3D Viewing. CMPT 361 Introduction to Computer Graphics Torsten Möller. Machiraju/Zhang/Möller

Transcription

1 3D Viewing CMPT 361 Introduction to Computer Graphics Torsten Möller

2 Reading Chapter 4 of Angel Chapter 6 of Foley, van Dam, 2

3 Objectives What kind of camera we use? (pinhole) What projections make sense Orthographic Perspective The viewing pipeline Viewing in OpenGL Shadows 3

4 3D Viewing Popular analogy: virtual camera taking pictures in a virtual world The process of getting an image onto the computer screen is like that of taking a snapshot. 4

5 3D Viewing (2) With a camera, one: establishes the view opens the shutter and exposes the film 5

6 3D Viewing (3) With a camera, one: establishes the view opens the shutter and exposes the film With a computer, one: chooses a projection type (not necessarily perspective) establishes the view clips the scene according to the view projects the scene onto the computer display 6

7 The ideal pinhole camera Single ray of light gets through small pinhole Film placed on side of box opposite to pinhole Projection 7

8 The pinhole camera Angle of view (always fixed) Depth of field (DOF): infinite every point within the field of view is in focus (the image of a point is a point) Problem: just a single ray of light & fixed view angle Solution: pinhole lens, DOF no longer infinite 8

9 The synthetic camera model Image formed in front of the camera Center of projection (COP): center of the lens (eye) 9

10 Types of projections Choose an appropriate type of projection (not necessarily perspective) Establishes the view: direction and orientation 10

11 3D viewing with a computer Clips scene with respect to a view volume Usually finite to avoid extraneous objects Projects the scene onto a projection plane In a similar way as for the synthetic camera model Everything in view volume is in focus Depth-of-field (blurring) effects may be generated 11

12 Where are we at? Clip against 3D view volume Project onto projection plane 12

13 Objectives What kind of camera we use? (pinhole) What projections make sense Orthographic Perspective The viewing pipeline Viewing in OpenGL Shadows 13

14 Projection: 3D 2D We study planar geometric projections: projecting onto a flat 2D surface project using straight lines Projection rays, called projectors, are sent through each point in the scene from the centre of projection (COP) - our pinhole Intersection between projectors and projection plane form the projection 14

15 Perspective and parallel projections Perspective: - Determined by COP Parallel: COP at infinity By direction of projectors (DOP) 15

16 COP in homogeneous coordinates Perspective projection: COP is a finite point: (x, y, z, 1) Parallel projection Direction of projection is a vector: (x, y, z, 1) (x, y, z, 1) = (a, b, c, 0) Points at infinity and directions correspond in a natural way 16

17 Perspective vs. parallel Perspective projection: Realistic, mimicking our human visual system Foreshortening: size of perspective projection of object varies inversely with the distance of that object from the center of projection Distances, angles, parallel lines are not preserved Parallel projection: Less realistic but can be used for measurements Foreshortening uniform, i.e., not related to distance Parallel lines are preserved (length preserving?) 17

18 18

19 Taxonomy of projections Angle between projectors and projection plane? Number of principal axes cut by projection plane 19

20 Parallel projections Used in engineering mostly, e.g., architecture Allow measurements Different uniform foreshortenings are possible, i.e., not related to distance to projection plane Parallel lines remain parallel Angles are preserved only on faces which are parallel to the projection plane same with perspective projection 20

21 Orthographic (parallel) projections Projectors are normal to the projection plane Commonly front, top (plan) and side elevations: projection plane perpendicular to z, y, and x axis Matrix representation (looking towards negative z along the z axis; projection plane at z = 0): 21

22 Orthographic projections Axonometric projection Projection plane not normal to any principal axis E.g., can see more faces of an axis-aligned cube 22

23 Orthographic projections Foreshortening: three scale factors, one each for x, y, and z axis Axonometric projection example: 23

24 Orthographic projection Axonometric projection Isometric Projection-plane normal makes same angle with each principal axis There are just eight normal directions of this kind All three principal axes are equally foreshortened, good for getting measurements Principal axes make same angle in projection plane Alternative: dimetric & trimetric (general case) 24

