Rendering Pipeline and Coordinates Transforms

Transcription

1 Rendering Pipeline and Coordinates Transforms Alessandro Martinelli 16 October 2013 Rendering Pipeline (3): Coordinates Transforms Rendering Architecture First Rendering Pipeline Second Pipeline: Illumination and Shading Third Pipeline: Coordinates Transforms Fourth Pipeline: Texturing Fifth Pipeline: Memory Structures Programmable Graphics Hardware Computer Graphics

2 Planar 2D Transforms Rendering Pipeline (3): Coordinates Transforms Rendering Architecture First Rendering Pipeline Second Pipeline: Illumination and Shading Third Pipeline: Coordinates Transforms Planar 2D Transforms Spatial 3D Transforms Homogeneous Coordinates Parallel Projection, Perspective Projection and Perspective Transforms The SceneGraph and OpenGL Transforms Fourth Pipeline: Texturing Fifth Pipeline: Memory Structures Programmable Graphics Hardware A. Martinelli Transforms 16/10/ / 39

3 Planar 2D Transforms Planar 2D Coordinates Transforms Planar Coordinates Transforms move a coordinates couple (x, y) into a new position (x,y ) And they are used to move virtual objects on the plane. (x,y) (x,y ) (1) A. Martinelli Transforms 16/10/ / 39

4 Planar 2D Transforms Planar 2D Coordinates Transforms Planar Coordinates Transforms move a coordinates couple (x, y) into a new position (x,y ) (x,y) (x,y ) (1) And they are used to move virtual objects on the plane. The Transforms we are considering are linear transforms, and they are described with this formulas: x = ax +by +e y = cx +dy +f (2) A. Martinelli Transforms 16/10/ / 39

5 Planar 2D Transforms Planar 2D Coordinates Transforms Planar Coordinates Transforms move a coordinates couple (x, y) into a new position (x,y ) (x,y) (x,y ) (1) And they are used to move virtual objects on the plane. The Transforms we are considering are linear transforms, and they are described with this formulas: x = ax +by +e y = cx +dy +f (2) With matrix notation a+b : [ [ x a b y = c d [ x y [ e + f You should remember that the matrix-vector product is the result of the dot product of each row in the matrix with the vector itself (3) A. Martinelli Transforms 16/10/ / 39

6 Planar 2D Transforms Planar Transforms: Reference Systems Each Planar Transform is related to a Reference System [ [ [ [ x a b x e y = + c d y f (4) A. Martinelli Transforms 16/10/ / 39

7 Planar 2D Transforms Planar Transforms: Reference Systems Each Planar Transform is related to a Reference System [ [ [ [ x a b x e y = + c d y f (4) a+b If If If [ x y [ x y [ x y [ 0 = 0 [ 1 = 0 [ 0 = 1 then then then [ [ x e y = f [ [ x a y = c [ [ x b y = d [ e + f [ e + f (5) [ b d [ e + f [ 0 1 [ 0 0 [ 1 0 [ e f [ a c [ e + f A. Martinelli Transforms 16/10/ / 39

8 Planar 2D Transforms Planar Transforms: Translation a+b Translation moves the Reference System of an Object without changing its axes [ [ [ [ x 1 0 x e y = + (6) 0 1 y f A. Martinelli Transforms 16/10/ / 39

9 Planar 2D Transforms Planar Transforms: Translation a+b Translation moves the Reference System of an Object without changing its axes [ [ [ [ x 1 0 x e y = + (6) 0 1 y f [ [ 0 e [ + 1 f 0 1 [ 0 0 [ 1 0 [ e f [ 1 0 [ e + f A. Martinelli Transforms 16/10/ / 39

10 Planar 2D Transforms Planar Transforms: Scaling a+b Scaling applies a change in the axes size [ [ [ x sx 0 x y = 0 s y y [ (7) A. Martinelli Transforms 16/10/ / 39

11 Planar 2D Transforms Planar Transforms: Scaling a+b Scaling applies a change in the axes size [ [ [ x sx 0 x y = 0 s y y [ 0 sy [ (7) [ 0 1 [ 0 0 [ 1 0 [ sx0 A. Martinelli Transforms 16/10/ / 39

12 Planar 2D Transforms Planar Transforms: Rotation around the origin (1/3) Planar Rotations rotate the axes with an angle α As for translations, rotations are rigid transforms, because they keep fixed dimensions and angles. The points (1,0) and (0,1) will move on the Unit Circumference A. Martinelli Transforms 16/10/ / 39

13 Planar 2D Transforms Planar Transforms: Rotation around the origin (1/3) Planar Rotations rotate the axes with an angle α As for translations, rotations are rigid transforms, because they keep fixed dimensions and angles. The points (1,0) and (0,1) will move on the Unit Circumference [ 0 1 [?? α [ 0 0 α [?? [ 1 0 Unit Circumference A. Martinelli Transforms 16/10/ / 39

14 Planar 2D Transforms Planar Transforms: Rotation around the origin (2/3) a+b The Point (1,0) got moved to (cos(α),sin(α)) The Point (0,1) got moved to (cos(α+ π 2 ),sin(α+ π 2 )) = ( sin(α),cos(α)) a+b A. Martinelli Transforms 16/10/ / 39

15 Planar 2D Transforms Planar Transforms: Rotation around the origin (2/3) a+b The Point (1,0) got moved to (cos(α),sin(α)) The Point (0,1) got moved to (cos(α+ π 2 ),sin(α+ π 2 )) = ( sin(α),cos(α)) a+b [ sin(α) cos(α) [ 0 1 α [ 0 0 α [ cos(α) sin(α) [ 1 0 Unit Circumference A. Martinelli Transforms 16/10/ / 39

