Chap 7, 2008 Spring Yeong Gil Shin

Transcription

1 Three-Dimensional i Viewingi Chap 7, 28 Spring Yeong Gil Shin

2 Viewing i Pipeline H d fi i d? How to define a window? How to project onto the window?

3 Rendering "Create a picture (in a synthetic camera) Specification of projection type Specification of viewing parameters: viewer's eye, viewing plane (viewing coordinates) Clipping in 3D: window, view volume Projection : the transformation of points from a coordinate system in n dimensions to a coordinate system in m dimensions where m<n Display: view port

4 General 3D Viewing i Pipeline Modeling coordinates (MC) World coordinates (WC) Viewing coordinates (VC) VRC, camera position Projection coordinates (PC) window, projection type Normalied coordinates (NC) Device coordinates (DC) view port in a screen

5 Projections Projection coordinate system: left-handed Core, DirectX why? h? - screen coordinate system is left-handed d right-handed - GKS, PHIGS, OpenGL, DirectX Projection plane (view plane): viewing surface where objects are projected. v n u

6 Viewing-Coordinate Parameters View reference point (VRP) The viewing origin in WC : Eye position, camera position View-plane (Projection plane) Locates on view ais : vp Perpendicular to view Viewing Coordinate : uvn Defined by P, N, VUP P = (, y, )

7 Viewing-Coordinate Parameters How we specify a view plane normal vector N [Way I] The origin of WC to a selected point position [Way II] The direction from a reference point P ref to the viewing origin: N = P P ref P ref : look-at point Viewing direction: N

8 Viewing-Coordinate Parameters View-up vector: VUP Specified in the world coordinates Used to establish the positive direction for the y view ais VUP should be perpendicular to N, but it can be difficult to a direction for VUP that is precisely perpendicular p to N y view VUP view N P = VRP view

9 Viewing-Coordinate Reference Frame (VRC) The camera orientation is determined by viewing reference frame view, y view, view (or uvn) The origin of the viewing reference frame : P (= VRP) uvn is called View Reference Coordinate (VRC) n = N = ( n, n y, n ) N VUP n u = VUP v = n u = ( u, u y, u = ( v, vy, v ) ) n VUP N v P View plane w.r.t. VRC u

10 World-to-Viewing Transformation Transformation from WC to VRC Translate the viewing-coordinate origin to the worldcoordinate origin Apply rotations to align the u, v, n aes with the world w, y w, w aes, respectively u v n

11 World-to-Viewing Transformation y v v v u u u y y T R = = n n n y y T R u n v P P u y u u u = =, P n P v T R M y y vc wc n n n v v v

12 Eample of VRC WC Given R VUP R R y R R = = VPN = VUP = R VUP VUP = 2.5 = det 2.5 = R R R R ( ) ( ) ( 6, 8, 7.5) i 6. j 8. = (.8,.6,.) VPN = (.48,.64,.6), k 7.5 = (.35,.48,.8) = 2.5 VUP ( ) R R R R = y y

13 Hierarchy of plane geometric projections plane geometric projections parallel perspective orthographic aonometric oblique trimetric cavalier cabinet dimetric isometric single-point two-point three-point

14 Parallel l Projections Parallel l direction of projection (DOP) Direction of projection (DOP) same for all points The parallel l projection of fthe point t( (,y,) on the y-plane gives ( + a,y + b,) When a = b =, the projection is said to be orthographic or orthogonal. Otherwise, it is oblique. Preserves relative dimension Orthographic parallel projections The direction of projection is normal to the projection plane Architectural, engineering drawings

15 Aonometric Parallel Projection Orthographic h parallel l projection that displays more than one face of an object Projection plane intersects each principal ais Classify by how many identical angles of a corner of a projected cube. Three: isometric Two: dimetric None : trimetric t i

16 Oblique Parallel Projections Parallel projection of which DOP (Direction of projection) is not perpendicular to the projection plane Only faces of the object parallel to the projection plane are shown true sie and shape DOP α A' B' Α Β C AB = A' B' B' C' = BC 2 α = arctan(/ 2) = C ' )

