1. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics CIS 636 Introduction to Computer Graphics William H. Hsu Department of Computing and Information Sciences, KSU KSOL course pages: http://snipurl.com/1y5gchttp://snipurl.com/1y5gc Course web site: http://www.kddresearch.org/Courses/CIS636http://www.kddresearch.org/Courses/CIS636 Instructor home page: http://www.cis.ksu.edu/~bhsuhttp://www.cis.ksu.edu/~bhsu Readings: Slides based on Viewing 1-3, http://www.cs.brown.edu/courses/cs123/lectures.htmhttp://www.cs.brown.edu/courses/cs123/lectures.htm Sections 2.6 – 2.7, Eberly 2 e – see http://snurl.com/1ye72http://snurl.com/1ye72 OpenGL API Documentation: http://www.opengl.org/documentation/http://www.opengl.org/documentation/ Lab 2: glFrustum(), gluPerspective() (http://snurl.com/1zt6z)http://snurl.com/1zt6z CG Basics 4 of 8: Detailed Introduction to Viewing

2. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Online Recorded Lectures for CIS 636 Introduction to Computer Graphics Project Topics for CIS 636 Computer Graphics Basics (8)  1. Mathematical Foundations – Week 2  2. Rasterizing and 2-D Clipping – Week 3  3. OpenGL Primer 1 of 3 – Week 3  4. Detailed Introduction to 3-D Viewing – Week 4  5. OpenGL Primer 2 of 3 – Week 5  6. Polygon Rendering – Week 6  7. OpenGL Primer 3 of 3 – Week 8  8. Visible Surface Determination – Week 9 Recommended Background Reading for CIS 636 Shared Lectures with CIS 736 (Computer Graphics)  Regular in-class lectures (35) and labs (7)  Guidelines for paper reviews – Week 7  Preparing term project presentations, demos for graphics – Week 11

3. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Lecture Outline History of Projections Normalizing and Viewing Transformations Perspective (to Parallel) Transformation Kinds of Projections  Parallel  Orthographic: top, front, side, other  Axonometric: isometric, dimetric, trimetric  Oblique: cavalier, cabinet  Other  Perspective: 1-point, 2-point, 3-point Viewing in OpenGL

4. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics History Geometrical Constructions Types of Projection Projection in Computer Graphics From 3-D to 2-D: Orthographic and Perspective Projection – Part 1 Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

5. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Parallel projections used for engineering and architecture because they can be used for measurements Perspective imitates our eyes or camera, looks more natural Logical Relationship among Types of Projections Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

6. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Same method as multiview orthographic projections, except projection plane not parallel to any of the coordinate planes; parallel lines are equally foreshortened Isometric: Angles between all three principal axes are equal (120º). The same scale ratio applies along each axis Dimetric: Angles between two of the principal axes are equal; need two scale ratios Trimetric: Angles different between the three principal axes; need three scale ratios Note: different names for different views, but all part of a continuum of parallel projections of the cube; these differ in where the projection plane is relative to its cube dimetric isometric dimetric orthographic Carlbom Fig. 3-8 Axonometric Projections Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

7. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Used for:  catalogue illustrations  patent office records  furniture design  structural design Pros:  don’t need multiple views  illustrates 3D nature of object  measurements can be made to scale along principal axes Cons:  lack of foreshortening creates distorted appearance  more useful for rectangular than curved shapes Construction of isometric projection: projection plane cuts each principal axis by 45° Example Carlbom Fig.2.2 Isometric Projection: Special Case of Axonometric Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

8. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Projectors are at an oblique angle to the projection plane; view cameras have accordion housing, used for skyscrapers Pros:  can present the exact shape of one face of an object (can take accurate measurements): better for elliptical shapes than axonometric projections, better for “mechanical” viewing  lack of perspective foreshortening makes comparison of sizes easier  displays some of object’s 3D appearance Cons:  objects can look distorted if careful choice not made about position of projection plane (e.g., circles become ellipses)  lack of foreshortening (not realistic looking) oblique perspective Oblique Projections Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

9. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Source: http://www.usinternet.com/users/rniederman/star01.htmhttp://www.usinternet.com/users/rniederman/star01.htm View Camera Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

10. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Construction of an oblique parallel projection Plan oblique projection of a city (Carlbom Fig. 2-6) Front oblique projection of a radio (Carlbom Fig. 2-4) Examples of Oblique Projections Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

11. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Rules for placing projection plane for oblique views Projection plane should be set parallel to one of:  most irregular of principal faces, or one which contains circular or curved surfaces  longest principal face of object  face of interest Projection plane parallel to circular faceProjection plane not parallel to circular face Example and Principles: Oblique View Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

12. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Cavalier: Angle between projectors and projection plane is 45º. Perpendicular faces are projected at full scale Cabinet: Angle between projectors and projection plane is arctan(2) = 63.4º. Perpendicular faces are projected at 50% scale cavalier projection of unit cube cabinet projection of unit cube Main Types of Oblique Projections Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

13. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics multiview orthographic cavalier cabinet Carlbom Fig. 3-2 Examples of Orthographic and Oblique Projections Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

14. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Assume object face of interest lies in principal plane, i.e., parallel to xy, yz, or zx planes. (DOP = Direction of Projection, VPN = View Plane Normal) 1) Multiview Orthographic –VPN || a principal coordinate axis – DOP || VPN –shows single face, exact measurements 2) Axonometric –VPN || a principal coordinate axis –DOP || VPN –adjacent faces, none exact, uniformly foreshortened (as a function of angle between face normal and DOP) 3) Oblique –VPN || a principal coordinate axis –DOP || VPN –adjacent faces, one exact, others uniformly foreshortened Parallel Projections: Summary Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

15. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Used for:  advertising  presentation drawings for architecture, industrial design, engineering  fine art Pros:  gives a realistic view and feeling for 3D form of object Cons:  does not preserve shape of object or scale (except where object intersects projection plane) Different from a parallel projection because  parallel lines not parallel to the projection plane converge  size of object is diminished with distance  foreshortening is not uniform Perspective Projections Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

16. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics One Point Perspective (z-axis vanishing point) Three Point Perspective (z, x, and y-axis vanishing points) Two Point Perspective (z, and x-axis vanishing points) z Vanishing Points [1] For right-angled forms whose face normals are perpendicular to the x, y, z coordinate axes, the number of vanishing points = number of principal coordinate axes intersected by projection plane Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

17. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics What happens if same form is turned so that its face normals are not perpendicular to x, y, z coordinate axes? Although projection plane only intersects one axis (z), three vanishing points were created Note: can achieve final results which are identical to previous situation in which projection plane intersected all three axes New viewing situation: cube rotated, face normals no longer perpendicular to any principal axes Note: unprojected cube depicted here with parallel projection Perspective drawing of rotated cube Vanishing Points [2] Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

18. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics We’ve seen two pyramid geometries for understanding perspective projection: Combining these 2 views: 1.perspective image is intersection of a plane with light rays from object to eye (COP) 2.perspective image is result of foreshortening due to convergence of some parallel lines toward vanishing points Vanishing Points and View Point [1] Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

19. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Project parallel lines AB, CD on xy plane Projectors from the eye to AB and CD define two planes, which meet in a line which contains the view point, or eye This line does not intersect the projection plane (XY), because parallel to it. Therefore there is no vanishing point Vanishing Points and View Point [2] Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

20. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Lines AB and CD (this time with A and C behind projection plane) projected on xy plane: A’B and C’D Note: A’B not parallel to C’D Projectors from eye to A’B and C’D define two planes which meet in a line containing view point This line intersects projection plane Point of intersection is vanishing point C A Vanishing Points and View Point [3] Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

21. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Analogous to size of film used in a camera Determines proportion of width to height of image displayed on screen Square viewing window has aspect ratio of 1:1 Movie theater “letterbox” format has aspect ratio of 2:1 NTSC television has aspect ratio of 4:3, HDTV is 16:9 Kodak HDTV 16:9 NTSC 4:3 Aspect Ratio Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

22. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Determines amount of perspective distortion in picture  parallel projection: none  wide-angle lens: lots In a frustum, two viewing angles  Width and height angles  We specify Height angle  Get Width angle from Aspect ratio * Height angle Choosing View angle  Analogous to photographer choosing a specific type of lens  e.g., wide-angle or telephoto (narrow angle) lens View Angle [1] Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

23. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Lenses made for distance shots often have a nearly parallel viewing angle and cause little perspective distortion, though they foreshorten depth Wide-angle lenses cause a lot of perspective distortion Resulting pictures View Angle [2] Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

24. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Real cameras have a roll of film that captures pictures Synthetic camera “film” is a rectangle on an infinite film plane that contains image of scene Why haven’t we talked about the “film” in our synthetic camera, other than mentioning its aspect ratio? How is the film plane positioned relative to the other parts of the camera? Does it lie between the near and far clipping planes? Behind them? Turns out that fine positioning of Film plane doesn’t matter. Here’s why:  for a parallel view volume, as long as the film plane lies in front of the scene, parallel projection onto film plane will look the same no matter how far away film plane is from scene  same is true for perspective view volumes, because the last step of computing the perspective projection is a transformation that stretches the perspective volume into a parallel volume In general, it is convenient to think of the film plane as lying at the eye point (Position) Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission. “Where’s My Film?”

25. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Some camera models take a Focal length Focal Length is a measure of ideal focusing range; approximates behavior of real camera lens Objects at distance of Focal length from camera are rendered in focus; objects closer or farther away than Focal length get blurred Focal length used in conjunction with clipping planes Only objects within view volume are rendered, whether blurred or not. Objects outside of view volume still get discarded Focal Length Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

26. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Non-oblique view volume: Oblique view volume: Look vector at an angle to film plane Limitations of this Camera Model Look vector perpendicular to film plane Can create the following view volumes:  perspective: positive view angle  parallel: zero view angle Cannot create oblique view volume Non-oblique vs. oblique view volumes: For example, view cameras with bellows are used to take pictures of (tall) buildings. Film plane is parallel to the façade, while camera points up This is oblique view volume, with façade undistorted Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

27. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Carlbom, Ingrid and Paciorek, Joseph, “Planar Geometric Projections and Viewing Transformations,” Computing Surveys, Vol. 10, No. 4 December 1978 Kemp, Martin, The Science of Art, Yale University Press, 1992 Mitchell, William J., The Reconfigured Eye, MIT Press, 1992 Foley, van Dam, et. al., Computer Graphics: Principles and Practice, Addison-Wesley, 1995 Wernecke, Josie, The Inventor Mentor, Addison-Wesley, 1994 References Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

28. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Projection in Computer Graphics Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

29. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics More concrete way to say the same thing as orientation  soon you’ll learn how to express orientation in terms of Look and Up vectors Look Vector  the direction the camera is pointing  three degrees of freedom; can be any vector in 3-space Up Vector  determines how the camera is rotated around the Look vector  for example, whether you’re holding the camera horizontally or vertically (or in between)  projection of Up vector must be in the plane perpendicular to the look vector (this allows Up vector to be specified at an arbitrary angle to its Look vector) Up vector Look vector Position projection of Up vector Look and Up Vectors Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

30. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics D = P 1 – P 0 = (x 1 – x 0, y 1 – y 0 ) Leave P E i as an arbitrary point on the clip edge; it’s a free variable and drops out Clip Edge i Normal N i PEiPEi P 0 -P E i left: x = x min (-1,0)(x min, y)(x 0 - x min,y 0 -y) right: x = x max (1,0)(x max,y)(x 0 - x max,y 0 -y) bottom: y = y min (0,-1)(x, y min )(x 0 -x,y 0 - y min ) top: y = y max (0,1)(x, y max )(x 0 -x,y 0 - y max ) Calculations for Parametric Line Clipping Algorithm Clipping: Parametric Clipping, Review Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

31. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Volume of space between Front and Back clipping planes defines what camera can see Position of planes defined by distance along Look vector Objects appearing outside of view volume don’t get drawn Objects intersecting view volume get clipped Front clipping plane Back clipping plane Clipping: Front and Back Clipping Planes [1] Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

32. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Reasons for Front (near) clipping plane: Don’t want to draw things too close to the camera  would block view of rest of scene  objects would be prone to distortion Don’t want to draw things behind camera  wouldn’t expect to see things behind the camera  in the case of perspective camera, if we decided to draw things behind camera, they would appear upside-down and inside-out because of perspective transformation Reasons for Back (far) clipping plane: Don’t want to draw objects too far away from camera  distant objects may appear too small to be visually significant, but still take long time to render  by discarding them we lose a small amount of detail but reclaim a lot of rendering time  alternately, scene may be filled with many significant objects  for visual clarity, we may wish to declutter scene by rendering those nearest camera and discarding rest Clipping: Front and Back Clipping Planes [2] Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

33. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Have you ever played a video game and all of the sudden some object pops up in the background (e.g. a tree in a racing game)? That’s the object coming inside the far clip plane. The old hack to keep you from noticing the pop-up is to add fog in the distance. A classic example of this is from Turok: Dinosaur Hunter Now all you notice is fog and how little you can actually see. This practically defeats the purpose of an outdoor environment! And you can still see pop-up from time to time. Thanks to fast hardware and level of detail algorithms, we can push the far plane back now and fog is much less prevalent Putting the near clip plane as far away as possible helps Z precision. Sometimes in a game you can position the camera in the right spot so that the front of an object gets clipped letting you see inside of it. Clipping: Front and Back Clipping Planes [3] Then (1997) Now (2008) Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

34. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics From Position, Look vector, Up vector, Aspect ratio, Height angle, Clipping planes, and (optionally) Focal length together specify a truncated view volume Truncated view volume is a specification of bounded space that camera can “see” 2D view of 3D scene can be computed from truncated view volume and projected onto film plane Truncated view volumes come in two flavors: parallel and perspective Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission. View Volume Specification

35. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Limiting view volume useful for eliminating extraneous objects Orthographic parallel projection has width and height view angles of zero Height Width Look vector Near distance Position Far distance Up vector Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission. Truncated View Volume (Cuboid) for Orthographic Parallel Projection

36. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Removes objects too far from Position  Otherwise would merge into “blobs” Removes objects too close to Position  Would be excessively distorted Position Near distance Far distance Height angle Width angle = Look vector Up vector Height angle Aspect ratio Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission. Truncated View Volume (Frustum) for Perspective Projection

37. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Reduce degrees of freedom Five steps to specifying view volume 1. Position the camera  and therefore its view/film plane 2. Point it at what you want to see  with camera in desired orientation 3. Define the field of view  for a perspective view volume, aspect ratio of film and angle of view: somewhere between wide angle, normal, and zoom  for a parallel view volume, width and height) 4. Choose perspective or parallel projection 5. Determine focal distance Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission. Stage One: Specifying a View Volume

38. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Placement of view volume (visible part of world) specified by camera’s position and orientation  Position (a point)  Look and Up vectors Shape of view volume specified by  horizontal and vertical view angles  front and back clipping planes Perspective projection: projectors intersect at Position Parallel projection: projectors parallel to Look vector, but never intersect (or intersect at infinity) Coordinate Systems  world coordinates – standard right-handed xyz 3-space  viewing reference coordinates – camera-space right handed coordinate system (u, v, n); origin at Position and axes rotated by orientation; used for transforming arbitrary view into canonical view Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission. Specifying a View Volume

39. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics We have now specified an arbitrary view using our viewing parameters Problem: map arbitrary view specification to 2D picture of scene. This is hard, both for clipping and for projection Solution: reduce to a simpler problem and solve Note: Look vector along negative, not positive, z-axis is arbitrary but makes math easier – Need: View specification from which it is easy to take pictures – Canonical view: from the origin, looking down negative z-axis – parallel projection – sits at origin: Position = (0, 0, 0) – looks along negative z-axis: Look vector = (0, 0, –1) – oriented upright: Up vector = (0, 1, 0) – film plane extending from –1 to 1 in x and y – think of the scene as lying behind window and we’re looking through the window up n v u Arbitrary View Volume too Complex Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

40. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Our goal is to transform our arbitrary view and the world to the canonical view volume, maintaining the relationship between view volume and world, then take picture  for parallel view volume, transformation is affine: made up of translations, rotations, and scales  in the case of a perspective view volume, it also contains a non-affine † perspective transformation that frustum into a parallel view volume, a cuboid  the composite transformation that will transform the arbitrary view volume to the canonical view volume, named the normalizing transformation, is still a 4x4 homogeneous coordinate matrix that typically has an inverse  easy to clip against this canonical view volume; clipping planes are axis- aligned!  projection using the canonical view volume is even easier: just omit the z- coordinate  for oblique parallel projection, a shearing transform is part of the composite transform, to “de-oblique” the view volume † Affine transformations preserve parallelism but not lengths and angles. The perspective transformation is a type of non-affine transformation known as a projective transformation, which does not preserve parallelism Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission. Normalizing to Canonical View Volume

41. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Problem of taking a picture has now been reduced to problem of finding correct normalizing transformation It is a bit tricky to find the rotation component of the normalizing transformation. However, it is easier to find the inverse of this rotational component (trust us) So we’ll digress for a moment and focus our attention on the inverse of the normalizing transformation, which is called the viewing transformation. The viewing transformation turns the canonical view into the arbitrary view, or (x, y, z) to (u, v, n) Viewing Transformation  Normalizing Transformation Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

42. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics We know the view specification: Position, Look vector, and Up vector We need to derive an affine transformation from these parameters that will translate and rotate the canonical view into our arbitrary view  the scaling of the film (i.e. the cross-section of the view volume) to make a square cross-section will happen at a later stage, as will clipping Translation is easy to find: we want to translate the origin to the point Position; therefore, the translation matrix is Rotation is harder: how do we generate a rotation matrix from the viewing specifications that will turn x, y, z, into u, v, n?  a digression on rotation will help answer this Building Viewing Transformation from View Specification Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission.

43. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics 3 x 3 rotation matrices We learned about 3 x 3 matrices that “rotate” the world (we’re leaving out the homogeneous coordinate for simplicity) When they do, the three unit vectors that used to point along the x, y, and z axes are moved to new positions Because it is a rigid-body rotation  the new vectors are still unit vectors  the new vectors are still perpendicular to each other  the new vectors still satisfy the “right hand rule” Any matrix transformation that has these three properties is a rotation about some axis by some amount! Let’s call the three x-axis, y-axis, and z-axis-aligned unit vectors e 1, e 2, e 3 Writing out: Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission. Rotation [1]

44. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Let’s call our rotation matrix M and suppose that it has columns v 1, v 2, and v 3 : When we multiply M by e 1, what do we get? Similarly for e 2 and e 3 : Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission. Rotation [2]

45. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Thus, for any matrix M, we know that Me 1 is the first column of M If M is a rotation matrix, we know that Me 1 (i.e., where e 1 got rotated to) must be a unit-length vector (because rotations preserve length) Since Me 1 = v 1, the first column of any rotation matrix M must be a unit vector Also, the vectors e 1 and e 2 are perpendicular… So if M is a rotation matrix, the vectors Me 1 and Me 2 are perpendicular… (if you start with perpendicular vectors and rotate them, they’re still perpendicular) But these are the first and second columns of M … Ditto for the other two pairs As we noted in the slide on rotation matrices, for a rotation matrix with columns v i  columns must be unit vectors: ||v i || = 1  columns are perpendicular: v i v j = 0 (i  j) Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission. Rotation [3]

46. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Therefore (for rotation matrices) We can write this matrix of v i v j dot products as where M T is a matrix whose rows are v 1, v 2, and v 3 Also, for matrices in general, M -1 M = I, (actually, M -1 exists only for “well-behaved” matrices) Therefore, for rotation matrices only we have just shown that M -1 is simply M T M T is trivial to compute, M -1 takes considerable work: big win! Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission. Rotation [4]

47. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Summary If M is a rotation matrix, then its columns are pairwise perpendicular and have unit length Inversely, if the columns of a matrix are pairwise perpendicular and have unit length and satisfy the right-hand rule, then the matrix is a rotation For such a matrix, Adapted from slides for Brown University CS 123, Computer Graphics © 2000 – 2007, A. van Dam. Used with permission. Rotation [5]

48. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Summary History of Projections Normalizing and Viewing Transformations Perspective (to Parallel) Transformation Kinds of Projections  Parallel  Orthographic: top, front, side, other  Axonometric: isometric, dimetric, trimetric  Oblique: cavalier, cabinet  Other  Perspective: 1-point, 2-point, 3-point Viewing in OpenGL

49. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Terminology World Coordinates Screen Coordinates Normalizing Transformation Viewing Transformation Duality Perspective (to Parallel) Transformation Projections  Parallel  Orthographic: top, front, side, other  Axonometric: isometric, dimetric, trimetric  Oblique: cavalier, cabinet  Other  Perspective: 1-point, 2-point, 3-point

50. Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics Next: OpenGL Tutorial 2 Four More Short OpenGL Tutorials from SIGGRAPH 2000 Vicki Shreiner: Animation and Depth Buffering  Double buffering  Illumination: light positioning, light models, attenuation  Material properties  Animation basics in OpenGL Vicki Schreiner: Imaging and Raster Primitives Ed Angel: Texture Mapping Dave Shreiner: Advanced Topics  Display lists and vertex arrays  Accumulation buffer  Fog  Stencil buffering  Fragment programs (to be concluded in Tutorial 3)