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Advanced Graphics &
Animation
3D Viewing Pipeline
Three-dimensional Viewing Pipeline
Transform into view coordinates
and Canonical view volume
Clip against canonical view
volume
Project on to view plane
Map into viewport
Transform to physical
Device coordinates
transform
clip
transform
World coordinates(3D)
View coordinates(3D)
View coordinates(3D)
View coordinates(2D)
Normalized device coordinates
Physical device coordinates
Parallel Projection
 Mostly used by drafters and engineers to create working drawings
of an object which preserves its scale and shape.
 The distance between the COP and the projection plane is infinite
i.e. The projectors are parallel to each other and have a fixed
direction.
P1
P2
P1‟
P2‟
Projection plane
• Orthographic projection:
here direction of projection is perpendicular to the
view plane.
• Axonometric projection:
when direction of projection is not parallel to any of
three principal axes.
Perspective Projection
• Generalization of the principles used by artists in drawing of scenes.
• It takes object representation in view space (L.H.S.) and produce a
projection on the view plane (canvas used by the artist).
• The projection of a 3D point onto the viewplane is the intersection of
the line from the point to the COP (eye position of the artist).
Cont.
• Distance between the COP and projection plane is finite.
• Perspective projection does not preserve object scale and shape.
P1
P2
P1‟
P2‟
Projection plane
COP
Perspective Anomalies
Perspective foreshortening
- the farther an object from the COP, the smaller it appears (i.e. its
projected size becomes smaller).
cop
View plane
Vanishing points
- there is an illusion that certain sets of parallel lines (that are not
parallel to the view plane ) appear to meet at some point on the view
plane.
- the vanishing point for any set of parallel lines that are parallel
to one of principal axis is referred to as a principal vanishing point
(PVP).
- the number of PVPs is determined by the number of principal
axes intersected by the view plane.
One principal vanishing point projection
- occurs when the projection plane is perpendicular to one of the
principal axes (x, y or z ).
Vanishing point
View plane is parallel to
XY-plane and intersects
Z-axis only.
Two principal vanishing point projection
X-axis
Vanishing point Z-axis
Vanishing point
View plane intersects
Both X and Z axis but
not the Y axis
• Three principal vanishing point intersection
View plane intersects all
Three of the principal axis
X, Y and Z axis
VP1
VP3
VP2
Deriving Perspective Projection
Assume
point vertex denoting COP : (xc,yc,zc)
point on the object : (x1,y1,z1)
representation of “projection ray” containing above two points
x = xc + ( x1-xc) u …………..eq 1
y = yc + ( y1-yc) u …………..eq 2
z = zc + ( z1-zc) u …………..eq 3
The projected point (x2, y2,D) will be the point where this line intersects the
XY plane . Putting z=0 for this intersection point in eq 3 .
u = - zc / z1-zc
Substituting into first two equations,
x2 =
y2 =
Value of D may be computed which is different from zero (to
preserve depth relationship between objects)
D = z1 / (z1 –zc)
z-z
zx-zx
c1
c11c
z-z
zy-zy
c1
c11c
Standard perspective projection
y
x
z
A (x,0,z)
A‟ (x‟,0,0)
P (x ,y, z)
P‟ (x‟,y‟,0)
C(0,0,-d)
do
(0,0,0)B (0,0,z)
z
Using similar triangles ABC and A‟OC,
x‟ = d.x / (z+d)
y‟ = d.y / (z+d)
z‟ = 0
matrix representation :
d100
0000
00d0
000d
 viewing based on synthetic camera analogy.
Specifying an arbitrary 3D view
By selecting different viewing parameters, user
can position the synthetic camera.
View reference
point
View-up vector
View plane
Effect of change of viewing parameters
 Imagine a string tied to „view reference point‟ on one end and to the
synthetic camera on the other end.
 By changing viewing parameters, we can swing the camera through
the arc or change the length of the string.