17 Oblique Parallel Projections cavalier projections DOP 45 cabinet projections DOP 63.4

18 Perspective vs. Parallel Projections Perspective Parallel

19 Perspective Projections Lines that are not parallel l to the projection plane converge to a a single point in the projection (the vanishing point) Lines parallel to one of the major ais come to a vanishing point, these are called (principle) ais vanishing points. Only three ais vanishing points in 3D space. Vanishing point

20 Projection Types

21 Projections Window a rectangular region on the projection plane Projection Reference Point (PRP) PRP is specified in the VRC system In general, (,,) Direction of projection (DOP): in a parallel projection, PRP CW Center of projection (COP): in a perspective projection, PRP = COP COP=PRP v u n window CW (Center of Window) Projection Plane

22 View Volume (View Frustum) 3D clipping region where we can see Defined by front and back clipping planes (which are parallel to view plane) (,,) (right,top,near) COP (left,bottom,near) far near,far are positive

23 Create a View Volume (left, top) far v near (right, bottom) u n CW is not aligned n-ais (left, right, top, bottom, near, far) n width far height v near u CW is aligned n-ais (width, height, ht near, far) n far CW is aligned n-ais v near (field of view, aspect, near, far) aspect = width/height u field of view

24 3D Viewing i ) Establishing a View Reference Coordinate System User supply the following parameters the view reference point (VRP) - in WC VPN - a vector in WC VUP - a vector in WC v n VRP u

25 3D Viewingi 2) View Mapping (WC VRC NPC) the projection type the Projection Reference Point (PRP) a view plane distance (= vp ) In DirectX, view plane is identical to near plane a back plane and a front plane distance 3) device-dependent dependent transformation v n PRP u

26 Planar Geometric Projections Mathematics of Planar Geometric Projections y P(,y,) P p ( p,y p,d) COP d Centre of projection at the origin Centre of projection at the origin Projection plane at =d

27 Planar Geometric Projections = = y d y d p p ; P(,y,) P (,y,d) = = = = d y y d y d d p p / ; / P p ( p,y p,d) d y P M Y X = = y d P M W Z per / = = d d y d y W X W Y W X d W p p p, /, / ),, (,, / d d W W W / /

28 Normaliing i Transformation Transform an arbitrary parallel- or perspectiveprojection view volume into the normalied or canonical view volume (in DirectX) or y or y - B - F - B - F parallel-projection canonical view volume - - perspective-projection canonical view volume

29 Normaliing i Transformation OpenGL

30 Normaliing transformation for parallel projections. Translate VRP to the origin of the WC 2. Rotate VRC such that n ais = ais, u ais = ais and v ais = y ais. (By & 2, transformation from WCS to VRC) 3. Shear so the direction of projection parallel to the ais. (not necessary for orthographic projections) 4. Translate and scale into the parallel-projection canonical view volume

31 Normaliing transformation for parallel projections [Step2] R ( θ ) R ( φ ) R ( α ) ( y VUP n Or use orthogonal matri properties v VRP VPN R u VPN VUP R = R = R y = R R VPN VUP R r r2 r3 ry r2y r3y R = r r r 2 3

32 Normaliing transformation for parallel projections [Step 3] shearing after step2, VRC = WC DOP = CW - PRP y DOP y VPN DOP - VPN -

33 Normaliing transformation for parallel projections -component of DOP is invariant. a b SH a b (, ) = DOP = CW PRP = dop wc DOP' = a dop = dop, a b DOP dopy b = dop ( dop dop ) wc = y dop

34 Normaliing transformation for parallel projections View volume after transformation steps to 3 y (u ma, v ma, B) - Step4 B - - or y F (u min, v min, F) -

35 Normaliing transformation for parallel projections [Step 4] translate and scale. Translate satethe front center of the view volume ou uma + umin v + vmin T = ma par T,, F Scale to the 2 2 sie 2 2 S par = S,, uma umin vma vmin F B

36 Normaliing transformation for perspective projection. Translate VRP to the origin of the WC: T( VRP) 2. Rotate VRC such that n ais = ais, u ais = ais and v ais = y ais 3. Translate such that PRP=(prp u, prp v, prp n ) is at the origin: T( PRP) 4. Shear so the center line of the view volume becomes the -ais 5. (Scale such that the view volume becomes the canonical perspective view volume.)

