Computer Graphics Chapter 7 Three-Dimensional Viewing Viewing

Transcription

1 Computer Graphics Chapter 7 Three-Dimensional Viewing

2 Outline Overview of Three-Dimensional Viewing Concepts The Three-Dimensional Viewing Pipeline Three-Dimensional Viewing-Coorinate Parameters Transformation from Worl to Viewing Coorinates Projection Transformations Orthogonal Projections Oblique Parallel Projections Perspective Projections The Viewport Transformation an Three-Dimensional Screen Coorinates Three-Dimensional Clipping Algorithms 744 / 3628 Computer Graphics 2

3 Overview Many processes in 3-D viewing, such as the clipping routines, are similar to those in the two-imensional viewing pipeline, but some tasks that are not present in 2-D viewing 744 / 3628 Computer Graphics 3

4 Viewing a Three-Dimensional Scene First set up a coorinate reference for the viewing, or camera, parameters Defines the position an orientation for a view plane (or projection plane) that correspons to a camera film plane Object escriptions are then transferre to the viewing reference coorinates an projecte onto the view plane 744 / 3628 Computer Graphics 4

5 Projections Parallel projection Getting the escription of a soli object onto a view plane Project points on the object surface along parallel lines Perspective projection Generating a view of a three-imensional scene Project points to the view plane along converging paths 744 / 3628 Computer Graphics 5

6 Depth Cueing Can easily ientify, for a particular viewing irection, which is the front an which is the back of each isplaye object A wire-frame object is isplaye without epth information One simple metho for inicating epth with wire-frame isplays is to vary the brightness of line segments accoring to their istances from the viewing position Depth cueing is applie by choosing a maximum an a minimum intensity value an a range of istances over which the intensity is to vary Another one is moeling the effect of the atmosphere on the perceive intensity of objects More istant objects appear immer to us than nearer objects ue to light scattering by ust particles, hae, an smoke 744 / 3628 Computer Graphics 6

7 Ientifying Visible Lines an Surfaces Clarify epth relationships in a wire-frame isplay using techniques other than epth cueing Simply to highlight the visible lines or to isplay them in a ifferent color Common engineering technique is to isplay the nonvisible lines as ashe lines Remove the nonvisible lines - also removes information about the shape of the back surfaces of an object Wire-frame representations inicate overall appearance, front an back 744 / 3628 Computer Graphics 7

8 Surface Renering By renering object surfaces using the lighting conitions in the scene an the assigne surface characteristics Also set backgroun illumination effects Surface properties of objects - surface is transparent or opaque an whether the surface is smooth or rough 744 / 3628 Computer Graphics 8

9 The Three-Dimensional Viewing Pipeline Analogous to the processes involve in taking a photograph p First, choose a viewing position corresponing to where we woul place a camera Then, ecie on the camera orientation Finally, the scene is croppe to the sie of a selecte clipping winow 744 / 3628 Computer Graphics 9

10 The Three-Dimensional Viewing Pipeline (2) In 3-D, the clipping winow is positione on a selecte view plane Scenes are clippe against an enclosing volume of space, which is efine by a set of clipping planes Clipping region is calle the view volume Its shape an sie epens on the imensions of the clipping winow, the type of projection There are also a few other tasks that must be performe, such as ientifying visible surfaces an applying the surface-renering proceures 744 / 3628 Computer Graphics

11 Taxonomy of Planar Geometric Projections Planar Geometric Projections Parallel Perspective Multiview Axonometric Oblique point 2 point 3 point Orthographic Dimetric Trimetric Isometric 744 / 3628 Computer Graphics

12 Orthogonal Projections Transformation of object escriptions to a view plane along lines that are all parallel to the view-plane normal vector (or, equivalently, an orthographic projection) Projection lines are perpenicular to the view plane Often use to prouce the front, sie, an top views of an object Lengths an angles are accurately epicte an can be measure from the rawings 744 / 3628 Computer Graphics 2