16 Planar 2D Transforms Planar Transforms: Rotation around the origin (3/3) a+b The set of all the 2D rotation matrices is defined as : [ [ [ x cos(α) sin(α) x y = sin(α) cos(α) y (8) Rotation matrices are also called orthogonal matrices, because they transform π 2 angles into π 2 angles (they indeed keep the dimensions of any angle). Rotation matrices are also called normal matrices, because they transform unit vectors into unit vectors (they indeed keep the size of any vector) A. Martinelli Transforms 16/10/ / 39

17 Planar 2D Transforms Applying Transforms(1/2) Let s suppose to apply 2 transforms which move the point (x,y) into the point (x 1,y 1 ) (T 1 ) and (x 1,y 1 ) into (x 2,y 2 ) (T 2 ). [ x1 = y 1 [ a1 b 1 c 1 d 1 [ x y [ [ [ [ [ e1 x2 a2 b + = 2 x1 e2 + f 1 y 2 c 2 d 2 y 1 f 2 (9) (x 2,y 2 ) is the result of applying 2 transforms on (x,y) [ [ ([ [ [ )[ x2 a2 b = 2 a1 b 1 x e1 x + y 2 c 2 d 2 c 1 d 1 y f 1 y The order in which this transforms are applied is relevant. [ e2 + f 2 (10) A. Martinelli Transforms 16/10/ / 39

18 Planar 2D Transforms Applying Transforms(2/2) Example [ [ a1 b 2 2 T 1 : 1 = 2 2 c 1 d , [ [ e1 0 = f 1 0 [ [ a2 b T 2 : 2 10 = c 2 d 2 01 [ [ e2 1, = f 2 0 (11) The first transform is a π 4 rotation, the second is a translation. A. Martinelli Transforms 16/10/ / 39

19 Planar 2D Transforms Applying Transforms(2/2) Example [ [ a1 b 2 2 T 1 : 1 = 2 2 c 1 d , [ [ e1 0 = f 1 0 [ [ a2 b T 2 : 2 10 = c 2 d 2 01 [ [ e2 1, = f 2 0 (11) The first transform is a π 4 rotation, the second is a translation. First case: T 1 before T 2 [ [ x2 1 0 = y ([ [ x y [ ) [ [ 2 = 2 x 2 2 y x y (12) A. Martinelli Transforms 16/10/ / 39

20 Planar 2D Transforms Applying Transforms(2/2) Example [ [ a1 b 2 2 T 1 : 1 = 2 2 c 1 d , [ [ e1 0 = f 1 0 [ [ a2 b T 2 : 2 10 = c 2 d 2 01 [ [ e2 1, = f 2 0 (11) The first transform is a π 4 rotation, the second is a translation. First case: T 1 before T 2 [ [ x2 1 0 = y ([ [ x y [ ) [ [ 2 = 2 x 2 2 y x y (12) First case: T 2 before T 1 [ [ 2 2 x2 = 2 2 y ([ [ x y [ ) [ [ 2 = 2 (x +1) 2 2 y 2 2 (x +1)+ 2 2 y (13) A. Martinelli Transforms 16/10/ / 39

21 Spatial 3D Transforms Rendering Pipeline (3): Coordinates Transforms Rendering Architecture First Rendering Pipeline Second Pipeline: Illumination and Shading Third Pipeline: Coordinates Transforms Planar 2D Transforms Spatial 3D Transforms Homogeneous Coordinates Parallel Projection, Perspective Projection and Perspective Transforms The SceneGraph and OpenGL Transforms Fourth Pipeline: Texturing Fifth Pipeline: Memory Structures Programmable Graphics Hardware A. Martinelli Transforms 16/10/ / 39

22 Spatial 3D Transforms Spatial 3D Transforms (1/3) 3D Transforms are described similarly to the 2D case [ x [ [ a b c x = d e f y g h i z y z A 3D Transform is related to a 3D Reference System (0,0,0) is moved to (l,m,n) (1,0,0) is moved to (l,m,n)+(a,d,g) (0,1,0) is moved to (l,m,n)+(b,e,h) (0,0,1) is moved to (l,m,n)+(c,f,i) a+b : +[ l m n (14) A. Martinelli Transforms 16/10/ / 39

23 Spatial 3D Transforms Spatial 3D Transforms (2/3) b e h + l m n c f i + l m n l m n a d g + l m n A. Martinelli Transforms 16/10/ / 39

24 Spatial 3D Transforms Spatial 3D Transforms (2/3) Translation a+b : [ x y z = [ [ x y z + [ tx t y t z (15) Scaling a+b : [ x y z = [ sx s y s z [ x y z +[ (16) gltranslatef(t x,t y,t z ); glscalef(s x,s y,s z ); A. Martinelli Transforms 16/10/ / 39

25 Spatial 3D Transforms Rotazione: Angoli di Eulero (1/2) While 2D rotations are easy, because of the unique rotation axes, 3D Rotations are more complex because the possible rotation axes are infinite. A. Martinelli Transforms 16/10/ / 39

26 Spatial 3D Transforms Rotazione: Angoli di Eulero (1/2) While 2D rotations are easy, because of the unique rotation axes, 3D Rotations are more complex because the possible rotation axes are infinite. Rotazioni di Eulero Euler Rotations are defined starting from Euler angles: α x,α y,α z. R = R x (α x )R y (α y )R z (α z ) (17) Where each rotation is the rotation around one of the 3 Cartesian axes. A. Martinelli Transforms 16/10/ / 39

