1 / 56

3D Viewing

3D Viewing. Camera Analogy. 3D is just like taking a photograph (lots of photographs!) It uses transformations. viewing volume. camera. model. tripod. Camera Analogy and Transformations. Projection transformations adjust the lens of the camera Viewing transformations

idalia
Download Presentation

3D Viewing

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 3D Viewing

  2. Camera Analogy • 3D is just like taking a photograph (lots of photographs!) • It uses transformations viewing volume camera model tripod

  3. Camera Analogy and Transformations • Projection transformations • adjust the lens of the camera • Viewing transformations • tripod–define position and orientation of the viewing volume in the world • Modeling transformations • moving the model • Viewport transformations • enlarge or reduce the physical photograph

  4. Coordinate Systems and Transformations • Steps in forming an image • specify geometry (world coordinates) • specify camera (camera coordinates) • project (window coordinates) • map to viewport (screen coordinates) • Each step uses transformations • Every transformation is equivalent to a change in coordinate systems (frames)

  5. v e r t e x Modelview Matrix Projection Matrix Perspective Division Viewport Transform Modelview Projection Modelview l l l Transformation Pipeline • other calculations here • material è color • shade model (flat) • polygon rendering mode • polygon culling • clipping normalized device eye object clip window

  6. Projection Transformation • Shape of viewing frustum • Perspective projection gluPerspective( fovy, aspect, zNear, zFar ) glFrustum(left,right,bottom,top,zNear,zFar) • Orthographic parallel projection glOrtho(left,right,bottom,top,zNear,zFar) gluOrtho2D( left, right, bottom, top ) • calls glOrtho with z values near zero

  7. Applying Projection Transformations • Typical use (orthographic projection) glMatrixMode( GL_PROJECTION ); glLoadIdentity(); glOrtho(left, right, bottom, top, zNear, zFar);

  8. tripod Viewing Transformations • Position the camera/eye in the scene • place the tripod down; aim camera • To “fly through” a scene • change viewing transformation andredraw scene gluLookAt( eyex, eyey, eyez, aimx, aimy, aimz, upx, upy, upz ) • up vector determines unique orientation

  9. VRP - AT n =  | VRP - AT | U  U = VUP  n, u = |U|  n x u = v Viewing Specification & Eye Coordinates • Suppose that modeling transformation has been done and now all objects are in WC. • We need viewing specification or camera information. • VRP: view reference point or eye position. • AT: a reference point for objects. (ATis a reference point toward which the eye is aimed. AT is typically somewhere in the middle of the scene.) • VUP: view up vector. • VPN: view plane normal. (orientation of projection point) VRP-AT • {VRP, u, v, n} defines eye coordinates (EC), which is also called camera coordinates or view coordinates. • In EC, camera is at the origin and looks in the –n direction. y y VUP v VRP AT u n z z x x VRP

  10. Eye Coordinates: Teapot Example • Let’s focus on the teapot centered at WC. v u n VRP

  11. y y v v u u x x VRP n z z n Viewing Transformation • We have two coordinates: WC={O,x,y,z} and EC={VRP,u,v,n}. • Let’s superimpose EC onto WC: a translation followed by a rotation. • Such a transformation (which is a rigid motion) simply reinterprets the model from WC into EC, and is called viewing transformation. y v u VRP n x z

  12. Viewing Transformation (cont’d) • viewing transformation in a single 4x4 matrix • Why from WC to EC? It’s for easy projection and clipping. You will see pretty soon! • Modeling and viewing transformations are often combined into a matrix Mmodelview. viewing transformation modeling transformation in WC in EC in LC modelview transformation in LC in EC

  13. v y u x z n View Volume • Thanks to the viewing transformation, all vertices are newly defined in EC={VRP,u,v,n}, and we can forget about the WC={O,x,y,z}. • Then, let’s use {x,y,z} and {u,v,n} interchangeably because we are familiar with {x,y,z}. • Suppose a huge scene. In general, we cannot see all objects in the scene. We need to specify how much of the scene we can see. View volume does it. =

  14. Perspective View Volume • We can imagine a window through which we can see the world. • The window is parallel to xy-plane. • Then, for perspective projection, the output image is made by the projectors (from COP) that pass through the window. • Such a perspective view volume makes an infinite pyramid. • We can bound near and far sides of the infinite pyramid. • The perspective view volume makes a truncated pyramid, frustum. • Only the objects inside the frustum are visible. y COP window z x window near plane far plane COP