13 Axonometric an Isometric Orthogonal Projections Axonometric projection isplays more than one face of an object Principal axes - coorinate axis in which the object is efine Isometric projection is generate by aligning the projection plane (or the object) so that t the plane intersects each principal axis at the same istance from the origin 744 / 3628 Computer Graphics 3

14 Orthogonal Projection Coorinates For any position (x, y, ) in viewing coorinates, the projection coorinates are x p = x, y p = y The -coorinate value is preserve for use in the visibility etermination proceures 744 / 3628 Computer Graphics 4

15 Orthogonal Projection Environment We will use a right- hane view system The eye or camera position is on the + axis, a istance from the origin The view irection is parallel to the axis The view plane is in the xy plane an passes through the origin y x 5

16 Orthogonal Parallel Projection A point in 3-space projects onto the viewing along rays parallel to the axis y (x, y, ) (x, y, ) What is (x, y, )? )? x 6

17 Orthogonal Parallel Projection Looking own the y axis: (x, y, ) (x, y, ) x So =, x = x 7

18 Orthogonal Parallel Projection Looking own the x axis: y (x, y, ) (x, y, ) So y =y 8

19 Orthogonal Parallel Projection Thus, for parallel, orthographic projections, x = x, y = y, = So, to perform a parallel projection on an object, we can use matrix multiplication What is M? M p' Mp i.e., we simply py rop the coorinate 9

20 Clipping Winow an Orthogonal- Projection View Volume Clipping winow is positione on the view plane with its eges parallel to the x view an y view axes The eges of the clipping winow specify the x an y limits These limits are use to form the top, bottom, an two sies of a clipping region calle the orthogonal-projection view volume 744 / 3628 Computer Graphics 2

21 Clipping Winow an Orthogonal- Projection View Volume Limit the extent of the orthogonal view volume in the view irection by selecting positions for one or two aitional bounary yplanes that are parallel to the view plane These two planes are calle the near-far clipping planes or front-back clipping planes 744 / 3628 Computer Graphics 2

22 Normaliation Transformation for an Orthogonal Projection Some graphics packages use a unit cube for this normalie view volume, with each of the x, y, an coorinates normalie in the range from to Another normaliation-transformation approach his to use a symmetric cube, with ih coorinates in the range from to 744 / 3628 Computer Graphics 22

23 Normaliation Transformation for an Orthogonal Projection Screen coorinates are often specifie in a left-hane Normalie coorinates also are often specifie in a left-hane Positive istances in the viewing i irection to be irectly interprete as istances from the screen 744 / 3628 Computer Graphics 23

24 Normaliation Transformation for an Orthogonal Projection Position (x min, y min, near ) is mappe to the normalie position (,, ), an position (x max, y max, far ) is mappe to (,, ) 744 / 3628 Computer Graphics 24

25 Viewport Transformation Clipping is applie with normaliation view volume After clipping visibility testing surface renering Viewport transformation is applie to generate the final screen isplay of the scene 744 / 3628 Computer Graphics 25

26 Oblique Parallel Projections The projection path is not perpenicular to the view plane, this mapping is calle an oblique parallel projection This projection can prouce combinations such as a front, sie, an top view of an object 744 / 3628 Computer Graphics 26

27 Oblique Parallel Projections in Drafting an Design An oblique parallel projection is often specifie with two angles, an A position (x, y, ) is oblique projecte to (x p,y p, vp )onaview plane Position (x, y, vp ) is the corresponing orthogonal-projection point Aviewplane view-plane line is with the length L is assigne a value between an 9 can vary from to / 3628 Computer Graphics 27

28 Oblique Parallel Projections in Drafting an Design The projection coorinates in terms of x, y, L, an are x p = x + L cos, y p = y + L sin tan = ( vp )/L L = L ( vp ) where L =cot x p = x + L ( ( vp ) ) cos, y p = y + L ( ( vp ) ) sin An orthogonal projection is obtaine when L = (9) The effect of an oblique parallel projection is to shear planes of constant an project them onto the view plane 744 / 3628 Computer Graphics 28