27 Spatial 3D Transforms Rotazione: Angoli di Eulero (1/2) While 2D rotations are easy, because of the unique rotation axes, 3D Rotations are more complex because the possible rotation axes are infinite. Rotazioni di Eulero Euler Rotations are defined starting from Euler angles: α x,α y,α z. R = R x (α x )R y (α y )R z (α z ) (17) Where each rotation is the rotation around one of the 3 Cartesian axes. R z (α z ) is the rotation around the z axes, that is the rotation around the origin in a+b the xy plane. [ x [ [ +cz s z 0 x c z = cos(α z ),s z = sin(α z ) y = +s z +c z 0 y (18) z z A. Martinelli Transforms 16/10/ / 39

28 Spatial 3D Transforms Rotazione: Angoli di Eulero (2/2) R x (α x ) is the rotation around the x axes, that is the rotation around the origin in a+b the yz plane. [ x [ [ x c x = cos(α x ),s x = sin(α x ) y = 0 +c x s x y (19) z 0 +s x +c x z A. Martinelli Transforms 16/10/ / 39

29 Spatial 3D Transforms Rotazione: Angoli di Eulero (2/2) R x (α x ) is the rotation around the x axes, that is the rotation around the origin in a+b the yz plane. [ x [ [ x c x = cos(α x ),s x = sin(α x ) y = 0 +c x s x y (19) z 0 +s x +c x z R y (α y ) is the rotation around the y axes, that is the rotation around the origin in a+b the zx plane plane. [ x [ [ +cy 0 +s y x c y = cos(α y ),s y = sin(α y ) y = y (20) z s y 0 +c y z A. Martinelli Transforms 16/10/ / 39

30 Spatial 3D Transforms Rotazione intorno ad asse generico Any rotation can be described as the rotation around some (generic) axes. D D 2 α D 1 The Rotation Matrix is built starting from the matrix C D describing a cross product with D a+b???? [ 0 Dz D y C = D z 0 D y D y D x 0, R(α,D) = I +sin(α)c +(1 cos(α))c 2 (21) glrotatef(α,d x,d y,d z ); Euler Transforms: glrotatef(α x,1,0,0);glrotatef(α y,0,1,0);glrotatef(α z,0,0,1); A. Martinelli Transforms 16/10/ / 39

31 Homogeneous Coordinates Rendering Pipeline (3): Coordinates Transforms Rendering Architecture First Rendering Pipeline Second Pipeline: Illumination and Shading Third Pipeline: Coordinates Transforms Planar 2D Transforms Spatial 3D Transforms Homogeneous Coordinates Parallel Projection, Perspective Projection and Perspective Transforms The SceneGraph and OpenGL Transforms Fourth Pipeline: Texturing Fifth Pipeline: Memory Structures Programmable Graphics Hardware A. Martinelli Transforms 16/10/ / 39

32 Homogeneous Coordinates Homogeneous Coordinates(1/2) Homogeneous Coordinates are a very smart instrument in order to describe coordinates transforms. A. Martinelli Transforms 16/10/ / 39

33 Homogeneous Coordinates Homogeneous Coordinates(1/2) Homogeneous Coordinates are a very smart instrument in order to describe coordinates transforms. The transform: is equivalent to: [ x y z = [ a b c d e f g h i [ x y z x = ax +by +cz +l y = dx +ey +fz +m z = gx +hy +iz +n +[ l m n (22) (23) A. Martinelli Transforms 16/10/ / 39

34 Homogeneous Coordinates Homogeneous Coordinates(1/2) Homogeneous Coordinates are a very smart instrument in order to describe coordinates transforms. The transform: is equivalent to: [ x y z = [ a b c d e f g h i [ x y z x = ax +by +cz +l y = dx +ey +fz +m z = gx +hy +iz +n +[ l m n Which may be written in this way (marking the products with ): x = a x +b y +c z +l 1 y = d x +e y +f z +m 1 z = g x +h y +i z +n 1 1 = 0 x +0 y +0 z +1 1 (22) (23) (24) A. Martinelli Transforms 16/10/ / 39

35 Homogeneous Coordinates Homogeneous Coordinates(2/2) the equation we have seen: x = a x +b y +c z +l 1 y = d x +e y +f z +m 1 z = g x +h y +i z +n 1 1 = 0 x +0 y +0 z +1 1 (25) A. Martinelli Transforms 16/10/ / 39

36 Homogeneous Coordinates Homogeneous Coordinates(2/2) the equation we have seen: x = a x +b y +c z +l 1 y = d x +e y +f z +m 1 z = g x +h y +i z +n 1 1 = 0 x +0 y +0 z +1 1 (25) Can be written (again) with matrix notation x y z = 1 a+b a b c l d e f m g h i n : x y z 1 (26) A. Martinelli Transforms 16/10/ / 39

37 Homogeneous Coordinates Homogeneous Coordinates(2/2) the equation we have seen: x = a x +b y +c z +l 1 y = d x +e y +f z +m 1 z = g x +h y +i z +n 1 1 = 0 x +0 y +0 z +1 1 (25) Can be written (again) with matrix notation x y z = 1 a+b a b c l d e f m g h i n : x y z 1 (26) (x,y,z,1) T is a vector written with Homogeneous Coordinates. Using Homogeneous Coordinates, we can write any linear 3D transform using a 4x4 matrix. A. Martinelli Transforms 16/10/ / 39

38 Homogeneous Coordinates Projective Geometry A vector written with Homogeneous Coordinates is obtained by adding the component w: (x,y,z,w) T (27) A. Martinelli Transforms 16/10/ / 39

