Viewing. Cliff Lindsay, Ph.D. WPI

Transcription

1 Viewing Cliff Lindsa, Ph.D. WPI

2 Building Virtual Camera Pipeline l Used To View Virtual Scene l First Half of Rendering Pipeline Related To Camera l Takes Geometr From ApplicaHon To RasteriaHon Stages Virtual Camera

3 3D Viewing and View Volume Previousl: Lookat( ) to set camera position ow: Set view volume

4 Different View Volume Shapes Orthogonal view volume (no foreshortening) Perspective view volume (ehibits foreshortening) l Different view volume => different look l Foreshortening? ear objects bigger

5 View Volume Parameters l eed to set view volume parameters l Projec>on tpe: perspechve, orthographic, etc. l Field of view and aspect raho l ear and far clipping planes

6 Field of View l View volume parameter l Determines how much of world in picture (verhcall) l Larger field of view = smaller the objects are drawn field of view (view angle) center of projection θ

7 ear and Far Clipping Planes l Onl objects between near and far planes drawn ear plane Far plane

8 Viewing Frustrum l ear plane + far plane + field of view = Viewing Frustum l Objects outside the frustum are clipped ear plane Far plane Viewing Frustum

9 SeCng up View Volume/Projec>on Tpe l Previous OpenGL projechon commands deprecated!! l PerspecHve view volume/projechon: l gluperspec>ve(fov, aspect, near, far) or l glfrustum(lex, right, boyom, top, near, far) l Orthographic: l glortho(lex, right, boyom, top, near, far) l Useful funchons, so we implement similar in mv.js: l Perspec>ve(fov, aspect, near, far) or l Frustum(leX, right, boyom, top, near, far) l Ortho(leX, right, boyom, top, near, far) What are these arguments? et!

10 Perspec>ve(fov, aspect, near, far) l Aspect raho used to calculate window width ear plane fov w h Aspect = w / h near far

11 Frustum(leK, right, bomom, top, near, far) l Can use Frustrum( ) in place of Perspec>ve() l Same view volume shape, different arguments left top bottom right near far near and far measured from camera

12 Ortho(leK, right, bomom, top, near, far) l For orthographic projechon left top bottom right near far near and far measured from camera

13 Eample Usage: SeCng View Volume/Projec>on Tpe void displa() { // clear screen glclear(gl_color_buffer_bit);.. // Set up camera position LookAt(,,,,,,,,);.. // set up perspective transformation Perspective(fov, aspect, near, far); }.. // draw something displa_all(); // our displa routine

14 Review l SeZng Up & Moving The Camera l Look At FuncHon l View Volumes l ear & Far Clipping Planes

15 Taonom of Planar Geometric Projec>ons planar geometric projections parallel perspective multiview aonometric orthographic point 2 point 3 point oblique isometric dimetric trimetric 5

16 Perspec>ve Projec>on l AXer sezng view volume, then projechon transform l ProjecHon? l Classic: Converts 3D object to corresponding 2D on screen l How? Draw line from object to projechon center l Calculate where each cuts projechon plane

17 Orthographic Projec>on l How? Draw parallel lines from each object verte l The projechon center is at infinite l In short, use (,) coordinates, just drop coordinates Projection of Triangle in 2D Triangle In 3D

18 Homogeneous Coordinate Representa>on p = p = p = w p = Vertices before Projection Vertices after Projection default orthographic projection M = p p = Mp Default Projection Matri In prachce, can let M = I, set the term to ero later

19 Default View Volume/Projec>on? l What if ou user does not set up projechon? l Default on most sstems is orthogonal (Ortho( )); l To project points within default view volume p = p = p = Vertices before Projection Vertices after Projection Projection of Triangle in 2D Triangle In 3D

20 The Problem with Classic Projec>on l Keeps (,) coordintates for drawing, drops l We ma need. Wh? p = p = p = Classic Projection Loses value Projection of Triangle in 2D VerteTriangle In 3D

21 ormalia>on: Keeps Value l Most graphics sstems use view normalia-on l ormalia>on: convert all other projechon tpes to orthogonal projechons with the default view volume Perspec>ve transform matri Ortho transform matri Default view volume Clipping against it

22 Parallel Projec>on l normalia>on find 44 matri to transform user- specified view volume to canonical view volume (cube) User- specified View Volume Canonical View Volume For Eampl: glortho(left, right, bottom, top, near, far)

23 Parallel Projec>on: Ortho l Parallel projechon: 2 parts. Transla>on: centers view volume at origin l Thus translahon factors: - (right + lex)/2, - (top + boyom)/2, - (far+near)/2 ( right + left) / 2 ( top + bottom) / 2 ( far + near) / 2

24 Parallel Projec>on: Ortho 2. Scaling: reduces user- selected cuboid to canonical cube (dimension 2, centered at origin) l 2 right left Scaling factors: 2/(right - lex), 2/(top - boyom), 2/(far - near) 2 top bottom 2 far near

25 Parallel Projec>on: Ortho Concatenating Translation Scaling, we get Ortho Projection matri 2 right left 2 top bottom 2 far near X ( right + left) / 2 ( top + bottom) / 2 ( far + near) / 2 P = ST = 2 right left 2 top bottom 2 near far right left right left top + bottom top bottom far + near far near

26 Final Ortho Projec>on l Set = l Equivalent to the homogeneous coordinate transformahon l Hence, general orthogonal projechon in 4D is M orth = P = M orth ST

27 Perspec>ve Projec>on l ProjecHon map the object from 3D space to 2D screen Perspective() Frustrum( )

28 Perspec>ve Projec>on + Projection plane (,,) (,, ) (,,) Ee (COP) ear Plane Based on similar triangles: = - = -

29 Perspec>ve Projec>on l l So (*,*) projechon of point, (,,) unto near plane is given as: ( *, * ) umerical eample: =, Q. Where on the viewplane does P = (,.5, -.5) lie for a near plane at =? *, =, =,.5 =.5.5 ( * ) (.666,.333)

30 Pseudodepth l Classical perspechve projechon projects (,) coordinates to (*, *), drops coordinates Map to same (*,*) Compare their values (,,) l But we need to find closest object (depth teshng)!!!

