1. Modeling and Viewing Modeling
Use modeling (local) coordinates and geometric transformations to build hierarchically more complex objects and scenes. The final scene is in world frame
Viewing
Model the camera: position , orientation, camera (view) reference frame, projection
Viewing transformations: from world to camera coordinates
Clipping/hidden surface removal: clip out from consideration parts outside of view volume
Projection transformations and hidden surface removal: from 3D viewing coord. to 2D projection coord. and normalized device coordinates
Viewport transformations: from normalized device coordinates to screen (device) coordinates
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2. Viewing and Projection
Transforming
Modeling transformations (affine), 3D to 3D
Viewing transformations (affine), 3D to 3D
Linear in homogeneous coordinates
Images of parallel lines stay parallel
Transformation matrix in homogeneous coordinates has last row,
Projection transformations (not affine), 3D to 2D
Images of parallel lines may intersect at infinity
Transformation matrix most general
Viewport transformations (affine), maps viewing window to viewport, 2D to 2D
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3. Viewing Terminology
Viewing volume: the region in 3D that can contain objects that are visible by the camera
Projection: math transformations that maps from 3D to 2D (or 4D to 3D, in homogeneous)
Projection plane: the plain containing the 2D image
Viewing window: the rectangle in the image plane that will be mapped to the screen eventually
Viewport: 2D rectangle within the display window on the screen that shows the viewing window
Clipping: cutting off from consideration parts outside the view volume (done easier if the view volume is mapped to a canonical view volume which is a cube)
Motivation:
-- object/picture often has parts
-- model once/instantiate multiple times
-- modify only a part
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4. Graphics functions Graphics systems support viewing by
Providing a viewing model whose parameters specify the camera
Providing functions for viewing, projection and viewport
Implementing viewing and projection transformations as matrix multiplications in homogeneous coordinates
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5. Viewing APIs
The the position of the eye or camera is called the view reference point (VRP)
A unit view plane normal (VPN), is in the viewing direction, it is perpendicular to the image plane. In open GL VPN is in direction opposite to the one in which camera is looking
Another vector called the view-up vector is a vector specifying which is the approximate “up” direction for the camera
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6. Camera Model: viewing VRP: Camera(eye) position,
(u,v,n): camera frame.
Viewing: specify VRP,n,VUP.
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7. Viewing Coordinate System: (u,v,n)
right handed
v, the y-axis of the view frame, is the perpendicular projection of VUP on the projection plane
n is the z-axis of the view frame
u = v × n
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8. Setting up the camera
Construct a scene and then look at it from a point of view, eye:
eye, the eyepoint, is the VRP specified in the world coordinates
Camera is pointed at a point at, the at point
These points determine VPN
vpn = eye – at
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9. Two Points of View
Hold camera frame fixed, move objects in front of the camera
glLoadIdentity();
glTranslatef(0,0,-d);
Keep objects stationary and move the camera away from the objects
glLookAt(0,0,d,0,0,0,0,1,0);
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10. gluLookAt Utility Routine in OpenGL
Defines a viewing transformation matrix M, and M postmultiplies CTM, i.e. CTM=CTM*M
Eye point: eyex, eyey, and eyez.
At point: atx, aty,and atz.
VUP: upx, upy, and upz
gluLookAt(eyex, eyey, eyez, atx, aty, atz, upx, upy, upz);
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11. Viewing Transformations
Given 3D model of a scene/object in 4D homogeneous coordinates P, with respect to the world coordinate system
Given camera coordinate system (position, VRP, and camera frame (u,v,n) )
Viewing transformation M converts coordinates of objects from world to camera coordinates
How?
