CS-184: Computer Graphics. Today

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1 CS-84: Computer Graphics Lecture 5: Projection Prof. James O Brien Universit of California, Berkele V Toda Windowing and Viewing Transformations Windows and viewports Orthographic projection Perspective projection 2

2 Screen Space Monitor has some number of piels e.g Some sub-region used for given program You call it a window Let s call it a viewport instead [24, 768] [24, 768] [69, 75] [6, 35] [,] [,] 3 Screen Space Ma not reall be a screen mage file =2.5 Printer (,2) (3,2) Other Little piel details Sometimes odd (,) (,) (,) (2,) (3,) =-.5 Upside down Heagonal =-.5 =3.5 4 From Shirle tetbook.

3 Screen Space Viewport is somewhere on screen You probabl don t care where Window Sstem likel manages this detail Sometimes ou care eactl where Viewport has a size in piels Sometimes ou care (images, tet, etc.) Sometimes ou don t (using high-level librar) 5 Canonical View Space Canonical view region 2D: [-,-] to [+,+] (,) (-,) reflect- (-,-) (,-) (-n /2,n /2) scale (-.5, n -.5) translate (n /2,-n /2) (n -.5, -.5) 6 From Shirle tetbook.

4 Canonical View Space Canonical view region 2D: [-,-] to [+,+] n 2 = n 2 2 n n 2 (-,-) (-n /2,n /2) (,) (n /2,-n /2) (-,) reflect- scale (-.5, n -.5) translate (,-) (n -.5, -.5) From Shirle tetbook. 7 Canonical View Space Canonical view region 2D: [-,-] to [+,+] Define arbitrar window and define objects Transform window to canonical region Do other things (we ll see clipping latter) Transform canonical to screen space Draw it. From Shirle tetbook. 8

5 Canonical View Space World Coordinates Canonical Screen Space (Meters) (Piels) Note distortion issues... 9 Projection Process of going from 3D to 2D Studies throughout histor (e.g. painters) Different tpes of projection Linear Orthographic Perspective Nonlinear } Man special cases in books just one of these two... Orthographic is special case of perspective...

6 Linear Projection Projection onto a planar surface Projection directions either Converge to a point Are parallel (converge at infinit) A 2D view Linear Projection Perspective Orthographic 2

7 Linear Projection Orthographic Perspective 3 Linear Projection Orthographic Perspective 4

8 Linear Projection A 2D view Note how different things can be seen Parallel lines meet at infinit Perspective Orthographic 5 Orthographic Projection No foreshortening Parallel lines sta parallel Poor depth cues 6

9 Canonical View Space Canonical view region 3D: [-,-,-] to [+,+,+] Assume looking down -Z ais Recall that Z is in our face -Z [,,] [-,-,-] 7 Orthographic Projection Convert arbitrar view volume to canonical [,,] [-,-,-] -Z 8

10 Orthographic Projection View vector Up vector Center far,bottom,left Right = view X up near,top,right Origin *Assume up is perpendicular to view. 9 Orthographic Projection Step : translate center to origin 2

11 Orthographic Projection Step : translate center to origin Step 2: rotate view to -Z and up to +Y 2 Orthographic Projection Step : translate center to origin Step 2: rotate view to -Z and up to +Y Step 3: center view volume 22

12 Orthographic Projection Step : translate center to origin Step 2: rotate view to -Z and up to +Y Step 3: center view volume Step 4: scale to canonical size 23 Orthographic Projection Step : translate center to origin Step 2: rotate view to -Z and up to +Y Step 3: center view volume Step 4: scale to canonical size M = S T 2 R T M = M o M v 24

13 Perspective Projection Foreshortening: further objects appear smaller Some parallel line sta parallel, most don t Lines still look like lines Z ordering preserved (where we care) 25 Perspective Projection mage from D. Forsth Pinhole a.k.a center of projection 26

14 Perspective Projection mage from D. Forsth Foreshortening: distant objects appear smaller 27 Perspective Projection Vanishing points Depend on the scene Not intrinsic to camera One point perspective 28

