matrix * Vector Rules A 3x3 matrix (mat3) can only multiply with a 3d vector (vec3) A 4x4 matrix (mat4) can only multiply with a 4d (vec4) Q. If we have a 4x4 matrix and a 3d point, how do we make our 3d point into a 4d point?
  • 14. 4d Vectors XYZ and W vec4 in GLSL For POINTS set the 4th component to 1.0 For VECTORS set the 4th component to 0.0 Q. Any idea why? vec4 (1.0, 5.0, -20.0, 0.0); vec4 (0.0, -1.0, 0.0, 1.0); This is a dirty trick!
  • 15. Now, What Was the Point in Going 4d? We can combine many matrices together by multiplication mat4 M = R * T * S; vec4 result = M * vec4 (vp.xyz, 1.0); Send fewer matrix uniforms updated over the bus Create a transformation pipeline (more on that soon)
  • 16. 4X4 Homogenous Matrix Sx 0.0 0.0 Tx 0.0 Sy 0.0 Ty 0.0 0.0 Sz Tz 0.0 0.0 0.0 1.0 rotation and scaling translation x, y, z Putting translation in the final column lets us do our sneaky trick...
  • 17. Matrix * Vector Q. Can you figure it out? What the 0 or 1 does at the end of a vec4?
  • 18. Matrix * Vector Q. Can you figure it out? What the 0 or 1 does at the end of a vec4?
  • 19. Matrix Multiplication a b c d e f g h AB = = (ae + bg) (af + bh) (ce + dg) (cf + dh) Each cell in result is = Sum of: A(our row, first col) * B(first row, our col) + ... ... ... + A(our row, last col) * B(last row, our col) A B
  • 20. Identity Matrix 1.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 Main diagonal
  • 21. Transpose Matrix Swaps between column-major and row-major layout Flip values over the main diagonal Q. Can y'all compute the transpose of this? 1.0 0.5 0.0 0.0 0.0 2.0 0.0 0.0 -5.0 0.0 1.0 -1.0 0.0 0.0 0.0 1.0
  • 22. Inverse Matrix Reverses any matrix transformation Or transform relative to another object Quite complicated to compute Work out determinant, then multiply with cofactors 1.0 0.0 0.0 2.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 Q. Guess?
  • 23. Most Important Homework 1.Find out how to build & use the following for 4x4 matrices on paper: Identity Scaling Translation Rotation around X axis, Y axis, and Z axis Matrix * Matrix 2.Spot the difference between row and column-major layouts Hint: column-major has the translation in part in a column 3.Know which order multiplication goes in R-to-L or L-to-R?
  • 24. Guidelines Get a 3d maths library for C/C++ or make your own Christophe Riccio's GLM http://glm.g-truc.net/ I made a simple one (Blackboard) Make a cheat-sheet (or grab mine off Blackboard) Know how the maths work for all of these operations If unusure textbooks and online sources! This stuff definitely comes up in job interview tests, especially certain famous companies starting with 'H'
  • 25. Next Extending the transformation pipeline to add a virtual camera viewing position and angle using perspective
  • 26. Notes I deliberately skipped some things Vector addition Unit vectors and normalisation Dot product of 2 vectors Cross product of 2 vectors I plan to introduce this where we actually use it (lighting) ...and because I ramble too much