39 Homogeneous Coordinates Projective Geometry A vector written with Homogeneous Coordinates is obtained by adding the component w: (x,y,z,w) T (27) Projective Geometry studies vectors written with Homogeneous Coordinates. To each vector written with Homogeneous Coordinates there is a 3D vector a+b like this : ( x (X,Y,Z) T = w, y w, z ) T (28) w A. Martinelli Transforms 16/10/ / 39

40 Homogeneous Coordinates Projective Geometry A vector written with Homogeneous Coordinates is obtained by adding the component w: (x,y,z,w) T (27) Projective Geometry studies vectors written with Homogeneous Coordinates. To each vector written with Homogeneous Coordinates there is a 3D vector a+b like this : ( x (X,Y,Z) T = w, y w, z ) T (28) w An Infinit Points Set (x,y,z,w) T will map to a unique (X,Y,Z). Points having w = 0 are called Point at infinity. A. Martinelli Transforms 16/10/ / 39

41 Homogeneous Coordinates Projective Geometry A vector written with Homogeneous Coordinates is obtained by adding the component w: (x,y,z,w) T (27) Projective Geometry studies vectors written with Homogeneous Coordinates. To each vector written with Homogeneous Coordinates there is a 3D vector a+b like this : ( x (X,Y,Z) T = w, y w, z ) T (28) w An Infinit Points Set (x,y,z,w) T will map to a unique (X,Y,Z). Points having w = 0 are called Point at infinity. Note Our Rendering Pipeline treats transforms as 4x4 matrices and vectors as 4-components vectors with Homogeneous Coordinates. Vertices and Vectors are transformed into 3 coordinates vectors applying the projective relation, BEFORE any rasterization process. A. Martinelli Transforms 16/10/ / 39

42 Parallel Projection, Perspective Projection and Perspective Transforms Rendering Pipeline (3): Coordinates Transforms Rendering Architecture First Rendering Pipeline Second Pipeline: Illumination and Shading Third Pipeline: Coordinates Transforms Planar 2D Transforms Spatial 3D Transforms Homogeneous Coordinates Parallel Projection, Perspective Projection and Perspective Transforms The SceneGraph and OpenGL Transforms Fourth Pipeline: Texturing Fifth Pipeline: Memory Structures Programmable Graphics Hardware A. Martinelli Transforms 16/10/ / 39

43 Parallel Projection, Perspective Projection and Perspective Transforms Projection Matrices During the Rendering Process, all the vertices are transformed using a 4x4 transform which is the projection matrix. This matrix resemble a camera placed somewhere in the 3D space. A. Martinelli Transforms 16/10/ / 39

44 Parallel Projection, Perspective Projection and Perspective Transforms Projection Matrices During the Rendering Process, all the vertices are transformed using a 4x4 transform which is the projection matrix. This matrix resemble a camera placed somewhere in the 3D space. Projection Matrix A 4x4 Homogeneous Coordinates Transform Matrix defining the position of the world 3D with respect to the camera space. A. Martinelli Transforms 16/10/ / 39

45 Parallel Projection, Perspective Projection and Perspective Transforms Projection Matrices During the Rendering Process, all the vertices are transformed using a 4x4 transform which is the projection matrix. This matrix resemble a camera placed somewhere in the 3D space. Projection Matrix A 4x4 Homogeneous Coordinates Transform Matrix defining the position of the world 3D with respect to the camera space.... so : not the position of the camera in the world, but its inverse matrix!!! A. Martinelli Transforms 16/10/ / 39

46 Parallel Projection, Perspective Projection and Perspective Transforms Projection Matrices During the Rendering Process, all the vertices are transformed using a 4x4 transform which is the projection matrix. This matrix resemble a camera placed somewhere in the 3D space. Projection Matrix A 4x4 Homogeneous Coordinates Transform Matrix defining the position of the world 3D with respect to the camera space.... so : not the position of the camera in the world, but its inverse matrix!!! This is because the rasterization process will be performed in the camera internal reference system. Our Virtual World got projected into the camera space, and the projection matrix tells us how this projection is performed. A. Martinelli Transforms 16/10/ / 39

47 Parallel Projection, Perspective Projection and Perspective Transforms Parallel Projection Matrix Parallel Projection Matrices We can build a simple projection matrix buy applying rigid transforms like rotations and translation, or a scaling. In this way we will create a World projection through parallel lines. A. Martinelli Transforms 16/10/ / 39

48 Parallel Projection, Perspective Projection and Perspective Transforms Parallel Projection Matrix Parallel Projection Matrices We can build a simple projection matrix buy applying rigid transforms like rotations and translation, or a scaling. In this way we will create a World projection through parallel lines. An Important case: The Identity Projection The 4x4 Identity matrix is not projecting at all. It is like a default projection on the z axes. A. Martinelli Transforms 16/10/ / 39

49 Parallel Projection, Perspective Projection and Perspective Transforms Parallel Projection Matrix Parallel Projection Matrices We can build a simple projection matrix buy applying rigid transforms like rotations and translation, or a scaling. In this way we will create a World projection through parallel lines. An Important case: The Identity Projection The 4x4 Identity matrix is not projecting at all. It is like a default projection on the z axes. Starting from the identity matrix and applying rotations, we will have a change in the projection direction. NOTE: the projection will still be performed using the z axes in the camera reference system A. Martinelli Transforms 16/10/ / 39

50 Parallel Projection, Perspective Projection and Perspective Transforms Perspective Projection (1/2) A Perspective Projection is a projection of a point P on a plane π (Projection Plane) along the line connecting P with a particular point F we will call the Eye. A. Martinelli Transforms 16/10/ / 39