15 Perspective Projection Vanishing points Depend on the scene Nor intrinsic to camera Two point perspective 29 Perspective Projection Vanishing points Depend on the scene Not intrinsic to camera Three point perspective 3

16 Perspective Projection View Frustum v n u 3 Perspective Projection Near n Far f Top t Y Up View Bottom b Center -Z Distance to image plane i 32

17 Perspective Projection Step : Translate center to origin Y -Z 33 Perspective Projection Step : Translate center to origin Step 2: Rotate view to -Z, up to +Y Y -Z 34

18 Perspective Projection Step : Translate center to origin Step 2: Rotate view to -Z, up to +Y Step 3: Shear center-line to -Z ais Y -Z 35 Perspective Projection Step : Translate center to origin Step 2: Rotate view to -Z, up to +Y Step 3: Shear center-line to -Z ais Step 4: Perspective i + f i ' i f -Z 36

19 37 Perspective Projection Step 4: Perspective Points at z=-i sta at z=-i Points at z=-f sta at z=-f Points at z= goto z=± Points at z=- goto z=-(i+f) and values divided b -z/i Straight lines sta straight Depth ordering preserved in [-i,-f ] Movement along lines distorted -Z ' + i f i f i 38 From Shirle tetbook. view plane e Perspective Projection WRONG ' + i f i f i

20 Perspective Projection Ee plane Top Near Far Some horizontal lines View vector ẑ 39 Perspective Projection Visualizing division of and but not z ẑ 4

21 Perspective Projection Motion in, ẑ 4 Perspective Projection Note that points on near plane fied ẑ 42

22 Perspective Projection Recall that points on far plane will sta there... ẑ 43 Perspective Projection When we also divide z points must remain on straight lines ẑ 44

23 Perspective Projection Lines etend outside view volume ẑ 45 Perspective Projection Motion in z ẑ 46

24 Perspective Projection Motion in z ẑ 47 Perspective Projection Motion in z ẑ 48

25 Perspective Projection Total motion ẑ 49 Perspective Projection Step : Translate center to orange Step 2: Rotate view to -Z, up to +Y Step 3: Shear center-line to -Z ais Step 4: Perspective Step 5: center view volume Step 6: scale to canonical size -Z 5

26 Perspective Projection Step : Translate center to orange Step 2: Rotate view to -Z, up to +Y Step 3: Shear center-line to -Z ais Step 4: Perspective Step 5: center view volume Step 6: scale to canonical size } } } M v M p M o M = M o M p M v -Z 5 Perspective Projection There are other was to set up the projection matri View plane at z= zero Looking down another ais etc... Functionall equivalent 52

27 53 r(t) = p +t d Vanishing Points Consider a ra: d p 54 Vanishing Points gnore Z part of matri X and Y will give location in image plane Assume image plane at z=-i ' ' = z w whatever

28 55 Vanishing Points ' = ' = z z w ' ' = z z w w / / / / 56 Vanishing Points Assume d z = + ' + + ' + = ' ' = t p td p t p td p z z z z w w / / / / = ±( ) d d t Lim

29 Vanishing Points t Lim d = ±' d ( All lines in direction d converge to same point in the image plane -- the vanishing point Ever point in plane is a v.p. for some set of lines Lines parallel to image plane ( d z = ) vanish at infinit What s a horizon? 57 Ra Picking Pick object b picking point on screen Compute ra from piel coordinates. 58

30 59 Ra Picking Transform from World to Screen is: nverse: What Z value? = w z w z W W W W M = ' w z w z W W W W M 6 r(t) = a w +t(b w a w ) b s = [s,s, f ] a s = [s,s, i] Ra Picking Recall that: Points at z=-i sta at z=-i Points at z=-f sta at z=-f r(t) = p +t d Depends on screen details, YMMV General idea should translate...

31 Suggested Reading Fundamentals of Computer Graphics b Pete Shirle Chapter 6 6

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