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12. Viewing Transformations
Let W=(O,ex,ey,ez) be the world coord. system
Let V=(VRP,u,v,n) be the camera coord. System
Let M be the change of frame matrix, mapping V to W, M = R.T(-VRP), where
T is a translation mapping VRP to O
R is a rotation aligning (u,v,n) with (ex,ey,ez), in 3D affine coordinates it represented by a matrix with rows u’,v’,n’
Then for a point P with modeling coordinates w=(xw,yw,zw,1)’ the viewing coordinates are
v =(vx,vy,vz,1)’, where
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13. Custom Utility Routine
You might need to define your own transformation routine
Flight simulator: Display the world from the pilot’s point of view
Pilot see the world in terms of roll, pitch, and heading
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14. Custom Utility Routine
The following routine could serve as the viewing transformation:
void pilotView{GLdouble planex, GLdouble planey,
               GLdouble planez, Gldouble roll,                 
GLdouble pitch, GLdouble heading) {
     glRotated(roll, 0.0, 0.0, 1.0);      
glRotated(pitch, 0.0, 1.0, 0.0);      
glRotated(heading, 1.0, 0.0, 0.0);      
glTranslated(-planex, -planey, -planez); }
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15. Custom Utility Routine
Orbiting the camera around an object that's centered at the origin
Use polar coordinates.
Let the distance variable define the radius of the orbit
The azimuth is the angle of rotation of the camera about the object in the x-y plane
elevation is the angle of rotation of the camera in the y-z plane
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16. Projections
Projecting: mapping from 3D viewing coordinates to 2D coordinates in projection plane. In homogeneous coordinates it is a map
from 4D viewing coordinates to 3D.
Projections
Parallel: orthogonal and oblique
Perspective
Canonical views: orthographic and perspective projections
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17. Perspective projection
Projectors intersect at COP
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18. Parallel Projections Projectors parallel. COP at infinity.
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19. Parallel projections: summary
Center of projection is at infinity.
Projectors are parallel.
Parallel lines stay parallel
There is no forshorthening
Distances and angles are transformed consistently
Used most often in engineering design, CAD systems. Used for top and side drawings from which measurements could be made.
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20. Orthographic Projection: projectors orthogonal to projection plane
same for all points
DOP
(direction of projectors)
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21. Orthographic Projections
DOP is perpendicular to the view plane
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22. Multiview Parallel Projection
Faces are parallel to the projection plane
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23. Isometric Projection
Projector makes equal angles with all three principal axes
All three axes are equally foreshortened
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24. Mechanical Drawing
isometric
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25. Oblique Parallel Projections
Most general parallel views
Projectors make an arbitrary angle with the projection plane
Angles in planes parallel to the projection plane are preserved
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26. Oblique Projections: projectors are not orthogonal to image plane
Cavalier
Angle between projectors and projection plane is 45°. Lines orthogonal to the projection plane
Retain their exact length. Perpendicular faces are projected at full scale
Cabinet
Angle between projectors and projection plane is arctan(2)=63.4°. Lines orthogonal to the
projection plane are projected at half length. Perpendicular faces are projected at 50% scale.
Looks like forshorthening.
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27. Perspective Projection
Most natural for people
In human vision, perspective projection of the world is created on the retina (back of the eye)
Used in CG for creating realistic images
Perspective projection images carry depth cues
Foreshorthening causes distant objects to appear smaller
Relative lengths and angles are not preserved
A perspective image cannot be used for metric measurements of the 3D world
Parallel lines not parallel to the image plane converge at a vanishing point
An axis (principal) vanishing point is a point of convergence for lines parallel to a principal axis of the object. We distinguish one-, two-, three-point projections.)