  15. Perspective View Volume (cont’d) • The frustum is defined by {l, r, b, t, n, f} which stands for {left, right, bottom, top, near, far}. We set n and fto be positive such that they are the perpendicular distances from COP to the two planes. • Note that the pyramid is not upright, which is unrealizable in the camera model but is useful for simulating human stereo vision, etc. z = -f -z y far plane near plane (r, t, -n) COP x (l, b, -n) z

  16. Viewing Examples: Model and EC • Given the model in the figure, suppose that VRP=(0,0,40), AT=(0,0,0), and VUP=(0,10,0). • Let’s see some view volume examples with this model. y 10 VUP v=y=10 -n=-z AT 10 x v=y v 20 -20 30 -10 u=x=10 COP u u=x z=40 z n n=z

  17. Perspective Projection Example I (60, 60, -40) •  example 1 • We got the following image, where the viewport corresponds to the view volume’s window. 60 l= 0, r = 6 b = 0, t = 6 n = 4, f= 40 far plane 40 20 y=10 viewport x=10 20 40 60 near plane -20 (r, t, -n) = (6, 6, -4) COP (l, b, -n) = (0, 0, -4)

  18. Perspective Projection Example II (60, 60, -40) • example 2 • What if the window size is reduced by 1/4? • We got the following image. 60 l= 0, r = 3 b = 0, t = 3 n = 4, f= 40 40 (30, 30, -40) 20 viewport 20 40 60 -20 (6, 6, -4) COP (3, 3, -4)

  19. (22.5, 22.5, -15) Perspective Projection Example III (60, 60, -40) 60 • example 3 • What if the far plane moves towards COP? •  We got the following image. l= 0, r = 6 b = 0, t = 6 n = 4, f= 15 40 20 viewport 20 40 60 -20 (6, 6, -4) COP

  20. y x z Parallel Projection: Oblique vs. Orthographic • Recall that view plane and window are parallel to xy-plane. • Let’s consider the cross-section of EC (seen along –y direction). • Orthographic projection is a special parallel projection where the view plane is orthogonal to DOP. •  perspective •  parallel • - orthographic • - oblique x z Assuming the DOP is parallel to xz-plane. x x x DOP DOP DOP z orthographic oblique z z

  21. DOP window Parallel View Volume • In parallel projection, the view volume is an infinite parallelepiped (not necessarily a box), which is parallel to DOP. • We can also bound near and far sides of the infinite parallelepiped to make the parallel view volume. (Imagine that the window slides along DOP to make both planes.) • Only the objects inside the view volume are visible. window near plane far plane

  22. Parallel View Volume (cont’d) • Oblique projection is much less common, and so let’s consider only orthographic projection where DOP is orthogonal to the view plane. • Note that DOP is along the z-axis. • Then, the view volume has to be aligned with xyz-axes. • Two end points in a diagonal are enough to define the view volume. • As in the perspective projection, we set nand fto be positive. • Note that the view volume is defined by {l, r, b, t, n, f} for both perspective projection and orthographic projection. Of course, the set {l, r, b, t, n, f} has different semantics in the two cases. (r, t, -f) DOP y parallel view volume (l, b, -n) x z

  23. y=10 -z y -20 -10 x=10 x z Orthographic Projection Examples • Let’s use the same example. • example 2 l= 0, r = 20 b = 0, t = 20 n = 15, f= 40 • example 3 l= 5, r = 20 b = 5, t = 20 n = 15, f= 40 • example 4 • example 1 l= 5, r = 20 b = 5, t = 20 n = 25, f= 40 l= 0, r = 20 b = 0, t = 20 n = 4, f= 40

  24. Culling & Clipping • Where are we now? • Objects are defined in EC through modeling and viewing transformations. • Projection type (parallel or perspective) is specified. • View volume is given as {l, r, b, t, n, f} in EC. • The view volume specifies how much of the scene we can see. So, polygons completely inside the view volume are processed for display. • How about the scene outside the view volume? • Polygons completely outside the view volume are culled. • Polygons intersecting the boundary of the view volume are clipped against the boundary, and then processed for display.

  25. Clipping: Teapot Example • Suppose that, in orthographic projection, we have a view volume intersecting the teapot. clipped polygon mesh to be rendered

  26. y -1 -z 1 -1 1 x -x -1 z 1 -y Projection Transformation • OK, we just specified the view volume {l, r, b, t, n, f} in EC. The next step is projection transformation. • Let’s first tackle the orthographic projection. • Operations can be easily implemented in canonical view volume(CVV), which is defined as a 2x2x2 cube with its center at the origin, i.e. x=y=z=[-1,1]. • Projection transformation or specifically orthographic transformation transforms the box view volume into CVV.

