UBI 516 Advanced Computer Graphics Three Dimensional Viewing

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UBI 516 Advanced Computer GraphicsThree
Dimensional Viewing

• Aydin Öztürk
• ozturk_at_ube.ege.edu.tr
• http//www.ube.ege.edu.tr/ozturk

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Overview

• Viewing a 3D scene
• Projections
• Parallel and perspective

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Overview

• Depth cueing and hidden surfaces
• Identifying visible lines and surfaces

4
Overview

• Surface rendering

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Overview

• Exploded and cutaway views

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Overview

• 3D and stereoscopic viewing

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3D Viewing Pipeline
MC
DC
ViewportTransformation
ModelingTransformation
NC
WC
Normalization Transformation and Clipping
ViewingTransformation
VC
PC
ProjectionnTransformation
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Viewing Coordinates

• Generating a view of an object in 3D is similar
  to photographing the object.
• Whatever appears in the viewfinder is projected
  onto the flat film surface.
• Depending on the position, orientation and
  aperture size of the camera corresponding views
  of the scene is obtained.

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Specifying The View Coordinates

• For a particular view of a scene first we
  establish viewing-coordinate system.
• A view-plane (or projection plane) is set up
  perpendicular to the viewing z-axis.
• World coordinates are transformed to viewing
  coordinates, then viewing coordinates are
  projected onto the view plane.

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Specifying The View Coordinates

• To establish the viewing reference frame, we
  first pick a world coordinate position called the
  view reference point.
• This point is the origin of our viewing
  coordinate system. If we choose a point on an
  object we can think of this point as the position
  where we aim a camera to take a picture of the
  object.

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Specifying The View Coordinates

• Next, we select the positive direction for the
  viewing z-axis, and the orientation of the view
  plane, by specifying the view-plane normal
  vector, N.
• We choose a world coordinate position P and this
  point establishes the direction for N.
• OpenGL establishes the direction for N using the
  point P as a look at point relative to the
  viewing coordinate origin.

yv
xv
xv
yw
zv
N
P0
P
xw
zw
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Specifying The View Coordinates

• Finally, we choose the up direction for the view
  by specifying view-up vector V.
• This vector is used to establish the positive
  direction for the yv axis.
• The vector V is perpendicular to N.
• Using N and V, we can compute a third vector U,
  perpendicular to both N and V, to define the
  direction for the xv axis.

yv
xv
V
yw
zv
N
P0
P
xw
zw
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Specifying The View Coordinates
V

• To obtain a series of views of a scene , we can
  keep the the view reference point fixed and
  change the direcion of N. This corresponds to
  generating views as we move around the viewing
  coordinate origin.

P0
N
N
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Transformation From World To Viewing Coordinates

• Conversion of object descriptions from world to
  viewing coordinates is equivalent to
  transformation that superimpoes the viewing
  reference frame onto the world frame using the
  translation and rotation.

yw
xw
zw
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Transformation From World To Viewing Coordinates

• First, we translate the view reference point to
  the origin of the world coordinate system

yw
xw
zw
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Transformation From World To Viewing Coordinates

• Second, we apply rotations to align the xv,, yv
  and zv axes with the world xw, yw and zw axes,
  respectively.

yw
xw
xv
zw
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Transformation From World To Viewing Coordinates

• If the view reference point is specified at word
  position (x0, y0, z0), this point is translated
  to the world origin with the translation matrix T.

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Transformation From World To Viewing Coordinates

• The rotation sequence requires 3 coordinate-axis
  transformation depending on the direction of N.
• First we rotate around xw-axis to bring zv into
  the xw -zw plane.

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Transformation From World To Viewing Coordinates

• Then, we rotate around the world yw axis to align
  the zw and zv axes.

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Transformation From World To Viewing Coordinates

• The final rotation is about the world zw axis to
  align the yw and yv axes.