51 Parallel Projection, Perspective Projection and Perspective Transforms Perspective Projection (1/2) A Perspective Projection is a projection of a point P on a plane π (Projection Plane) along the line connecting P with a particular point F we will call the Eye. Esempio Considering the Eye F = (0,0,0) and the plane π : z = d π : z = dp = (x,y,z) P pr = (x,y,z ) F = (0,0,0) (0,0,d) A. Martinelli Transforms 16/10/ / 39

52 Parallel Projection, Perspective Projection and Perspective Transforms Perspective Projection (1/2) A Perspective Projection is a projection of a point P on a plane π (Projection Plane) along the line connecting P with a particular point F we will call the Eye. Esempio Considering the Eye F = (0,0,0) and the plane π : z = d F = (0,0,0) π : z = dp = (x,y,z) P pr = (x,y,z ) (0,0,d) The line connecting (0,0,0) with P is the set of points (x,y,z ) satisfing this constraints: x x = y y = z z (29) A. Martinelli Transforms 16/10/ / 39

53 Parallel Projection, Perspective Projection and Perspective Transforms Perspective Projection (1/2) A Perspective Projection is a projection of a point P on a plane π (Projection Plane) along the line connecting P with a particular point F we will call the Eye. Esempio Considering the Eye F = (0,0,0) and the plane π : z = d F = (0,0,0) π : z = dp = (x,y,z) P pr = (x,y,z ) (0,0,d) The line connecting (0,0,0) with P is the set of points (x,y,z ) satisfing this constraints: x x = y y = z z (29) The intersection with π is given by adding the constraint z = d. So x = dx z y = dy z z = d a+b : (30) A. Martinelli Transforms 16/10/ / 39

54 Parallel Projection, Perspective Projection and Perspective Transforms Perspective Projection (2/2) From the last slide: x = dx z y = dy z z = d (31) A. Martinelli Transforms 16/10/ / 39

55 Parallel Projection, Perspective Projection and Perspective Transforms Perspective Projection (2/2) From the last slide: This can be rewritten as: x = dx z y = dy z z = d x = d x+0 y+0 z x+0 y+1 z+0 1 y = 0 x+d y+0 z x+0 y+1 z+0 1 z = 0 x+0 y+d z x+0 y+1 z+0 1 (31) (32) A. Martinelli Transforms 16/10/ / 39

56 Parallel Projection, Perspective Projection and Perspective Transforms Perspective Projection (2/2) From the last slide: This can be rewritten as: x = dx z y = dy z z = d x = d x+0 y+0 z x+0 y+1 z+0 1 y = 0 x+d y+0 z x+0 y+1 z+0 1 z = 0 x+0 y+d z x+0 y+1 z+0 1 (31) (32) a+b Which may be written in homogeneus coordinates x = d x +0 y +0 z +0 1 y = 0 x +d y +0 z +0 1 z = 0 x +0 y +d z +0 1 w = 0 x +0 y +1 z +0 1, x y z w = : d d d x y z 1 (33) A. Martinelli Transforms 16/10/ / 39

57 Parallel Projection, Perspective Projection and Perspective Transforms Perspective Transform (1/2) a+b???? The matrix: d d d (34) Is a degenerated rank-0 matrix. Indeed: it s been built to transform all the 3D points of the space into points on the plane π. A. Martinelli Transforms 16/10/ / 39

58 Parallel Projection, Perspective Projection and Perspective Transforms Perspective Transform (1/2) a+b???? The matrix: d d d (34) Is a degenerated rank-0 matrix. Indeed: it s been built to transform all the 3D points of the space into points on the plane π. The truth is: we need to keep the informations about z values, beacuse we need them for out rendering algorithms (Z-Buffering). We would also like to control Clipping depth values (di default -1 e 1). A. Martinelli Transforms 16/10/ / 39

59 Parallel Projection, Perspective Projection and Perspective Transforms Perspective Transform (1/2) a+b???? The matrix: d d d (34) Is a degenerated rank-0 matrix. Indeed: it s been built to transform all the 3D points of the space into points on the plane π. The truth is: we need to keep the informations about z values, beacuse we need them for out rendering algorithms (Z-Buffering). We would also like to control Clipping depth values (di default -1 e 1). A usefull matrix would look like this: s x s y α β (35) A. Martinelli Transforms 16/10/ / 39

60 Parallel Projection, Perspective Projection and Perspective Transforms Perspective Transform (2/2) a+b???? α and β are functions based on values z near and z far. And they are evaluated starting from the equation z : z αz +β = z (36) A. Martinelli Transforms 16/10/ / 39

61 Parallel Projection, Perspective Projection and Perspective Transforms Perspective Transform (2/2) a+b???? α and β are functions based on values z near and z far. And they are evaluated starting from the equation z : z αz +β = (36) z When z = z near, z = 1. When z = z far, z = 1. 1 = αz near +β, 1 = αz far +β (37) z near z far A. Martinelli Transforms 16/10/ / 39

62 Parallel Projection, Perspective Projection and Perspective Transforms Perspective Transform (2/2) a+b???? α and β are functions based on values z near and z far. And they are evaluated starting from the equation z : z αz +β = (36) z When z = z near, z = 1. When z = z far, z = 1. 1 = αz near +β, 1 = αz far +β (37) z near z far Solving: α = z far +z near, β = 2z farz near (38) z far z near z far z near s x s y z far+z near 2z far z near (39) z far z near z far z near gluperspective(angle y,rap xy,z near,z far ); Applies a perspective projection. angle y,rap xy are angular openings parameters of the Viewing Frustum, on which they are evaluated s x ed s y A. Martinelli Transforms 16/10/ / 39