- changing the view distance is equivalent to how far away
from the object the camera is when it takes the picture.
- changing the view reference point will change the part of the
object that is shown at the origin.
Cont.
- changing the view plane normal is equivalent to
taking photograph of object from different orientations.
- changing view-up is equivalent to twisting the camera
in our hands. It fixes the camera angle.
View Volume
- The view volume bounds that portion of the
3D space that is to be clipped out and
projected onto the view plane.
View Volume for Perspective Projection
- its shape is semi-infinite pyramid with apex at the
view point and edge passing through the corners of the
window.
cop
View
window
Front clipping
plane
Back clipping
plane
Frustum view volume
View Volume for Parallel Projection
-It's shape is an infinite parallelepiped with sides parallel to the
direction of projection.
Parallelepiped
Viewed volume
View
window
Front clipping
plane
Back clipping
plane
Producing a Canonical view volume for a
perspective projection
cop
View
window
Front
clip
Back clip
View frustum
centerlineGeneral shape
for the Perspective
View volume
View volume
Step 1: shear the view volume so that centerline of the frustum is
perpendicular to the view plane and passes through the center of the
view window.
Frustum centerline
View volume
Step2: scale view volume inversely proportional to the distance from
the view window, so that shape of view volume becomes rectangular
parallelepiped.
View volume
Converting object coordinates to view plane
coordinates
 similar to the process of rotation about an arbitrary axis
zw
Yw
xw
World coordinate system
Yv
Xv
VRP
View plane (eye)
coordinate system
Steps:
1. Translate origin to view reference point (VRP).
2. Translate along the view plane normal by view distance.
3. Align object coordinate‟s z-axis with view plane coordinates z-
axis (the view plane normal).
a)- Rotate about x-axis to place the line (ie. Object coordinates
z-axis) in the view plane coordinates xz-plane.
b)- Rotate about y-axis to move the z axis to its proper position.
c)- Rotate about the z-axis until x and y axis are in place in the
view plane coordinates.
Ref:
„Computer Graphics‟ by S. Harrington (pp. 279-284)

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3 d viewing

  • 2. Three-dimensional Viewing Pipeline Transform into view coordinates and Canonical view volume Clip against canonical view volume Project on to view plane Map into viewport Transform to physical Device coordinates transform clip transform World coordinates(3D) View coordinates(3D) View coordinates(3D) View coordinates(2D) Normalized device coordinates Physical device coordinates
  • 3. Parallel Projection  Mostly used by drafters and engineers to create working drawings of an object which preserves its scale and shape.  The distance between the COP and the projection plane is infinite i.e. The projectors are parallel to each other and have a fixed direction. P1 P2 P1‟ P2‟ Projection plane
  • 4. • Orthographic projection: here direction of projection is perpendicular to the view plane. • Axonometric projection: when direction of projection is not parallel to any of three principal axes.
  • 5. Perspective Projection • Generalization of the principles used by artists in drawing of scenes. • It takes object representation in view space (L.H.S.) and produce a projection on the view plane (canvas used by the artist). • The projection of a 3D point onto the viewplane is the intersection of the line from the point to the COP (eye position of the artist).
  • 6. Cont. • Distance between the COP and projection plane is finite. • Perspective projection does not preserve object scale and shape. P1 P2 P1‟ P2‟ Projection plane COP
  • 7. Perspective Anomalies Perspective foreshortening - the farther an object from the COP, the smaller it appears (i.e. its projected size becomes smaller). cop View plane
  • 8. Vanishing points - there is an illusion that certain sets of parallel lines (that are not parallel to the view plane ) appear to meet at some point on the view plane. - the vanishing point for any set of parallel lines that are parallel to one of principal axis is referred to as a principal vanishing point (PVP). - the number of PVPs is determined by the number of principal axes intersected by the view plane.
  • 9. One principal vanishing point projection - occurs when the projection plane is perpendicular to one of the principal axes (x, y or z ). Vanishing point View plane is parallel to XY-plane and intersects Z-axis only.