37 Normaliing transformation for perspective projection After step,2,3 Step 4 v n u PRP v n u

38 Normaliing transformation for perspective projection [Step 4] shearing shear so that CW PRP is into ais SH per = Sh par Another Way: VRP' = SH per T(-PRP) [ ] T component of VRP': vrp = prp n PRP

39 Normaliing transformation for perspective projection [Step 5] scale. Scale and y to give the sloped planes bounding the view-volume unit slope. Scale the window so its half-height eg tand half-width are both vrp scale : y scale : 2 vrp' ( u ma u min ) 2 vrp' ( v ma v min )

40 Normaliing transformation for perspective projection 2. Scale uniformly all three aes such that the back clipping plane =vrp +B becomes. scale factor: /(vrp +B) Perspective scale transformation S per = 2vrp' 2vrp' S,, ( uma umin )( vrp' + B) ( vma vmin )( vrp' + B) ( vrp' + B)

41 DirectX: Viewing Transformation How to define VRC Eye-Point (= VRP) Look-At Position Look-At Position Eye-Point N Up-Vector ( v) v Eye n y LHS Up-vector Look-at Synta D3DXMATRIX *D3DXMatriLookAtLH( D3DXMATRIX *pout, CONST D3DXVECTOR3 *peye, CONST D3DXVECTOR3 *pat, CONST D3DXVECTOR3 *pup ); Parameters pout : Pointer to the D3DXMATRIX structure that is the result of the operation. peye : Pointer to the D3DXVECTOR3 structure that defines the eye point. pat : Pointer to the D3DXVECTOR3 structure that defines the camera look-at target. pup : Pointer to the D3DXVECTOR3 structure that defines the current world's up, usually [,, ]. u

42 DirectX: Perspective Projection Transformation When CW aligns n-ais Synta D3DXMATRIX *D3DXMatriPerspectiveFovLH( D3DXMATRIX *pout, FLOAT fovy, FLOAT Aspect, FLOAT n, FLOAT f ); Parameters pout : Pointer to the D3DXMATRIX structure that is the result of the operation. fovy : Field of view in the y direction, in radians. Aspect : Aspect ratio, defined as view space width divided by height. n : Z-value of the near view-plane. n f : Z-value of the far view-plane. v u near far

43 DirectX: Perspective Projection Transformation When CW does not align n-ais Synta D3DXMATRIX *D3DXMatriPerspectiveOffCenterLH (D3DXMATRIX *pout, FLOAT l, FLOAT r, FLOAT b, FLOAT t, FLOAT n, FLOAT f ); Parameters pout : Pointer to the D3DXMATRIX structure that is the result of the operation. l : Minimum -value of the view volume. r : Maimum -value of the view volume. n b : Minimum y-value of the view volume. t : Maimum y-value of the view volume. n : Minimum -value of the view volume. f : Maimum -value of the view volume. far v u near

44 DirectX: Orthographic Parallel Projection Transformation Synta D3DXMATRIX *WINAPI D3DXMatriOrthoLH( D3DXMATRIX *pout, FLOAT w, FLOAT h, FLOAT n, FLOAT f ); Parameters pout : Pointer to the D3DXMATRIX structure that contains the resulting matri. w : Width of the view volume. h : Height of the view volume. n : Minimum -value of the view volume which is referred to as -near. f : Maimum -value of the view volume which is referred to as -far. w n v u near far h