  27. y -1 -z 1 -1 1 x -x -1 z Tpar = 1 -y Projection Transformation (cont’d) • Obviously we need scaling. However, scaling is about origin. • As the 1st step, let’s translate the view volume’s center to the origin. • We know that the view volume’s center is ((l+r)/2,(b+t)/2,-(n+f )/2)). • So, the translation is Tpar = T(-(l+r)/2,-(b+t)/2,(n+f )/2)) y x=[-1,1] y=[-1,1] z=[-1,1] x=[l,r] y=[b,t] z=[-n,-f] x z y x Tpar makes  z

  28. Spar = Projection Transformation (cont’d) • Note that the CVV has the size of 2x2x2. So, let’s scale! • Width along x “r - l” should be scaled into 2. • So, scaling factor along x direction Sx = 2/(r-l) • Similarly, Sy = 2/(t-b) and Sz = 2/(f-n) • Mortho=Spar • Tpar is the orthographic (projection) transformation from the box view volume into CVV, or from EC to clip coordinates (CC). • After orthographic transformation, clipping is done. Then, the clipped model is within CVV whose dimensions are all 1 along x/y/z axes. It’s called that the model is in normalized devicecoordinates(NDC). • Some people say we have a 3D film (generated by projection). y x z r - l

  29. Projection Transformation (cont’d) • Orthographic transformation with teapot example. v u n translation scaling CVV in NDC

  30. Projection Transformation (cont’d) • Given {l, r, b, t, n, f}, the orthographic transformation Mortho is computed as Spar• Tpar. • Mortho can be simply rewritten as follows. x = (2/(r-l))x - (r+l)/(r-l) y = (2/(t-b))y - (t+b)/(t-b) z = (2/(f-n))z + (f+n)/(f-n)

  31. Projection Transformation (cont’d) + • Some systems such as OpenGL use -z direction (0 0 -1) as DOP. (Compare this with positive-z projection direction in EC.) For this purpose, 3D reflection is required. reflection wrt xy-plane

  32. Window Coordinates and Viewport • A window at your screen (not the window associated with the view volume) is associated with window coordinates (WdC). • A viewport is defined at WdC. • The viewport is not necessarily the entire window, and is defined by its lower left corner (xmin & ymin), width, and height. • Viewport transformation transforms the x- and y-coordinates of CVV to coincide with the viewport. viewport window height (xmin,ymin) width

  33. viewport y oy y -1 height/2 z=-1 y -z ymin 1 x -1 1 x -x ox z xmin -1 x z -width/2 1 -y width/2 z -height/2 z=1 Viewport Transformation • The viewport transformation is a scaling followed by a translation. • For scaling, Sx=width/2 andSy=height/2. • Translation moves the center at the origin to the viewport center (ox,oy)=(xmin+width/2, ymin+height/2). • So, Mviewport= T(ox,oy,0) • S(Sx, Sy,1). Simply, x=x*(width/2)+ox and y=y*(height/2)+oy. Don’t do matrix multiplication. translation scaling

  34. Viewport Transformation (cont’d) • Viewport transformation with teapot example • Final images CVV

  35. Orthographic Projection Summary • The modeling transformation moves from LC to WC. • Viewing specifies the location/orientation of the viewer or camera, and viewing transformation moves from WC to EC. • The two transformations are combined into Mmodelview. • The view volume is defined by {l, r, b, t, n, f}. • The orthographic transformationMortho changes the view volume into CVV. It’s said the model is then in clip coordinates(CC). • Clipping is done. • The viewport transformationMviewport transforms vertices in CVV into 3D points in WdC. orthographic transformation modeling & viewing transformation in EC in LC viewport transformation in CC in WdC clipping in NDC

  36. y -1 -z 1 -1 1 x -x -1 z 1 -y Projection Transformation for Perspective Projection • Now, let’s tackle the perspective projection • Note that the modelview transformation does not depend on the projection type. What distinguishes the perspective projection from the parallel projection is the projection transformation. • Perspective projection transformation: from the frustum to CVV. • It is often simply called a perspective transformation. -z z=-f y far plane (r, t, -n) near plane COP x (l, b, -n) z

  37. -z -z z: unaffected x= x + pz should be x x 3D Shearing • Let’s see the frustum along -y direction • So, shearing along x is needed. OK, but is that it? No … -z z=-f y far plane (r, t, -n) near plane COP x (l, b, -n) z

