Lecture 2: Geometric Transformation Yuhao Zhu CSC 292/572, Fall 2020 http://yuhaozhu.com Mobile Visual Computing Logistics ‣ Written assignment 0 is up and is due Sept. 2 11:30 AM. ‣ Course schedule: https://www.cs.rochester.edu/courses/572/fall2020/ schedule.html. You will find reading assignments and slides. ‣ Don’t forget to sign up for the news flash presentation. Link sent in BB. ‣ Come and introduce yourself in the first 2 weeks during my office hour. ‣ Start thinking and talking to me about your final project idea. !2 Geometric Transformation ‣ Perhaps the single most important math for this course and for understanding all visual computing applications. ‣ This lecture is an introduction and overview of the concepts that will be used in later lectures, not a systematic treatment of geometric transformation. ‣ Assumes basic understanding of linear algebra. A nice and gentle introduction to linear algebra could be found in Chapter 5 of Fundamentals of Computer Graphics (log in using your UR NetID). ‣ Required reading: Chapter 6 of Fundamentals of Computer Graphics. !3 Geometric Transformation: What Is It? Scaling Rotation Shearing Translation There are many more: reflection, dilation, projection, etc. https://www.ck12.org/geometry/translations/lesson/Geometric-Translations-BSC-GEOM/ FCG 4e !4 Geometric Transformation: What Is It? ‣ Change the position of all the points in the space in the same way. ‣ A point P [x, y, z]. Think of it as a 1x3 matrix ‣ Transformation: change P [x, y, z] to P’ [x’, y’, z’] ‣ Mathematically, transforming P to P’ is multiplying P with a 3x3 matrix T. ‣ Different transformations require different matrices. T00, T01, T02 [ [x, y, z] x = [x’, y’, z’] [ T10, T11, T12 T20, T21, T22 !5 Point Transformation using Matrix Multiplication ‣ What should T be like if we want to keep x the same before and after the transformation — regardless of where P [x, y, z] is. x’ = xT00 + yT10 + zT20 = x, for ∀ x, y, z T00, T01, T02 [ [x, y, z] x = [x’, y’, z’] [ T10, T11, T12 T20, T21, T22 !6 Point Transformation using Matrix Multiplication ‣ What should T be like if we want to keep x the same before and after the transformation — regardless of where P [x, y, z] is. x’ = xT00 + yT10 + zT20 = x, for ∀ x, y, z 1 0 0 T100 , T01, T02 [ [x, y, z] x = [x’, y’, z’] [ T010 , T11, T12 T020, T21, T22 !6 Point Transformation using Matrix Multiplication ‣ What should T be like if we want to keep y the same before and after the transformation — regardless of where P [x, y, z] is. y’ = xT01 + yT11 + zT21 = y, for ∀ x, y, z 0 1 0 T00, T001 , T02 [ [x, y, z] x = [x’, y’, z’] [ T10, T111 , T12 T20, T021, T22 !7 Point Transformation using Matrix Multiplication ‣ What should T be like if we want to keep z the same before and after the transformation — regardless of where P [x, y, z] is. z’ = xT02 + yT12 + zT22 = z, for ∀ x, y, z 0 0 1 T00, T01, T002 [ [x, y, z] x = [x’, y’, z’] [ T10, T11, T012 T20, T21, T122 !8 Identity Matrix ‣ What should T be like if we want to keep a point unchanged before and after the transformation — regardless of where P [x, y, z] is? ‣ That matrix is called the identity matrix T100 , T001 , T002 [ [x, y, z] x = [x’, y’, z’] [ T010 , T111 , T012 T020, T021, T122 !9 Scaling ‣ Changing from P [x, y, z] to P’ [S0·x, S1·y, S2·z] ‣ The “scaling factor”: [S0, S1, S2] ‣ How should the transformation matrix look like? [ T00, T01, T02 [ [x, y, z] x = [x’, y’, z’] [ T10, T11, T12 T20, T21, T22 !10 Scaling ‣ Changing from P [x, y, z] to P’ [S0·x, S1·y, S2·z] ‣ The “scaling factor”: [S0, S1, S2] ‣ How should the transformation matrix look like? x’ = xT00 + yT10 + zT20 = S0x [ T00, T01, T02 [ [x, y, z] x = [x’, y’, z’] [ T10, T11, T12 T20, T21, T22 !10 Scaling ‣ Changing from P [x, y, z] to P’ [S0·x, S1·y, S2·z] ‣ The “scaling factor”: [S0, S1, S2] ‣ How should the transformation matrix look like? x’ = xT00 + yT10 + zT20 = S0x [ TS000 , T01, T02 [ [x, y, z] x = [x’, y’, z’] [ T010 , T11, T12 T020, T21, T22 !10 Scaling ‣ Changing from P [x, y, z] to P’ [S0·x, S1·y, S2·z] ‣ The “scaling factor”: [S0, S1, S2] ‣ How should the transformation matrix look like? x’ = xT00 + yT10 + zT20 = S0x [ TS000 , T001 , T02 [ [x, y, z] x = [x’, y’, z’] [ T010 , TS111 , T12 T020, T021, T22 !10 Scaling ‣ Changing from P [x, y, z] to P’ [S0·x, S1·y, S2·z] ‣ The “scaling factor”: [S0, S1, S2] ‣ How should the transformation matrix look like? x’ = xT00 + yT10 + zT20 = S0x [ TS000 , T001 , T002 [ [x, y, z] x = [x’, y’, z’] [ T010 , TS111 , T012 T020, T021, TS222 !10 Scaling ‣ Changing from P [x, y, z] to P’ [S0·x, S1·y, S2·z] ‣ The “scaling factor”: [S0, S1, S2] ‣ How should the transformation matrix look like? x’ = xT00 + yT10 + zT20 = S0x Scaling matrix is a [ [ diagonal matrix TS000 , T001 , T002 [x, y, z] x = [x’, y’, z’] [ T010 , TS111 , T012 T020, T021, TS222 !10 Scaling Even scaling 0.5, 0 x’ = 0.5x [x, y] x = [x’, y’] [ 0, 0.5 [ y’ = 0.5y Uneven scaling 0.5, 0 x’ = 0.5x [x, y] x = [x’, y’] [ 0, 1.5 [ y’ = 1.5y FCG 4e !11 Rotation ‣ What should the transformation matrix be to rotate P around the z-axis by �, regardless what P is? z y P [x, y, z] P’ [x’, y’, z] z Keep z the same, rotate within the x-y plane [x’, y’] y [x, y] � � x x Rotation !12 Rotation sin α = y / r cos α = x / r y [x’, y’] r [x, y] � α x Rotation !13 Rotation sin α = y / r cos α = x / r sin(α + �) = y’ / r = sin α * cos � + cos α * sin � y = y / r * cos � + x / r * sin � [x’, y’] r [x, y] � α x Rotation !13 Rotation sin α = y / r cos α = x / r sin(α + �) = y’ / r = sin α * cos � + cos α * sin � y = y / r * cos � + x / r * sin � y’ / r = y / r * cos � + x / r * sin � [x’, y’] r [x, y] � α x Rotation !13 Rotation sin α = y / r cos α = x / r sin(α + �) = y’ / r = sin α * cos � + cos α * sin � y = y / r * cos � + x / r * sin � y’ / r = y / r * cos � + x / r * sin � [x’, y’] y’ = y * cos � + x * sin � r [x, y] � α x Rotation !13 Rotation sin α = y / r cos α = x / r sin(α + �) = y’ / r = sin α * cos � + cos α * sin � y = y / r * cos � + x / r * sin � y’ / r = y / r * cos � + x / r * sin � [x’, y’] y’ = y * cos � + x * sin � r [x, y] � cos(α + �) = x’ / r = cos α * cos � - sin α * sin � α x = x / r * cos - y / r * sin Rotation � � !13 Rotation sin α = y / r cos α = x / r sin(α + �) = y’ / r = sin α * cos � + cos α * sin � y = y / r * cos � + x / r * sin � y’ / r = y / r * cos � + x / r * sin � [x’, y’] y’ = y * cos � + x * sin � r [x, y] � cos(α + �) = x’ / r = cos α * cos � - sin α * sin � α x = x / r * cos - y / r * sin Rotation � � x’ / r = x / r * cos � - y / r * sin � !13 Rotation sin α = y / r cos α = x / r sin(α + �) = y’ / r = sin α * cos � + cos α * sin � y = y / r * cos � + x / r * sin � y’ / r = y / r * cos � + x / r * sin � [x’, y’] y’ = y * cos � + x * sin � r [x, y] � cos(α + �) = x’ / r = cos α * cos � - sin α * sin � α x = x / r * cos - y / r * sin Rotation � � x’ / r = x / r * cos � - y / r * sin � x’ = x * cos � - y * sin � !13 Rotation Matrix (Around Z-axis) x’ = x cos � - y sin � y’ = x sin � + y cos � z’ = z T00, T01, T02 [ [x, y, z] x = [x’, y’, z’] [ T10, T11, T12 T20, T21, T22 !14 Rotation Matrix (Around Z-axis) x’ = x cos � - y sin � = xT00 + yT10 + zT20, for ∀ x, y, z y’ = x sin � + y cos � = xT01 + yT11 + zT21, for ∀ x, y, z z’ = z = xT02 + yT12 + zT22, for ∀ x, y, z T00, T01, T02 [ [x, y, z] x = [x’, y’, z’] [ T10, T11, T12 T20, T21, T22 !14 Rotation Matrix (Around Z-axis) x’ = x cos � - y sin � = xT00 + yT10 + zT20, for ∀ x, y, z y’ = x sin � + y cos � = xT01 + yT11 + zT21, for ∀ x, y, z z’ = z = xT02 + yT12 + zT22, for ∀ x, y, z cosT00 �, Tsin01 �, T020 [ [x, y, z] x -sinT10 �, Tcos11 �, T120 = [x’, y’, z’] [ 0 0 1 T20, T21, T22 !14 Rotation Matrix (Around Z-axis) x’ = x cos � - y sin � = xT00 + yT10 + zT20, for ∀ x, y, z y’ = x sin � + y cos � = xT01 + yT11 + zT21, for ∀ x, y, z z’ = z = xT02 + yT12 + zT22, for ∀ x, y, z cosT00 �, Tsin01 �, T020 [ [x, y, z] x -sinT10 �, Tcos11 �, T120 = [x’, y’, z’] [ 0 0 1 T20, T21, T22 z indeed doesn’t change! !14 Rotation Matrix [ 1 0 0 [ cos � 0 -sin � Around X 0 cos � sin � Around Y 0 1 0 [ 0 -sin � cos � [ sin � 0 cos � cos � sin � 0 [ Derive the rest in Around Z -sin � cos � 0 your homework. [ 0 0 1 !15 Fixed after lecture Orthogonal Matrix QT = Q-1 ‣ Rotation matrix is orthogonal matrix, whose Q QT = I transpose is the same as inversion. ▹ The length of each row is 1.