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28. Vanishing Points
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29. Vanishing Points
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30. Early Perspective Giotto
Not systematic—parallel lines do not converge to a single "vanishing" point
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31. Math of Projections:Overview
Math of perspective projection, standard configuration
OpenGL perspective projections
Math of orthographic projection
OpenGL orthographic projections
Viewport transformations and setting them in OpenGL
Summary
Viewing transformations
Orthographic projection canonical viewing volume
Perspective projection canonical viewing volume
Hidden surface removal
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32. Perspective Viewing
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33. Perspective Projections-z -z
Viewing direction orthogonal
To projection plane
Genaral perspective:
Viewing direction is
not orthogonal to
projection plane
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34. Perspective Projections
The graphics system applies a 4 x 4 projection matrix after the model-view matrix
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35. Math of perspective projection
The discussion here is carried out with respect to the camera (view) reference frame, (VRP,u,v,n)
The projection transformation maps 3D points to 2D points in the projection plane
Standard configuration:
COP=VRP
Projection plane is orthogonal to z-axis, at z=d
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36. Math of perspective projection
Standard configuration:
Let O be COP
Projection plane has equation z=d, d <0
3D point P has homogeneous coordinates (x,y,z,1)
It is mapped to point Q (xp,yp,d) in the projection plane
Q is on the segment PO, thus
Q=cP+(1-c)O, where 0<c<1
Thus, xp = c.x, yp = c.y, d = c.z  c = d/z
Thus, xp = (d/z).x, yp = (d/z).y , in affine coordinates
Note that projection is not linear in affine coordates.
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37. Math of perspective projection
P=p(x,y,z,1)  Q=q(d.x/z, d.y/z, d, 1), in homogeneous coordinates.
Perspective projection is not linear in affine coord.
Perspective projection is linear in homogeneous
From homogeneous to affine coordinates:
(a,b,c,w)  (a/w, b/w,c/w)
Thus (x,y,z,d/z) in homogeneous becomes
(x/(z/d), y/(z/d),d,1) ==> (d.x/z, d.y/z,1) in proj. plane
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38. Clipping Object parts outside of the of the view volume are clipped
First we will discuss the openGL functions that set up the projection transformations
Next we will discuss the viewport transformation and setting the viewport in OpenGL
Last, we will go back to projection, and see how the graphics systems carry out efficiently the more general perspective projection by reducing it to the standard perspective, and then to the canonical orthographic
You can set up any projection you want (parallel or perspective) by setting up the the projection matrix directly
Although, more often we use affine transformations to reduce more general projections to the canonical ones
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39. Perspective Viewing in OpenGL
Depth of projection plane (size of clipping window)
inside the pyramid does not matter.
All that matters is object size relative to the window.
The projection plane depth does not affect relative size.
Thus in CG systems usually far or near plane is selected
to be the projection plane. Near in openGL.
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40. Projections in OpenGL Objects not in the view volume are clipped.
frustum
only fixed point
Objects not in the view volume are clipped.
View (projection) plane is front clipping plane.
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41. OpenGL Perspective glFrustum(left, right, bottom, top, near, far);
glMatrixMode(GL_PROJECTION);
glLoadIdentity( );
glFrustum(left, right, bottom, top, near, far);
The relationship between front(near) plane and
COP(origin) define the steepness of the frustum
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42. OpenGL Perspective gluPerspective(fovy, aspect, near, far);
fov is the angle between the top and bottom planes
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43. OpenGL Orthographic Projection
glOrtho(left, right, bottom, top, near, far);
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44. Standard Orthographic Projection
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45. Viewport Transformation
viewport transformation corresponds to the stage where the size of the developed photograph is chosen
The viewport is measured in pixels, in screen window coordinates, which reflect the position of pixels on the screen relative to the lower left corner of the window
vertices outside the viewing volume have been clipped
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46. Viewport Transformation
viewport is the rectangular region of the window where the image is drawn
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47. Defining the Viewport
The window manager, not OpenGL, is responsible for opening a window on the screen
By default the viewport is set to the entire pixel rectangle of the window that's opened
Use the glViewport() command to choose a smaller drawing region
void glViewport(GLint x, GLint y, GLsizei width, GLsizei height);
lower left corner
size of viewport rectangle
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48. Defining The Viewport By default, the initial viewport values are
(0, 0, winWidth, winHeight),
where winWidth and winHeight are the size of the window.