63 Parallel Projection, Perspective Projection and Perspective Transforms Viewing Frustum A Viewing Frustum is the set of points in the space 3D which, once projected, will be transformed into the Unit Volume Points When we use Perspective Projections, its shape is the one of a Pyramidal Frustum with a Rectangular Base Perspective 3D World Transform Camera Space Viewing Frustum A. Martinelli Transforms 16/10/ / 39

64 The SceneGraph and OpenGL Transforms Rendering Pipeline (3): Coordinates Transforms Rendering Architecture First Rendering Pipeline Second Pipeline: Illumination and Shading Third Pipeline: Coordinates Transforms Planar 2D Transforms Spatial 3D Transforms Homogeneous Coordinates Parallel Projection, Perspective Projection and Perspective Transforms The SceneGraph and OpenGL Transforms Fourth Pipeline: Texturing Fifth Pipeline: Memory Structures Programmable Graphics Hardware A. Martinelli Transforms 16/10/ / 39

65 The SceneGraph and OpenGL Transforms The SceneGraph The SceneGraph The Scene Graph is a tree: Whose nodes are coordinates transforms (4x4 matrices) TPr T1 House A. Martinelli Transforms 16/10/ / 39

66 The SceneGraph and OpenGL Transforms The SceneGraph The SceneGraph The Scene Graph is a tree: Whose nodes are coordinates transforms (4x4 matrices) Whose leaves are 3D Objects TPr T1 House A. Martinelli Transforms 16/10/ / 39

67 The SceneGraph and OpenGL Transforms The SceneGraph The SceneGraph The Scene Graph is a tree: Whose nodes are coordinates transforms (4x4 matrices) Whose leaves are 3D Objects Whose Projection Matrix (TPr in the example) is on the root of the tree TPr T1 House A. Martinelli Transforms 16/10/ / 39

68 The SceneGraph and OpenGL Transforms The SceneGraph The SceneGraph The Scene Graph is a tree: Whose nodes are coordinates transforms (4x4 matrices) Whose leaves are 3D Objects Whose Projection Matrix (TPr in the example) is on the root of the tree The Vertices of the Objects must be transformed appling sequentially all the matrices node after node from the leaves to the root. The last transform is always the Projection. TPr T1 House A. Martinelli Transforms 16/10/ / 39

69 The SceneGraph and OpenGL Transforms Rendering Pipeline: Coordinates Transforms GL PROJECTION (P) is a 4x4 Projection Matrix. GL MODELVIEW (M) is a 4x4 Matrix describing the 3D reference system where an object should be placed. The Vertex Pipeline implements the overall coordinates transform: v out = PMv in (40) OpenGL functions will allow to modify only one matrix at a time: Transforms Functions glloadidentity: setup the Identity (null) transform gltranslatef(t x,t y,t z ): translation glscalef(s x,s y,s z ): scaling glrotatef(α,d x,d y,d z ): rotation around the (d x,d y,d z ) direction Apart LoadIdentity this functions are Multiplicative: Each of them evaluates a 4x4 matrix Each of them will multiply the previous GL PROJECTION or GL MODELVIEW with the new matrix. A. Martinelli Transforms 16/10/ / 39 a+b :

70 The SceneGraph and OpenGL Transforms OpenGL: Coordinates Transforms (A very important example) We will call M the GL MODELVIEW, T i each matrix written from gltranslate, R i each matrix written from glrotate, I the Identity Matrix. A. Martinelli Transforms 16/10/ / 39

71 The SceneGraph and OpenGL Transforms OpenGL: Coordinates Transforms (A very important example) We will call M the GL MODELVIEW, T i each matrix written from gltranslate, R i each matrix written from glrotate, I the Identity Matrix. In the following code: glloadidentity(); : M = I gltranslatef(t x1,t y1,t z1 ); : M = IT 1 glrotatef(α 2,d x2,d y2,d z2 ); : M = IT 1 R 2 gltranslatef(t x3,t y3,t z3 ); : M = IT 1 R 2 T 3 A. Martinelli Transforms 16/10/ / 39

72 The SceneGraph and OpenGL Transforms OpenGL: Coordinates Transforms (A very important example) We will call M the GL MODELVIEW, T i each matrix written from gltranslate, R i each matrix written from glrotate, I the Identity Matrix. In the following code: glloadidentity(); : M = I gltranslatef(t x1,t y1,t z1 ); : M = IT 1 glrotatef(α 2,d x2,d y2,d z2 ); : M = IT 1 R 2 gltranslatef(t x3,t y3,t z3 ); : M = IT 1 R 2 T 3 A vertex v is applied M before being drawn v = Mv (41) which is the same as v = IT 1 R 2 T 3 v (42) A. Martinelli Transforms 16/10/ / 39

73 The SceneGraph and OpenGL Transforms OpenGL: Coordinates Transforms (A very important example) We will call M the GL MODELVIEW, T i each matrix written from gltranslate, R i each matrix written from glrotate, I the Identity Matrix. In the following code: glloadidentity(); : M = I gltranslatef(t x1,t y1,t z1 ); : M = IT 1 glrotatef(α 2,d x2,d y2,d z2 ); : M = IT 1 R 2 gltranslatef(t x3,t y3,t z3 ); : M = IT 1 R 2 T 3 A vertex v is applied M before being drawn which is the same as That is: v got applied (in the correct order) Transform T 3 Transform R 2 v = Mv (41) v = IT 1 R 2 T 3 v (42) Transform T 1 The order of appliance of each transform is the opposite to the order in which they are written in OpenGL A. Martinelli Transforms 16/10/ / 39