29 Oblique Parallel Projections in Drafting an Design The view plane is positione at the front face of a cube The back plane of the cube is sheare an overlappe with the front plane in the projection to the viewing surface A sie ege of the cube connecting the front an back planes is projecte into a line of length L That makes an angle with a horiontal line in the projection plane 744 / 3628 Computer Graphics 29

30 Cavalier an Cabinet Oblique Parallel Projections Typical choices for angle are 3 an 45 Two commonly use values for are tan= an tan=2 =45 an the views obtaine are calle cavalier projections All lines perpenicular to the projection plane are projecte with no change in length tan=2, ( 63.4) the resulting view is calle a cabinet projection All lines perpenicular to the viewing surface are projecte at half their length Cavalier Cabinet 744 / 3628 Computer Graphics 3

31 Clipping Winow an Oblique Parallel-Projection View Volume Clipping winow on the view plane with coorinate positions (xw min, yw min ) an (xw max, yw max ) Aing a near plane an a far plane The finite oblique parallelprojection view volume is an oblique parallelepipe l 744 / 3628 Computer Graphics 3

32 Oblique Parallel-Projection Transformation Matrix This matrix shifts the values of the x an y coorinates by an amount proportional to the istance from the view plane, which is at position vp on the view axis A general oblique parallel l projection matrix represents a -axis shearing transformation All coorinate positions within the oblique viewing volume are sheare by an amount proportional p to the istance from the view plane 744 / 3628 Computer Graphics 32

33 Perspective Projections In the real-worl, we see things in perspective Parallel lines o not look parallel They converge at some point By projecting objects to the view plane along converging paths to a position calle the projection reference point (or center of projection) pojec o Objects are isplaye with foreshortening effects Distant objects are smaller than the projections of objects of the same sie that are closer to the view plane 744 / 3628 Computer Graphics 33

34 Perspective Projection Points project through the focal point (e.g., i t) t th i l eyepoint) onto the view plane: Projection lines converge y x Virtual Image Plane 34

35 Perspective Projection Center of projection (COP) is no longer at infinity. Projection rays form a view frustum A pyrami with the tip at the COP eye view plane 35

36 Perspective Projection We will start with the projection plane parallel to the XY plane an perpenicular to the axis Lines parallel to the X or Y axis remain parallel X an Y istances become shorter as Z becomes more negative, e.g. a cube viewe in perspective: y x 36

37 Perspective Projection Computation Assume the projection plane is normal to the Z axis, locate at Z =. Assume the center of projection (COP, eyepoint) is locate at Z = What is p =(x, y, )? )? y projection plane p = (x,y,) p = (x, y, ) x Center of Projection 37

38 Perspective Projection Computation Looking own the y axis: - x p = (x, y, ) By similar triangles: x' x x x view plane eye x p = (x, y, ) x ' x' x' x x 38

39 Perspective Projection Computation Looking own the x axis: eye y p = (x, y, ) - p = (x, y, ) y y view plane By similar triangles: y' y y y ' y y' y y' 39

40 Perspective Projection Computation So, we have y ' x y y ' x x ' what is? y x,, ) ' ',, ' ( so we have, ' y x y x Can we put this into matrix form? 4

41 Perspective Projection Computation We want: So, x y ax ex by fy c g x a b c x y e f g h y i j k l m n o p, h, so, so, ix jy k l, so, i, j, k, l mx ny o p, so, m, n, o, p a e, f, b, c, g,, h 4

42 Perspective Projection Computation So, the matrix that will give us the correct perspective is: perspective is: x y M ba This works but what is the problem with it? Answer: The entries in the matrix are point epenent! i e every point will have to have a ifferent matrix 42 i.e., every point will have to have a ifferent matrix

43 Perspective Projection Computation How can we make the matrix not epen on the points? the points? The Cartesian point we want is,, y x which is equivalent to y x,, 43

