SE 313 – Computer Graphics

1. SE 313 – Computer Graphics Lecture 7: Mathematical Basis for 3D Transformations Lecturer: GazihanAlankuş Please look at the last three slides for assignments (marked with TODO)

2. Plan for Today • One hour lecture • Two hour quiz and lab

3. Exam Talk • Next week is the midterm exam • I can ask about anything that we have done here. • Slides may not be enough.

4. Our Goal • Understand the mathematical basis of 3D transformations.

5. What we know already • Math • Point, vector, magnitude, dot product, cross product, etc. • Transformations • Translation, rotation, scale • They have a mathematical basis

6. What will we transform? • Objects are made of points • Pairs of such points define vectors • Points • Translate, rotate? (around origin), scale? (wrt. origin) • Vectors • Translate?(no!), rotate, scale

7. Translation • How can we move an object? • Add numbers to the coordinates of all of its points • Explanation on the board

8. Rotation • How can we rotate an object (around the origin)? • Trigonometry! • Ok but how can we make these calculations more systematically? • Matrix-vector multiplication! • M x v (the column vector is on the right) • Explanation on the board

9. Scale • How can we scale an object (wrt the origin)? • Multiplication! • Explanation on the board

10. Combining Transformations • Of the same kind • Translation • Just add the numbers that you will add for each • Scale • Just multiply the numbers that you will multiply for each • Rotate? • Guess what, matrix multiplication from linear algebra is actually useful

11. Combining Transformations • What if I want to do “rotate, translate, and then rotate” like we did last week? • How will you combine element-wise addition, element-wise multiplication and matrix multiplication? • You can’t. Unless you make them be the same operation.

12. Scaling as Matrix Multiplication • Instead of doing element-wise multiplication, why not get an identity matrix, multiply the diagonals of that identity matrix with the scale factors and then use it? • Explanation on the board

13. Scaling as Matrix Multiplication • Now we can combine rotation and scale however we wish! • R x S x R x R x S x S x R x S • The result is still a 3 by 3 matrix! Ready to be applied to a vector or combined with whatever else you’ve got!

14. Translation as a matrix multiplication? • Think about it for a minute.

15. Translation as a matrix multiplication? • It is possible if we add one more dimension • The fourth column holds the translation nicely – very clever hack • Explanation on the board

16. Homogeneous Coordinates • Four entities in a column vector represent three dimensions. The last entry is always 1. If not, divide all with the last entry. • The fourth dimension does not have any significant meaning. It’s just a hack. • 3x3 rotation/scale matrices -> 4x4 matrices with the rest coming from identity • Transformation column vectors -> 4x4 identity matrices, with the column vector pasted to the fourth column.

17. Combining Transformations Using Homogeneous Coordinates • Mr x Mt x Ms x Mt x Mr x Mr x Mt • Still a 4x4 matrix, ready to be applied to your 3 dimensional column vectors with a 1 in the end (for points) • For vectors, you put a 0 in the end. Remember slide #4, you cannot translate vectors! • The math works nicely.

18. Combining Transformations and the Order of Matrix Multiplication • When you read the matrix multiplication from left to right, it is the world-to-local order of applying transformations

19. Combinations of Translations and Rotations • Mt x Mr x Mr x Mt x Mt x Mr • Whichever order you apply, this is equal to Mt’ x Mr’ for some t’ and r’ • The 3x3 part of the matrix is the rotation • The last column is the translation

20. More interesting facts • When you apply a transformation to an object that was straight in the origin, the object’s local coordinate frame moves with the object. The location and the x, y, z vectors of this local frame can be represented in global coordinates. • The x, y, z vectors of the local coordinate frame are the first three columns of the transformation matrix! • The location of this local coordinate frame is the fourth column of the transformation matrix. • You can use this information to construct transformation matrices.

21. Quaternions • 3x3 matrices have too much data. We can represent it with three numbers (three rotations). But this does not play well with mathematics. • Instead, we represent them with four numbers. Quaternions happen to be a good mathematical construct for this. • , blah blahblah. DON’T CARE! We don’t need them.

22. Quaternions • What we care about is that they represent rotations as an angle and an axis • http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation • They encode it in a funny way though: • Or, the four tuple: • We still do not care as strongly. Just know that, for you, quaternion means rotation. • You can use it instead of 3x3 matrices • You can create it using an axis and an angle • You can extract from it an axis and an angle • With quaternion multiplication, you can combine them just like you combine matrices • Most 3D libraries enable you to rotate a vector with a quaternion

23. Quizzes and Lab • For two hours, the assistant will administer • A closed-book quiz (15 minutes, on paper) • An open-book quiz (rest of class, on paper) • The blender implementation of the last two questions of the open-book quiz (graded by demonstrating to the assistant)