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Transformation From World To Viewing Coordinates

• The complete transformation from world to viewing
  coordinate transformation matrix is obtaine as
  the matrix product

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Transformation From World To Viewing Coordinates

• Another method for generating the
  rotation-transformation matrix is to calculate
  uvn vectors and obtain the composite rotation
  matrix directly. Given the vectors and
  , these unit vectors are calculated as

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Transformation From World To Viewing Coordinates

• This method also automatically adjusts the
  direction for so that is
  perpendicular to . The rotation matrix for
  the viewing transformation is then

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Transformation From World To Viewing Coordinates

• The matrix for translating the viewing origin to
  the world origin is

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Transformation From World To Viewing Coordinates

• The composite matrix for the viewing
  transformation is then

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Transformation From World To Viewing Coordinates
An Example For 2d System
y
2
P(5,5)
y'
x'
2
0 2 4 6

T300
P0(4,3)
0 2 4 6

x
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Transformation From World To Viewing Coordinates
An Example For 2d System

• Translation

y
0 2 4 6

P
2
x'
2
y'
T300
2 4 6

x
P0
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Transformation From World To Viewing Coordinates
An Example For 2d System

• Rotation

0 2 4 6

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Transformation From World To Viewing Coordinates
An Example For 2d System

• New coordinates

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Transformation From World To Viewing Coordinates
An Example For 2d System

• Alternative Method

y
0 1 2 3

P
1
x'
y'
1
n
v
T300
x
1 2 3
P0
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Projections

• Once WC description of the objects in a scene are
  converted to VC we can project the 3D objects
  onto 2D view-plane.
• Two types of projections
• -Parallel Projection
• -Perspective Projection

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Classical Viewings

• Hand drawings Determined by a specific
  relationship between the object and the viewer.

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Parallel Projections

• Coordinate Positions are transformed to the view
  plane along parallel lines.

View Plane
P2
P'2
P1
P'1
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Parallel Projections

• Orthographic parallel projection The
  projection is perpendicular to the view plane.
• Oblique parallel projecion The
  parallel projection is not perpendicular to the
  view plane.

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Orthographic Parallel Projection

• The orthographic transformation

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Orthographic Parallel Projection
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Oblique Parallel Projection

• The projectors are still ortogonal to the
  projection plane
• But the projection plane can have any orientation
  with respect to the object.
• It is used extensively in architectural and
  mechanical design.

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Oblique Parallel Projection

• Preserve parallel lines but not angles
• Isometric view Projection plane is placed
  symmetrically with respect to the three principal
  faces that meet at a corner of object.
• Dimetric view Symmetric with two faces.
• Trimetric view General case.

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Oblique Parallel Projection

• Preserve parallel lines but not angles
• Isometric view Projection plane is placed
  symmetrically with respect to the three principal
  faces that meet at a corner of object.
• Dimetric view Symmetric with two faces.
• Trimetric view General case.

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Oblique Parallel Projection
yv
(xp, yp)
a
(x, y, z)
L
f
xv
(x, y)
zv
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Oblique Parallel Projection

• The oblique transformation

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Oblique Parallel Projection
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Perspective Projections

• First discovered by Donatello, Brunelleschi, and
  DaVinci during Renaissance
• Objects closer to viewer look larger
• Parallel lines appear to converge to single point

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Perspective Projections

• In perspective projection object positions are
  transformed to the view plane along lines that
  converge to a point called the projection
  reference point (or center of projection)

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Perspective Projections

• In the real world, objects exhibit perspective
  foreshortening distant objects appear smaller
• The basic situation

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Perspective Projections

• When we do 3-D graphics, we think of the screen
  as a 2-D window onto the 3-D world

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Perspective Projections

• The geometry of the situation is that of similar
  triangles. View from above

View plane
(xp, yp)
d
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Perspective Projections

• Desired result for a point x, y, z, 1T
  projected onto the view plane

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Perspective Projections
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Perspective Projections
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Projection Matrix

• We talked about geometric transforms, focusing on
  modeling transforms
• Ex translation, rotation, scale, gluLookAt()
• These are encapsulated in the OpenGL modelview
  matrix
• Can also express projection as a matrix
• These are encapsulated in the OpenGL projection
  matrix

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View Volumes

• When a camera used to take a picture, the type of
  lens used determines how much of the scene is
  caught on the film.
• In 3D viewing, a rectangular view window in the
  view plane is used to the same effect. Edges of
  the view window are parallel to the xv-yv axes
  and window boundary positions are specified in
  viewing coordinates.

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View Volumes
View volume
View volume (frustum)
zv
window
Back Plane
window
Front Plane
Back Plane
Projection Reference Point
Front Plane
Parallel Projection
Parallel Projection
Perspective Projection
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Clipping

• An algorithm for 3D clipping identifies and saves
  all surface segments within the view volume for
  display.
• All parts of object that are outside the view
  volume are discarded.