44 Perspective Projection Computation Solution: use homogeneous points Our Cartesian point is: x, y, A homogeneous point that is equivalent to our esire Cartesian point is: x y can we come up with a matrix that gives us what we nee, but is point inepenent? 44

45 Perspective Projection Computation We want: So, x a b c x y e f g h y i j k l p m n o x ax by c, so, a, b, c, y ex fy g h, so, e, f, g, h ix jy k l, so, i, j, k, l mx ny o p, so, m, n, o, p 45

46 Perspective Projection Computation So, the new matrix we get is: M M per This gives us correct results an is point This gives us correct results, an is point inepenent 46

47 Focal Length is the focal length of the camera. What happens if we cut the focal length in half? What happens if we cut the focal length in half? Instea of we get,, y x,, y x What happens if we move all the objects twice as far away? 2 2 far away? Instea of we get,, y x, 2, 2 y x Decreasing the focal length is equivalent to making everything more istant. 47

48 Focal Length What woul happen if the focal length was really big? big? M per What oes this look like? Orthographic Projection O t og ap c oject o 48

49 Example Project the triangle with vertices P = (35, 7, -3) P 2 = (36, 35, -5) P 3 = (26, 26, -32) onto the view plane, LookFrom = (,, 4) y view plane P 2 P 3 P x 49

50 Computing the Projection Point P = M per P, with P = (35,7, -3): ' P p 3 p x (4 ) 7 (4 ),,,, ,4, y p 3 p 5

51 Computing the Projection Point P 2 = M per P 2 with P 2 = (36, 35, -5): ' P 2 p 2 36 p 3 35 y x (4 ) 35 (4 ),,,, ,6, p 5

52 Computing the Projection Point P 3 = M per P 3 with P 3 = (26, 26, -32): ' P 2 p 2 26 p 3 26 y x ,, ,7, (4 ) 26 (4 ),, p 52

53 Final Projection The results of projecting the polygon onto the view plane: View Plane P 3 = (7, 7, ) P 2 = (6, 6, ) P =(2, 4, ) 53

54 Perspective-Projection View Volume By aing near an far clipping planes that are perpenicular to the view axis (an parallel to the view plane) Perspective projection view volume form a truncate pyrami, or frustum, view volume Usually, both the near an far clipping planes are on the same sie of the projection reference point The far plane farther from the projection point than the near plane along the viewing i irection i 744 / 3628 Computer Graphics 54

55 Symmetric Perspective- Projection Frustum A line from the projection reference point pass through the center of the clipping winow If this centerline is perpenicular to the view plane, we have a symmetric frustum x wmin = x prp with/2, x wmax = x prp + with/2 y wmin = y prp height/2, y wmax = y prp + height/2 744 / 3628 Computer Graphics 55

56 Symmetric Perspective- Projection Frustum The cone of vision can be reference with a fiel-of-view angle, which is a measure of the sie of the camera lens tan(θ/2) = (height/2) eg )/( ( prp vp ) height = 2( prp vp ) tan(θ/2) prp vp = (height/2) cot(θ/2) =with cot(θ/2) / (2 aspect) 744 / 3628 Computer Graphics 56

57 Effects of Fiel-of-View Angles 744 / 3628 Computer Graphics 57

58 Symmetric Perspective- Projection Frustum When viewing volume is symmetric frustum Map locations insie the frustum to orthogonal projection coorinates within a rectangular parallelepipe Each projection line is converte by the perspective transformation to a line that is perpenicular p to the view plane 744 / 3628 Computer Graphics 58

59 Oblique Perspective-Projection Frustum Centerline of a perspectiveprojection view volume is not perpenicular to the view plane Can be converte to a symmetric frustum by applying the -axis shearing-transformation matrix Ajusts the centerline so that it is perpenicular to the view plane 744 / 3628 Computer Graphics 59

60 Normalie Perspective-ProjectionProjection Transformation Coorinates Centerline is now the view axis Scaling transforme frustum view volume into normalie view volume 744 / 3628 Computer Graphics 6