45 DirectX: Oblique Parallel Projection Transformation Synta D3DXMATRIX *D3DXMatriOrthoOffCenterLH (D3DXMATRIX *pout, FLOAT l, FLOAT r, FLOAT b, FLOAT t, FLOAT n, FLOAT f ); Parameters pout : Pointer to the D3DXMATRIX structure that is the result of the operation. l : Minimum -value of view volume. r : Maimum -value of view volume. b : Minimum y-value of view volume. t : Maimum y-value of view volume. n : Minimum -value of the view volume. f : Maimum -value of the view volume. far v near u n

46 3D Clipping i For orthographic projection, view volume is a bo For perspective p projection, view volume is a frustum left Near clipping plane Far clipping plane. right Need to calculate intersection with 6 planes

47 3D Clipping i Clipping is efficiently done on the normalied view volume. The canonical parallel projection view volume is defined by:, y, Clip primitives against this view volume

48 3D Region Coding for Clipping 3-D Etension of 2-D Cohen-Sutherland Algorithm For parallel-projection canonical view volume : Bit - point is above view volume: y > Bit 2 - point is below view volume: y < Bit 3 - point is right of view volume: > Bit 4 - point is left view volume: < Bit 5 - point is behind view volume: < Bit 6 - point is in front of view volume: > B or y F

49 3D Region Coding for Clipping

50 3D Region Coding for Clipping For perspective-projection canonical view volume y + F y=- B - - Bit - Point is above view volume y > Bit 2 - Point is below view volume y < Bit 3 - Point is right of view volume > Bit 4 - Point is left of view volume < Bit 5 - Point is behind view volume < - y= Bit 6 - Point is in front of view volume > min

51 Clipping and Homogeneous Coordinates Efficient to transform frustum into perspective canonical view volume unit slope planes Even better to transform to parallel canonical view volume Clipping i must be done in homogeneous coordinates We do not need to [X,Y,Z,W] [,y,,] for the clipped region Points in homogeneous coordinate can appear with W and cannot be clipped properly in 3D

52 Clipping and Homogeneous Coordinates 3D parallel projection volume is defined by:, y, Replace by X/W,Y/W,Z/W: X/W, Y/W, Z/W Corresponding plane equations are : X= W, X=W, Y= W, Y=W, Z= W, Z= If W>, multiplication li by W does not change sign. W>: W X W, W Y W, W Z However if W<, need to change sign : W<: W X W, W Y W, W Z

53 Clipping and Homogeneous Coordinates For the canonical parallel projection volume:,, yy, To clip to = (left): Homogeneous coordinate: Clip to X/W = Homogeneous plane: W+X = Point is visible if W+X >

54 Clipping and Homogeneous Coordinates The intersection of the line segment with a clipping plane: P 2 = ( α)p + αp and w + = [( α) w + αw2 ] + [( α) + α2] = α = (w + + w 2 ) (w ) Repeat for remaining boundaries : other Near and Far clipping planes

55 Clipping and Homogeneous Coordinates (eample) P = [ 2, y*, *,2] P = [, y2*, 2*, 2 ] 2 OB region P 2 P α = P (w + w 2 + ) (w 2 2 ) ( 2) ( 2) 2 = = = [ 2, y * 8 ( * *), * 8 ( * 3 + y 9 2 y * *), 2 ] 3 Projected - coordinate of P* = w * =

56 Points in Homogeneous Coordinates W Region A P =[ 2 3 4] T W= Projection of P and dp 2 onto W= plane X or Y Region B P 2 =[ 2 3 4] T Need to consider both regions when Performing clipping

57 Lines in Homogeneous Coordinates Could clip twice once for region B, once for region A. Epensive Check for negative W values and negate points before clipping P W Projection of points W= X P2

58 Lines in Homogeneous Coordinates P W Clipped lines W=X W= X W= X P2

59 3D Polygon Clipping Algorithms Bounding bo or sphere test for early rejection Sutherland-Hodgman and Weiler-Atherton algorithms can be generalied

60 What s Net