25 Parallel Projection (4) Axonometric Projection - Isometric Angles between the projection of the axes are equal i.e. 120º Alternative - dimetric & trimetric 25

26 Types of Axonometric Projections 26

27 Oblique (parallel) projections Projectors are not normal to projection plane Most drawings in the text use oblique projection 27

28 Oblique projections Two angles are of interest: Angle α between the projector and projection plane The angle φ in the projection plane Derive the projection matrix 28

29 Derivation of oblique projections 29

30 Common oblique projections Cavalier projections Angle α = 45 degrees Preserves the length of a line segment perpendicular to the projection plane Angle φ is typically 30 or 45 degrees Cabinet projections Angle α = 63.7 degrees or arctan(2) Halves the length of a line segment perpendicular to the projection plane more realistic than cavalier 30

31 Projections, continued 31

32 Perspective projections Mimics our human visual system or a camera Project in front of the center of projection Objects of equal size at different distances from the viewer will be projected at different sizes: nearer objects will appear bigger 32

33 Types of perspective projections Any set of parallel lines that are not parallel to the projection plane converges to a vanishing point, which corresponds to point at infinity in 3D One-, two-, three-point perspective views are based on how many principal axes are cut by projection plane 33

34 Vanishing points 34

35 Simple perspective projection COP at z = 0 Projection plane at z = d Transformation is not invertible or affine. Derive the projection matrix. 35

36 Simple perspective projection How to get a perspective projection matrix? Homogeneous coordinates come to the rescue 36

37 Summary of simple projections COP: origin of the coordinate system Look into positive z direction Projection plane perpendicular to z axis 37

38 Objectives What kind of camera we use? (pinhole) What projections make sense Orthographic Perspective The viewing pipeline Viewing in OpenGL Shadows 38

39 General viewing and projections With scene specified in world coordinate system Position and orientation of camera Projection: perspective or parallel Clip objects against a view volume which one? Normalize the view volume (easier to do this way) The rest is orthographic projection 39

40 Perspective Projections Need to describe the viewing VRP - view reference point VRP 40

41 Perspective Projections Need to describe the viewing VRP - view reference point VN - view normal VN VRP 41

42 Perspective Projections Need to describe the viewing VRP - view reference point VN - view normal VUP - up vector VUP VN v VRP u 42

43 Perspective Projections Need to describe the viewing VRP - view reference point VN - view normal VUP - up vector COP - centre of projection VUP VN v COP VRP u 43

44 Clipping Projection defines a view volume add front and back clipping plane make view volume finite why is this useful for a perspective projection? 44

45 Clipping (2) How should we clip a scene against the view volume? We could clip against the actual coordinates, but is there an easier way? 45

46 Clipping (3) Canonical view volume Parallel: x = -1, x = 1, y = -1, y = 1, z = 0, z = -1 Perspective: x = z, x = -z, y = z, y = -z, z = -z_min, z = -1 46

47 Viewing Process 47

48 Normalization - Parallel translate the VRP to the origin 48

49 Normalization - Parallel (2) rotate the view reference coordinates such that the VPN becomes the z axis, u becomes the x axis and v becomes the y axis 49

50 Normalization - Parallel (3) shear so that the direction of projection is parallel to the z axis 50

51 Normalization - Parallel (4) translate and scale into the canonical view volume 51

52 Normalization - Parallel (5) translates the VRP to the origin rotate the view reference coordinates such that the VPN becomes the z axis, u becomes the x axis and v becomes the y axis shear so that the direction of projection is parallel to the z axis translate and scale into the canonical view volume 52

53 Normalization - Parallel (6) Projection matrix: M = S T SH R T 53

54 Normalization - Perspective translates the VRP to the origin 54

55 Normalization - Perspective (2) rotate the view reference coordinates such that the VPN becomes the z axis, u becomes the x axis and v becomes the y axis. 55