61 The Viewport Transformation an Three- Dimensional Screen Coorinates Positions throughout the 3-D view volume also have a epth ( coorinate) The epth information is for the visibility testing an surface-renering algorithms The normalie values within the symmetric cube are renormalie on the range from to. In normalie coorinates, the norm = face of the symmetric cube correspons to the clipping-winow area, screen = The lower-left corner of the viewport screen area is at position (xvmin, yvmin, ) an the upper-right corner is at (xvmax, yvmax, ) 744 / 3628 Computer Graphics 6

62 3-D Clipping Algorithms Apply three-imensional clipping algorithms to the normalie bounaries of the view volume For the symmetric cube, the equations for the 3-D clipping planes x wmin =, x wmax = y wmin =, y wmax = wmin =, wmax = 744 / 3628 Computer Graphics 62

63 Clipping in Three-Dimensional Homogeneous Coorinates Each coorinate position enters the viewing pipeline, it is converte to 4-D representation (x, y, ) (x, y,, ) M represents the concatenation of all the various transformations If the homogeneous parameter h oes have the value, the homogeneous coorinates are the same as the Cartesian projection coorinates 744 / 3628 Computer Graphics 63

64 Three-Dimensional Region Coes Use a six-bit region coe Bit positions in this region-coe example are numbere from right to left, referencing left, right, bottom, top, near, an far clipping planes 744 / 3628 Computer Graphics 64

65 Three-Dimensional Point an Line Clipping First, test the line enpoint region coes for trivial acceptance or rejection of the line For a line segment with enpoints P = (x h, y h, h, h ) P 2 = (x h2, y h2, h2, h 2 ) P = P + (P 2 P )u for u x h = x h + (x h2 x h )u y h = y h + (y h2 y h )u h = h +( ( h2 h )u h = h + (h 2 h )u Using the enpoint region coes can etermine which clipping i planes are intersecte t 744 / 3628 Computer Graphics 65

66 Three-Dimensional Point an Line Clipping (2) For example, this line intersects the right clipping plane x max = x p = x h / h = [x h +(x h2 x h )u] / [h+(h2 h)u] h) = u = (x h h) )/[( [(x h h) (x) ( h2 h2)] Next, etermine the values y p an p on this clipping plane, using the calculate value for u Test the remaining section of the line against the other clipping planes for possible rejection or for further intersection calculations 744 / 3628 Computer Graphics 66

67 Three-Dimensional Polygon Clipping First, test a polyheron for trivial acceptance or rejection using its coorinate extents Clip eges to obtain new vertex lists for the object surfaces Polygon surfaces are often ivie into triangular Use the Sutherlan-Hogman approach For concave polygons, apply splitting methos to obtain a set of triangles 744 / 3628 Computer Graphics 67

68 Three-Dimensional Curve Clipping First, check to etermine whether the coorinate extents of a curve object Solve the simultaneous set of surface equations an the clipping-plane equation Most graphics packages o not inclue clipping routines for curve objects Instea, curve surfaces are approximate as a set of polygon patches, an the objects are then clippe using polygon-clipping routines 744 / 3628 Computer Graphics 68

69 Arbitrary Clipping Planes Some graphics packages, clip a threeimensional scene using aitional planes that can be specifie in any spatial orientation Assuming that objects behin the plane are to be clippe Plane parameter: Ax + By + C + D < If (A, B, C, D) = (.,.,., 8.), then any coorinate position satisfying x + 8. <. (or, x < 8.) ) is clippe from the scene 744 / 3628 Computer Graphics 69

70 Arbitrary Clipping Planes (2) 744 / 3628 Computer Graphics 7

71 En of Chapter / 3628 Computer Graphics 7

72 Viewing Transformations Projection: take a point from m imensions to n imensions where n < m imensions where n < m There are essentially two types of viewing transforms: Orthographic: parallel projection Points project irectly onto the view plane In eye/camera space (after viewing transformation): rop Perspective: convergent projection Points project through the origin onto the view plane In eye/camera space (after viewing transformation): ivie by 95