Title: 3D vIEWING
1CA 302 Computer Graphics and Visual Programming
Aydin Öztürk aydin.ozturk_at_ege.edu.tr http//www.u
be.ege.edu.tr/ozturk
2Overview
- Viewing a 3D scene
- Projections
- Parallel and perspective
3Overview
- Depth cueing and hidden surfaces
- Identifying visible lines and surfaces
4Overview
5Overview
- Exploded and cutaway views
6Overview
- 3D and stereoscopic viewing
73D Viewing Pipeline
MC
DC
ViewportTransformation
ModelingTransformation
NC
WC
Normalization Transformation and Clipping
ViewingTransformation
VC
PC
ProjectionTransformation
8Viewing 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.
9Specifying The View Coordinates
xv
yv
- 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.
yw
zv
P0(x0 , y0 , z0)
xw
zw
10Specifying The View Coordinates
- To establish the viewing reference frame, we
first pick a world coordinate position P0(x0
, y0 , z0) calle the view point or viewing
position (sometimes the eye position or the
camera position. - 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.
11Specifying 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.
yv
xv
xv
yw
zv
N
P0
P
xw
zw
12Specifying The View Coordinates
- We choose a world coordinate position P on an
object (we can - think of this point as the position where we aim
a camera to take a picture of the object and this
point establishes the direction for N.
yv
xv
xv
yw
zv
N
P0
P
xw
zw
13Specifying The View Coordinates
- 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
14Specifying 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 and is
defined by selecting a positon relative to the
world coordinate origin.
yv
xv
V
yw
zv
N
P0
P
xw
zw
15Specifying The View Coordinates
- 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
16Specifying The View Coordinates
- Usually, it can be difficult to determine a
direction for V that is precisely prependicular
to N. - Therefore, the user defined orientation of V is
projected onto a plane that is prependicular to
N. - Any direction for the view-up vector V can be
chosen. A convenient choice is often in a
direction parallel to the world yw axis that is
V(0,1,0)
yv
xv
V
yw
zv
N
P0
P
xw
zw
17Specifying The View Coordinates
- 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.
V
P0
N
N
18Transformation 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
19Transformation From World To Viewing Coordinates
- First, we translate the view reference point to
the origin of the world coordinate system
yw
xw
zw
20Transformation 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
21Transformation 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.
22Transformation 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.
23Transformation From World To Viewing Coordinates
- Then, we rotate around the world yw axis to align
the zw and zv axes.
24Transformation From World To Viewing Coordinates
- The final rotation is about the world zw axis to
align the yw and yv axes.
25Transformation From World To Viewing Coordinates
- The complete transformation from world to viewing
coordinate transformation matrix is obtaine as
the matrix product
26Transformation 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 N and V
, these unit vectors are calculated as
27Transformation From World To Viewing Coordinates
- This method also automatically adjusts the
direction for V so that v is
perpendicular to n. The rotation matrix for the
viewing transformation is then
28Transformation From World To Viewing Coordinates
- The matrix for translating the viewing origin to
the world origin is
29Transformation From World To Viewing Coordinates
- The composite matrix for the viewing
transformation is then
30Transformation From World To Viewing Coordinates
An Example For 2D System
y
P(5,5)
y'
2
x'
T300
2
0 2 4 6
P0(4,3)
x
0 2 4 6
31Transformation From World To Viewing Coordinates
An Example For 2d System
y
0 2 4 6
P
y'
x'
2
2
T300
P0
x
2 4 6
32Transformation From World To Viewing Coordinates
An Example For 2d System
0 2 4 6
33Transformation From World To Viewing Coordinates
An Example For 2d System
34Transformation From World To Viewing Coordinates
An Example For 2d System
y
0 1 2 3
P
1
x'
y'
1
n
v
T300
x
1 2 3
P0
35Projections
- 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
36Classical Viewings
- Hand drawings Determined by a specific
relationship between the object and the viewer.
37Parallel Projections
- Coordinate Positions are transformed to the view
plane along parallel lines.
View Plane
P'2
P2
P1
P'1
38Parallel 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.
39 Orthographic Parallel Projection
- The orthographic transformation
40 Orthographic Parallel Projection
41 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.
42Oblique 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.
43 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.
44Oblique Parallel Projection
yv
(xp, yp)
a
(x, y, z)
L
f
xv
(x, y)
zv
45Oblique Parallel Projection
- The oblique transformation
46Oblique Parallel Projection
47Perspective Projections
- First discovered by Donatello, Brunelleschi, and
DaVinci during Renaissance - Objects closer to viewer look larger
- Parallel lines appear to converge to single point
48Perspective 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)
49Perspective Projections
- In the real world, objects exhibit perspective
foreshortening distant objects appear smaller - The basic situation
50Perspective Projections
- When we do 3-D graphics, we think of the screen
as a 2-D window onto the 3-D world
51Perspective Projections
- The geometry of the situation is that of similar
triangles. View from above
View plane
(xp, yp)
d
52Perspective Projections
- Desired result for a point x, y, z, 1T
projected onto the view plane
53Perspective Projections
54Perspective Projections
55Projection 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
56View 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.
57View Volumes
View volume
View volume (frustum)
window
zv
window
Front Plane
Back Plane
Projection Reference Point
Front Plane
Parallel Projection
Parallel Projection
Perspective Projection
58Clipping
- 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.
59Clipping 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. -
60Clipping 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.
61Clipping 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.
62Clipping Polygon Surface
Viev volume
63Steps 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.
64 Normalized View Volumes
(Xvmax,Yvmax, Zvmax)
65OpenGL Projection Commands
66OpenGL 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
67Projections 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
68Projections in OpenGL
69Perspective 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)
70Perspective 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)
71Parallel 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)