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Clipping Lines

• To clip a line against the view volume, we need
  to test the relative position of the line using
  the view volumes boundary plane equation.
• An end point (x,y,z) of a line segment is outside
  a boundary plane if
• where A, B, C and D are the plane parameters
  for that boundary.

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Clipping Polygon Surface

• To clip a polygon surface, we can clip the
  individual polygon edges.
• First we test the coordinate extends against each
  boundary of the view volume to determine whether
  the object is completely inside or completely
  outside of that boundary.
• If the object has intersection with the boundary
  then we apply intersection calculations.

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Clipping Polygon Surface

• The projection operation can take place before
  the view- volume clipping or after clipping.
• All objects within the view volume map to the
  interior of the specified projection window.
• The last step is to transform the window contents
  to a 2D view port.

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Clipping Polygon Surface
Viev volume
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Steps For Normalized View Volumes

• A scene is constructed by transforming object
  descriptions from modeling coordinates to wc.
• The world descriptions are converted to viewing
  coordinates.
• The viewing coordinates are transformed to
  projection coordinates which effectively converts
  the view volume into a rectangular
  parallelepiped.
• The parallelepiped is mapped into the unit cube
  called normalized projection coordinate system.
• A 3D viewport within the unit cube is
  constructed.
• Normalized projection coordinates are converted
  to device coordinates for display.

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Normalized View Volumes
y
x
(Xwmax, ywmax, zback)
(Xvmax, yvmax, zvmax)
z
(Xwmin, ywmin, zfront)
Parallelepiped View Volume
(Xvmin, yvmin, zwmin)
Unit Cube
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Orthogonal Projection Normalization
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Oblique Projection Normalization

• Angles of projection
• ? for x axis
• ? for y axis
• Shearing matrix H(?, ?)

?
?
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Oblique Projection Normalization
Finished ? No, this is a sheared view volume, so
we have to apply orthogonal transformation
PPorth STH
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Perspective Projection Normalization

• Perspective Normalization is Trickier

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Perspective Projection Normalization

• Consider N
• After multiplying
• p Np

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Perspective Projection Normalization

• After dividing by w, p -gt p

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Perspective Projection Normalization

• Quick Check

• If x z
• x -1
• If x -z
• x 1

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Perspective Projection Normalization

• What about z?
• if z zmax
• if z zmin
• Solve for a and b such that zmin -gt -1 and zmax
  -gt1
• Resulting z is nonlinear, but preserves
  ordering of points
• If z1 lt z2 z1 lt z2

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Perspective Projection Normalization

• We did it. Using matrix, N
• Perspective viewing frustum transformed to cube
• Orthographic rendering of cube produces same
  image as perspective rendering of original
  frustum

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OpenGL Projection Commands
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OpenGL Look-At Function

• OpenGL utility function
• VRP eyePoint (eyex, eyey, eyez)
• VPN ( atPoint eyePoint ) (atx, aty, atz)
  (eyex, eyey, eyez)
• VUP upPoint eyePoint (upx, upy, upz)

gluLookAt(eyex, eyey, eyez, atx, aty, atz, upx,
upy, upz)
look-at positioning
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Projections in OpenGL

• Angle of view, field of view
• Only objects that fit within the angle of view of
  the camera appear in the image
• View volume, view frustum
• Be clipped out of scene
• Frustum truncated pyramid

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Projections in OpenGL
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Perspective in OpenGL

• Specification of a frustum
• near, far positive number !!
• ? zmax far
• ? zmin near

glMatrixMode(GL_PROJECTION) glLoadIdentity(
) glFrustum(xmin, xmax, ymin, ymax, near, far)
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Perspective in OpenGL

• Specification using the field of view
• fov angle between top and
• bottom planes
• fovy the angle of view in the
• up (y) direction
• aspect ratio width / height

glMatrixMode(GL_PROJECTION) glLoadIdentity(
) gluPerspective(fovy, aspect, near, far)
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Parallel Viewing in OpenGL

• Orthographic viewing function
• OpenGL provides only this parallel-viewing
  function
• near lt far !!
• ? no restriction on the sign
• ? zmax far
• ? zmin near

glMatrixMode(GL_PROJECTION) glLoadIdentity(
) glOrtho(xmin, xmax, ymin, ymax, near, far)
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Optional Clipping Planes

• glClipPlane(id, PlaneParameters)glEnable(id) /
  / id GL_CLIP_PLANE0, GL_CLIP_PLANE1, ... //
  PlaneParameters A,B,C and D of the plane. .
  .glDisable(id)