31 Perspec>ve Transforma>on l Perspec>ve transforma>on maps actual distance of perspechve view volume to range [ to ] (Pseudodepth) for canonical view volume Actual depth Actual view volume We want perspective Transformation and OT classical projection!! ear Far Pseudodepth Canonical view volume Set scaling Pseudodepth = a + b et solve for a and b -

32 Perspec>ve Transforma>on using Pseudodepth l Choose a, b so as varies from ear to Far, pseudodepth * varies from to (canonical cube) l Boundar condihons l l ( *, *, *) * = - when = - * = when = - F a + b =,, ear Actual depth Actual view volume Z Far Pseudodepth Canonical view volume Z* -

33 Transforma>on of : Solve for a and b l Solving: a + b * = l Use boundar condihons l * = - when = - () l * = when = - F..(2) l Set up simultaneous equahons a + b = = a + b...() af + b = F = af + b...(2) F

34 Transforma>on of : Solve for a and b = a + F = af + b...() b...(2) l MulHpl both sides of () b - = a b...(3) l Add eqns (2) and (3) F + = a af a = F + F = ( F + )...(4) F l ow put (4) back into (3)

35 Transforma>on of : Solve for a and b l Put soluhon for a back into eqn (3) l So b F F + = ) ( b...(3) a = F F b + = ) ( F F F F F F F F b = + = + = 2 ) ( ) ( 2 2 F F a + = ) ( F F b = 2

36 What does this mean? l Original point in original view volume, transformed into * in canonical view volume a + b * = l where Original verte value ear Actual view volume Far ( F + ) a = F F b = 2 F Transformed verte * value - Canonical view volume

37 Homogenous Coordinates l Want to epress projechon transform as 44 matri l Previousl, homogeneous coordinates of P = (P,P,P) => (P,P,P,) l Introduce arbitrar scaling factor, w, so that P = (wp, wp, wp, w) (ote: w is non- ero) l For eample, the point P = (2,4,6) can be epressed as l (2,4,6,) l or (4,8,2,2) where w=2 l or (6,2,8,3) where w = 3, or. l To convert from homogeneous back to ordinar coordinates, first divide all four terms b w and discard 4 th term

38 Perspec>ve Projec>on Matri l Recall PerspecHve Transform l We have: l In matri form: + + = ) ( b a w b a w w w w w w w b a ( ) + = b a,, * *, *, * = * = b a + * = Perspective Transform Matri Original verte Transformed Verte Transformed Verte after dividing b 4 th term

39 Perspec>ve Projec>on Matri l In perspechve transform matri, alread solved for a and b: l So, we have transform matri to transform values + + = ) ( b a wp b ap w wp wp w wp wp wp b a F F a + = ) ( F F b = 2

40 Perspec>ve Projec>on l ot done et!! Can now transform! l l l l Also need to transform the = (lex, right) and = (boyom, top) ranges of viewing frustum to [-, ] Similar to Orthographic, we need to translate and scale previous matri along and to get final projechon transform matri we translate b l l (right + lex)/2 in - (top + boyom)/2 in Scale b: l l 2/(right lex) in 2/(top boyom) in top bottom left - right

41 Perspec>ve Projec>on l Translate along and to line up center with origin of CVV l l (right + lex)/2 in - (top + boyom)/2 in l MulHpl b translahon matri: ( right + left) / 2 ( top + bottom) / 2 top Line up centers Along and bottom left - right

42 Perspec>ve Projec>on l To bring view volume sie down to sie of of CVV, scale b l l 2/(right lex) in 2/(top boyom) in l MulHpl b scale matri: 2 right left 2 top bottom top Scale sie down along and bottom left - right

43 Perspec>ve Projec>on Matri glfrustum(left, right, bottom, top,, F) = near plane, F = far plane ) ( 2 2 F F F F bottom top bottom top bottom top left right left right left right ) / ( 2 ) / ( 2 2 b a bottom top left right bottom top left right Scale Final Perspective Transform Matri Translate Previous Perspective Transform Matri

44 ormalia>on Transforma>on distorted object projects correctl original clipping volume original object new clipping volume Top View of before & aker normalia>on

45 Implementa>on l Set modelview and projechon matrices in applicahon program l Pass matrices to shader void displa( ){... model_view = LookAt(ee, at, up); projection = Ortho(left, right, bottom,top, near, far); // pass model_view and projection matrices to shader gluniformmatri4fv(matri_loc,, GL_TRUE, model_view); gluniformmatri4fv(projection_loc,, GL_TRUE, projection);... }

46 Implementa>on l And the corresponding shader in vec4 vposition; in vec4 vcolor; Out vec4 color; uniform mat4 model_view; Uniform mat4 projection; void main( ) { gl_position = projection*model_view*vposition; color = vcolor; }

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