The aspect ratio of a viewport should generally equal the aspect ratio of the viewing volume
If the two ratios are different, the projected image will be distorted as it's mapped to the viewport
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49. Viewport Distortion
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50. Summary World to View coordiantes Camera position: VRP
Viewing transformation
Translate VRP to origin: T(-VRP)
Rotate, aligning view frame (u,v,n) with world frame
The composite transformation R.T will have the effect of transforming the coordinates of vertices from world to view coordinates
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51. Summary Canonical view orthographic projection VRP at the origin
Looking in negative z direction (COP does not matter)
View-up vector is (0,1,0)
View volume is a cube of side 2 center at origin,
(left, right, bottom, top, near, far)=(-1,1,-1,1,1,-1)
(1,1,-1)
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(-1,-1,1)

52. Summary Perspective projection standard viewing VRP at origin
Looking in negative z-direction, look at is (0,0,-1)
COP coincides with VRP, viewing direction is orthogonal to view plane
View-up vector is (0,1,0)
View volume is a regular frustum
(left, right, top, bottom, near, far)=(-1,1,-1,1,1,z_far),
The near plane is at –1, the far is at -z.
Side clipping planes make 45 deg angles with z axis
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53. Canonical Viewing for Perspective
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54. Hidden Surface Removal
z-buffer algorithm – as the polygons are rasterized we keep track of the distance from the VRP to the closest point on each projector. We display only the closest point
Requires a depth or z buffer to store the necessary depth information
glutInitDisplayMode(GLUT_DOUBLE | GLUT_RGB | GLUT_DEPTH);
glEnable(GL_DEPTH_TEST);
glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT);
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55. Hidden Surface Removal
Display only visible surfaces (remove hidden)
Painter’s algorithm: sort polygons according to z, project in
order of decreasing depth
Pixel based: back track rays from pixels to first intersection
Z-buffer algorithm: based on relative depth after projection
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56. Perspective Normalization
How can we apply general perspective projection?
Derive the perspective projection matrix (general 4x4 matrix), load that as a PROJECTION matrix
Trick: Using series of affine transformations to alter the world, so that the image of the distorted world under standard (canonical) projection is the same as the image of the undistorted world under the original general projection
Graphics systems go even farther: perspective projections are implemented internally as orthographic projections (of the distorted world) with respect to the canonical view volume (2x2x2 cube). This way the clipping is very efficient, and z-buffer algorithm is supported by preserving relative depth in the process of transforming objects.
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57. Projection Normalization Idea
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58. Projection Normalization
The distortion is described by a homogeneous-coordinate matrix
Concatenate this matrix with an orthogonal-projection matrix to yield a resulting projection matrix
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59. General Perspective Projection
Projection converts points in 3-D space to points on the projection plane
Three steps (the graphics system implements them):
Converts the viewing volume (general frustum) to the canonical perspective view volume
Next converts the canonical perspective view volume to the canonical orthographic view volume
Applys orthographic projection matrix
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60. The “most” canonical view volume
Orthographic projection with respect to most canonical view volume
Canonical view volume is a 2x2x2 cube whose center is at the origin (default view volume)
How to clip?: simple
How to project?: no division
(1,1,-1)
(-1,-1,1)
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61. General Orthogonal Projection
glMatrixMode(GL_PROJECTION);
glLoadIdentity( );
glOrtho(-1.0, 1.0, -1.0, 1.0, -1.0, 1.0);
determine which objects are clipped out
Projection Matrix maps a view volume to the canonical view volume
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62. Projection Matrix
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63. Oblique Projection degree of obliqueness Viewing, projections
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64. Oblique Projection
shearing
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65. Perspective Projection Matrices
Look for a transformation allows a simple canonical projection by distorting the vertices of an object
Three steps: 1) select canonical viewing volume, 2) introduce the perspective normalization transformation and 3) derive the perspective projection matrix
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66. Perspective Projection Matrices
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67. Perspective Projection Matrices
N is called the perspective normalization matrix. It converts a perspective projection to an orthogonal projection
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68. Projection and Shadows
Shadows are not geometric objects but important for realism
A point is in a shadow if it is not illuminated by any light source
A shadow polygon is a flat shadow that results from projecting the original polygon onto a surface
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69. Projection and Shadows
use a modelview matrix
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