56 Normalization - Perspective (3) Translate so that the center of projection is at the origin 56

57 Normalization - Perspective (4) shear so that the centre line of the view volume becomes the z axis 57

58 Normalization - Perspective (5) scale into the canonical view volume 58

59 Normalization - Perspective (6) translates the VRP to the origin rotate the view reference coordinates such that the VPN becomes the z axis, u becomes the x axis and v becomes the y axis. translate so that the center of projection is at the origin shear so that the centre line of the view volume becomes the z axis scale into the canonical view volume 59

60 Normalization - Perspective (7) Projection matrix (perspective): M = P pers S SH T R T Projection matrix (parallel): M = P parall S SH R T 60

61 Objectives What kind of camera we use? (pinhole) What projections make sense Orthographic Perspective The viewing pipeline Viewing in OpenGL Shadows 61

62 Computer Viewing There are three aspects of the viewing process, all of which are implemented in a pipeline, Positioning the camera Setting the model-view matrix Selecting a lens Setting the projection matrix Clipping Setting the view volume Angel/Shreiner/Möller 62

63 The OpenGL Camera In OpenGL, initially the object and camera frames are the same Default model-view matrix is an identity The camera is located at origin and points in the negative z direction OpenGL also specifies a default view volume that is a cube with sides of length 2 centered at the origin Default projection matrix is an identity Angel/Shreiner/Möller 63

64 Default Projection Default projection is orthogonal 2 clipped out z=0 Angel/Shreiner/Möller 64

65 Moving the Camera Frame If we want to visualize an object with both positive and negative z values we can either Move the camera in the positive z direction Translate the camera frame Move the objects in the negative z direction Translate the world frame Both of these views are equivalent and are determined by the model-view matrix Want a translation Translate(0.0,0.0,-d); d > 0 Angel/Shreiner/Möller 65

66 Moving the Camera We can move the camera to any desired position by a sequence of rotations and translations Example: side view Rotate the camera Move it away from origin Model-view matrix C = TR Angel/Shreiner/Möller 66

67 The LookAt Function The GLU library contained the function glulookat to form the required modelview matrix through a simple interface Note the need for setting an up direction Replaced by LookAt() in mat.h Can concatenate with modeling transformations Example: isometric view of cube aligned with axes mat4 mv = LookAt(vec4 eye, vec4 at, vec4 up); Angel/Shreiner/Möller 67

68 glulookat LookAt(eye, at, up) Angel/Shreiner/Möller 68

69 Other Viewing APIs The LookAt function is only one possible API for positioning the camera Others include View reference point, view plane normal, view up (PHIGS, GKS-3D) Yaw, pitch, roll Elevation, azimuth, twist Direction angles Angel/Shreiner/Möller 69

70 OpenGL Orthogonal Viewing Ortho(left,right,bottom,top,near,far) near and far measured from camera Angel/Shreiner/Möller 70

71 OpenGL Perspective Frustum(left,right,bottom,top,near,far) Angel/Shreiner/Möller 71

72 Using Field of View With Frustum it is often difficult to get the desired view Perpective(fovy,aspect,near,far) often provides a better interface front plane aspect = w/h Angel/Shreiner/Möller 72

73 Objectives What kind of camera we use? (pinhole) What projections make sense Orthographic Perspective The viewing pipeline Viewing in OpenGL Shadows 73

74 Application of projection: shadows Essential component for realistic rendering Naturally use projections Can produce hard shadows Only handles shadows on a plane To shadow on a polygonal face, need clipping More advanced shadow algorithms exist, e.g., soft shadows and penumbra (not easy to do) Blinn 74 74

75 Shadow via projection A shadow polygon is obtained through projection where the center of projection is a light source 75

76 Shadow polygon: parallel projection Project shadow on z = 0 Light at (directional light) Derive the projection matrix (xp, yp, zp) (xs, ys, 0) 76

77 Shadows: parallel projection Light at infinity: hence projection: Now draw the object on the ground plane 77

78 Shadows: perspective projection Project on plane z = 0 Point light source Derive the matrix 78

79 Local Light: Shadows: perspective projection projection matrix: 79

80 Shadow of a teapot 80

81 Shadows requires no extra memory easily handles any number of light sources only shadows onto ground plane cannot handle objects which shadow other complex objects every polygon is rendered N+1 times, where N is number of light sources 81