74 The SceneGraph and OpenGL Transforms SceneGraphs and OpenGL (1/3) A good way to desing the Projection Matrix (once before drawing anything) glmatrixmode(gl PROJECTION);From this point, we work on P Setting ModelView on each Object ( Objects) A. Martinelli Transforms 16/10/ / 39

75 The SceneGraph and OpenGL Transforms SceneGraphs and OpenGL (1/3) A good way to desing the Projection Matrix (once before drawing anything) glmatrixmode(gl PROJECTION);From this point, we work on P glloadidentity();p is set to the Identity Setting ModelView on each Object ( Objects) A. Martinelli Transforms 16/10/ / 39

76 The SceneGraph and OpenGL Transforms SceneGraphs and OpenGL (1/3) A good way to desing the Projection Matrix (once before drawing anything) glmatrixmode(gl PROJECTION);From this point, we work on P glloadidentity();p is set to the Identity glscale( size/width, size/height, eps ); Terms correcting Viewport Transforms Setting ModelView on each Object ( Objects) A. Martinelli Transforms 16/10/ / 39

77 The SceneGraph and OpenGL Transforms SceneGraphs and OpenGL (1/3) A good way to desing the Projection Matrix (once before drawing anything) glmatrixmode(gl PROJECTION);From this point, we work on P glloadidentity();p is set to the Identity glscale( size/width, size/height, eps ); Terms correcting Viewport Transforms gluperspective( angle, rapport,z near,z far ); Basic Perspective Transform Setting ModelView on each Object ( Objects) A. Martinelli Transforms 16/10/ / 39

78 The SceneGraph and OpenGL Transforms SceneGraphs and OpenGL (1/3) A good way to desing the Projection Matrix (once before drawing anything) glmatrixmode(gl PROJECTION);From this point, we work on P glloadidentity();p is set to the Identity glscale( size/width, size/height, eps ); Terms correcting Viewport Transforms gluperspective( angle, rapport,z near,z far ); Basic Perspective Transform gltranslatef( t xmondo,t ymondo,t zmondo ); World Translation Setting ModelView on each Object ( Objects) A. Martinelli Transforms 16/10/ / 39

79 The SceneGraph and OpenGL Transforms SceneGraphs and OpenGL (1/3) A good way to desing the Projection Matrix (once before drawing anything) glmatrixmode(gl PROJECTION);From this point, we work on P glloadidentity();p is set to the Identity glscale( size/width, size/height, eps ); Terms correcting Viewport Transforms gluperspective( angle, rapport,z near,z far ); Basic Perspective Transform gltranslatef( t xmondo,t ymondo,t zmondo ); World Translation glrotatef( α Mondo,D xmondo,d ymondo,d zmondo ); World Rotation Setting ModelView on each Object ( Objects) A. Martinelli Transforms 16/10/ / 39

80 The SceneGraph and OpenGL Transforms SceneGraphs and OpenGL (1/3) A good way to desing the Projection Matrix (once before drawing anything) glmatrixmode(gl PROJECTION);From this point, we work on P glloadidentity();p is set to the Identity glscale( size/width, size/height, eps ); Terms correcting Viewport Transforms gluperspective( angle, rapport,z near,z far ); Basic Perspective Transform gltranslatef( t xmondo,t ymondo,t zmondo ); World Translation glrotatef( α Mondo,D xmondo,d ymondo,d zmondo ); World Rotation Setting ModelView on each Object ( Objects) glmatrixmode(gl MODELVIEW);From this point, we work on M A. Martinelli Transforms 16/10/ / 39

81 The SceneGraph and OpenGL Transforms SceneGraphs and OpenGL (1/3) A good way to desing the Projection Matrix (once before drawing anything) glmatrixmode(gl PROJECTION);From this point, we work on P glloadidentity();p is set to the Identity glscale( size/width, size/height, eps ); Terms correcting Viewport Transforms gluperspective( angle, rapport,z near,z far ); Basic Perspective Transform gltranslatef( t xmondo,t ymondo,t zmondo ); World Translation glrotatef( α Mondo,D xmondo,d ymondo,d zmondo ); World Rotation Setting ModelView on each Object ( Objects) glmatrixmode(gl MODELVIEW);From this point, we work on M glloadidentity();setting M to Identity A. Martinelli Transforms 16/10/ / 39

82 The SceneGraph and OpenGL Transforms SceneGraphs and OpenGL (1/3) A good way to desing the Projection Matrix (once before drawing anything) glmatrixmode(gl PROJECTION);From this point, we work on P glloadidentity();p is set to the Identity glscale( size/width, size/height, eps ); Terms correcting Viewport Transforms gluperspective( angle, rapport,z near,z far ); Basic Perspective Transform gltranslatef( t xmondo,t ymondo,t zmondo ); World Translation glrotatef( α Mondo,D xmondo,d ymondo,d zmondo ); World Rotation Setting ModelView on each Object ( Objects) glmatrixmode(gl MODELVIEW);From this point, we work on M glloadidentity();setting M to Identity gltranslatef( t xoggetto,t yoggetto,t zoggetto ); Object Positioning A. Martinelli Transforms 16/10/ / 39