  38. y z: unaffected y = y + qz -z should be 3D Shearing (cont’d) y • Similarly, let us see along -x. • Hxy(p,q) makes the vector CC, which connects COP and the near plane’s center ((l+r)/2, (b+t)/2, -n), be (0, 0, -n). -z -z z=-f y far plane near plane (r, t, -n) COP x (l, b, -n) z x x+pz 1 0 p 0 Hxy(p,q) in summary z: unaffected x = x + pz y = y + qz y y+qz 0 1 q 0 = z z 0 0 1 0 1 1 0 0 0 1

  39. 3D Shearing (cont’d) • Let’s compute p and q. • If the frustum happens to be upright, CCx=CCy=0, and so p=q=0, which means that Hxy(p,q)= I and shearing has no effect. CC = near-plane-center - COP should be

  40. Shearing followed by Scaling • 3D shearing has made the skewed frustum upright. • Now, let’s make the frustum’s side faces be of 45° slope. • Such a task can be performed by scaling. y y y To be done! 45° Done! -z -z -z original frustum upright frustum 45°-slope frustum

  41. Scaling • Note that shearing does not change the size of the near plane. • Both (r-l)/2 and (t-b)/2 should be the same as n for making a 45°-slope frustum. We then need scaling! • Sx= n/[(r-l)/2] = 2n/(r-l) • Sy= n/[(t-b)/2] = 2n/(t-b) • Sz = 1. (r-l)/2 -z y (t-b)/2 near plane n x COP z

  42. Shearing & Scaling: Teapot Example scaling shearing

  43. Distortion • From the 45° frustum into CVV: Let’s represent (x2 y2 z2)in terms of(x1 y1 z1). y p -z1 y z=-f 1 n (x1,y1,z1) (x2,y2,z2) z z 1 -1 z1 -n z=-n -1 The point p (whose y value is -z1) should go to y=1. -z1:1 = y1: y2 y2= -y1/z1 Similarly, x2= -x1/z1

  44. Distortion (cont’d) • We found that x2=-x1/z1 and y2=-y1/z1 . • How about z2? • We are to use matrix multiplication to represent projection. • z2= (ax1+by1+cz1+d)/(ex1+fy1+gz1+h) • (ex1+fy1+gz1+h)<- this part should be the same for x, y, and z • Z value should be independent of x, y value • Therefore z2= (cz1+d)/z1 • z2=A+B/z1. • OK, we found that z2=A+B/z1. • Let’s compute A and B. should be for the formula to be planar

  45. Distortion (cont’d) • Let’s compute A and B for z2=A+B/z1. • Note that z1=[-f, -n] and z2=[-1,1]. • So, if z1= -f, then z2= -1. Similarly, if z1= -n, then z2=1. • Solving the two equations, we get A=(n+f)/(n-f) and B=2nf/(n-f). • (x2 y2 z2 1) = (-x1/z1 -y1/z1A+B/z11) = (x1 y1 -Az1-B -z1) • Multiplying every component by -z1, we can represent the distortion transformation “by a matrix multiplication.” • It’s possible thanks to homogeneous coordinate system. called Mdistort not (0 0 0 1) !!

  46. y y z z x x Why called Distortion? • Mdistort distorts the model, and is not affine in the sense that parallelism is not preserved. • Distortion transformation moves COP to the  position of z-axis. • The model is distorted, but leads to a correct image as we are now in (orthographic) CVV. into CVV model distortion

  47. Distortion: Teapot Example

  48. Perspective Division • In homogeneous coordinates, each vertex should be divided by its w-component to get a corresponding Cartesian coordinates. • It’s called perspective division. • An example n=1 f=2 A=(n+f)/(n-f) B=2nf/(n-f). y Q=(0 1 -1/3) Q=(0 3/2 –3/2) P=(0 1 1) 3/2 1 1/2 P=(0 1 -1) -z -2 R=(0 1/2 –3/2) R=(0 1/3 -1/3)

  49. Perspective Division (cont’d) • The distortion transformation converts (x y z 1)into (x y -Az-B -z), and the perspective division is done by w=-z, which actually represents the distance from the eye. • As the distance from the eye increases, 1/w approaches 0, and so x/w and y/w also approach 0. This is how foreshortening is achieved. a b model distortion b a

  50. Perspective Division (cont’d) • Observe that Q and R are at the center of the frustum along z-axis but Q and R are not at the center of CVV. • As the transformed z-coordinates move farther away from the near plane, the locations become increasingly less precise. Such a non-linearity caused by perspective division leads to some problem for hidden-surface removal. Let’s revisit this later. y Q=(0 1 -1/3) Q=(0 3/2 –3/2) P=(0 1 1) 3/2 1 1/2 P=(0 1 -1) -z -2 R=(0 1/2 –3/2) R=(0 1/3 -1/3)

More Related