  • 10. Two principal vanishing point projection X-axis Vanishing point Z-axis Vanishing point View plane intersects Both X and Z axis but not the Y axis
  • 11. • Three principal vanishing point intersection View plane intersects all Three of the principal axis X, Y and Z axis VP1 VP3 VP2
  • 12. Deriving Perspective Projection Assume point vertex denoting COP : (xc,yc,zc) point on the object : (x1,y1,z1) representation of “projection ray” containing above two points x = xc + ( x1-xc) u …………..eq 1 y = yc + ( y1-yc) u …………..eq 2 z = zc + ( z1-zc) u …………..eq 3 The projected point (x2, y2,D) will be the point where this line intersects the XY plane . Putting z=0 for this intersection point in eq 3 . u = - zc / z1-zc
  • 13. Substituting into first two equations, x2 = y2 = Value of D may be computed which is different from zero (to preserve depth relationship between objects) D = z1 / (z1 –zc) z-z zx-zx c1 c11c z-z zy-zy c1 c11c
  • 14. Standard perspective projection y x z A (x,0,z) A‟ (x‟,0,0) P (x ,y, z) P‟ (x‟,y‟,0) C(0,0,-d) do (0,0,0)B (0,0,z) z
  • 15. Using similar triangles ABC and A‟OC, x‟ = d.x / (z+d) y‟ = d.y / (z+d) z‟ = 0 matrix representation : d100 0000 00d0 000d
  • 16.  viewing based on synthetic camera analogy. Specifying an arbitrary 3D view
  • 17. By selecting different viewing parameters, user can position the synthetic camera. View reference point View-up vector View plane
  • 18. Effect of change of viewing parameters  Imagine a string tied to „view reference point‟ on one end and to the synthetic camera on the other end.  By changing viewing parameters, we can swing the camera through the arc or change the length of the string. - changing the view distance is equivalent to how far away from the object the camera is when it takes the picture. - changing the view reference point will change the part of the object that is shown at the origin.
  • 19. Cont. - changing the view plane normal is equivalent to taking photograph of object from different orientations. - changing view-up is equivalent to twisting the camera in our hands. It fixes the camera angle.
  • 20. View Volume - The view volume bounds that portion of the 3D space that is to be clipped out and projected onto the view plane.
  • 21. View Volume for Perspective Projection - its shape is semi-infinite pyramid with apex at the view point and edge passing through the corners of the window. cop View window Front clipping plane Back clipping plane Frustum view volume
  • 22. View Volume for Parallel Projection -It's shape is an infinite parallelepiped with sides parallel to the direction of projection. Parallelepiped Viewed volume View window Front clipping plane Back clipping plane
  • 23. Producing a Canonical view volume for a perspective projection cop View window Front clip Back clip View frustum centerlineGeneral shape for the Perspective View volume View volume
  • 24. Step 1: shear the view volume so that centerline of the frustum is perpendicular to the view plane and passes through the center of the view window. Frustum centerline View volume
  • 25. Step2: scale view volume inversely proportional to the distance from the view window, so that shape of view volume becomes rectangular parallelepiped. View volume
  • 26. Converting object coordinates to view plane coordinates  similar to the process of rotation about an arbitrary axis zw Yw xw World coordinate system Yv Xv VRP View plane (eye) coordinate system
  • 27. Steps: 1. Translate origin to view reference point (VRP). 2. Translate along the view plane normal by view distance. 3. Align object coordinate‟s z-axis with view plane coordinates z- axis (the view plane normal). a)- Rotate about x-axis to place the line (ie. Object coordinates z-axis) in the view plane coordinates xz-plane. b)- Rotate about y-axis to move the z axis to its proper position. c)- Rotate about the z-axis until x and y axis are in place in the view plane coordinates.
  • 28. Ref: „Computer Graphics‟ by S. Harrington (pp. 279-284)