83 The SceneGraph and OpenGL Transforms SceneGraphs and OpenGL (1/3) A good way to desing the Projection Matrix (once before drawing anything) glmatrixmode(gl PROJECTION);From this point, we work on P glloadidentity();p is set to the Identity glscale( size/width, size/height, eps ); Terms correcting Viewport Transforms gluperspective( angle, rapport,z near,z far ); Basic Perspective Transform gltranslatef( t xmondo,t ymondo,t zmondo ); World Translation glrotatef( α Mondo,D xmondo,d ymondo,d zmondo ); World Rotation Setting ModelView on each Object ( Objects) glmatrixmode(gl MODELVIEW);From this point, we work on M glloadidentity();setting M to Identity gltranslatef( t xoggetto,t yoggetto,t zoggetto ); Object Positioning glrotatef( α Oggetto,D xoggetto,d yoggetto,d zoggetto ); Object Orientation A. Martinelli Transforms 16/10/ / 39

84 The SceneGraph and OpenGL Transforms SceneGraphs and OpenGL (1/3) A good way to desing the Projection Matrix (once before drawing anything) glmatrixmode(gl PROJECTION);From this point, we work on P glloadidentity();p is set to the Identity glscale( size/width, size/height, eps ); Terms correcting Viewport Transforms gluperspective( angle, rapport,z near,z far ); Basic Perspective Transform gltranslatef( t xmondo,t ymondo,t zmondo ); World Translation glrotatef( α Mondo,D xmondo,d ymondo,d zmondo ); World Rotation Setting ModelView on each Object ( Objects) glmatrixmode(gl MODELVIEW);From this point, we work on M glloadidentity();setting M to Identity gltranslatef( t xoggetto,t yoggetto,t zoggetto ); Object Positioning glrotatef( α Oggetto,D xoggetto,d yoggetto,d zoggetto ); Object Orientation glscalef( S Oggetto,S Oggetto,S Oggetto ); Object Resizing A. Martinelli Transforms 16/10/ / 39

85 The SceneGraph and OpenGL Transforms SceneGraphs and OpenGL (2/3) Open GL offers an important tool to build SceneGraphs: the Matrices Stack Usually, a Matrices Stack works on the glmatrixmode(gl MODELVIEW); Lo Stack delle Matrici glpushmatrix(); : create a copy of M push it on the stack glpopmatrix(); : retrieve (pop) the last matrix from the stack and use it to set M, overwriting it Nota P (GL PROJECTION) and M (GL MODELVIEW) can be both managed with push and pop and they have 2 separated stacks A. Martinelli Transforms 16/10/ / 39

86 The SceneGraph and OpenGL Transforms SceneGraphs and OpenGL (3/3) T1 House TPr T2 TRot Carousel Tv Coach Codice After Projection has been set up glmatrixmode(gl MODELVIEW); glloadidentity();m = I glpushmatrix();m = I gltranslatef(t2 x,t2 y,t2 z );M = IT 2 glrotatef(α rot,0,1,0);m = IT 2 T rot glpushmatrix();m = IT 2 T rot gltranslatef(tv x,tv x,tv x );M = IT 2 T rot T V drawcoach(); glpopmatrix();m = IT 2 T rot drawcarousel(); glpopmatrix();m = I gltranslatef(t1 x,t1 x,t1 x );M = IT 1 drawhouse(); A. Martinelli Transforms 16/10/ / 39

87 The SceneGraph and OpenGL Transforms Rendering Pipeline and Coordinates Transorms (1/2) A Coordinates Transforms Block is part of the Vertex Pipeline A. Martinelli Transforms 16/10/ / 39

88 The SceneGraph and OpenGL Transforms Rendering Pipeline and Coordinates Transorms (1/2) A Coordinates Transforms Block is part of the Vertex Pipeline It applies P and M as we have already seen A. Martinelli Transforms 16/10/ / 39

89 The SceneGraph and OpenGL Transforms Rendering Pipeline and Coordinates Transorms (1/2) A Coordinates Transforms Block is part of the Vertex Pipeline It applies P and M as we have already seen Note: About Transforming Normal Vectors The Normals we use in the Illumination Block should be transformed A. Martinelli Transforms 16/10/ / 39

90 The SceneGraph and OpenGL Transforms Rendering Pipeline and Coordinates Transorms (1/2) A Coordinates Transforms Block is part of the Vertex Pipeline It applies P and M as we have already seen Note: About Transforming Normal Vectors The Normals we use in the Illumination Block should be transformed For example: rotating a model should bring to rotated normals. A. Martinelli Transforms 16/10/ / 39

91 The SceneGraph and OpenGL Transforms Rendering Pipeline and Coordinates Transorms (1/2) A Coordinates Transforms Block is part of the Vertex Pipeline It applies P and M as we have already seen Note: About Transforming Normal Vectors The Normals we use in the Illumination Block should be transformed For example: rotating a model should bring to rotated normals. Normals should be transformed in the world, so only Modelview is applied. A. Martinelli Transforms 16/10/ / 39

92 The SceneGraph and OpenGL Transforms Rendering Pipeline and Coordinates Transorms (1/2) A Coordinates Transforms Block is part of the Vertex Pipeline It applies P and M as we have already seen Note: About Transforming Normal Vectors The Normals we use in the Illumination Block should be transformed For example: rotating a model should bring to rotated normals. Normals should be transformed in the world, so only Modelview is applied. But a direct application of the Modelview is wrong, because it would take into account scaling factors on each axes, so Normals are applied a modified version of the modelview. A. Martinelli Transforms 16/10/ / 39

93 The SceneGraph and OpenGL Transforms Rendering Pipeline and Coordinates Transorms (2/2) Vertex Stream Rastering Module ymax (x,c,z)min (x,c,z)max y ymin menti( (x,y),(c,z) ) Primitive Manager Z-Buffering read(z-test) Clipping, Viewport Transform write Color Buffer Depth Buffer A. Martinelli Transforms 16/10/ / 39