2 DIMENSIONAL
VIEWING
Ceng 477
Introduction to Computer Graphics
Fall 2010-2011
Computer Engineering
METU
Viewing Pipeline Revisited
Modeling
Transformations
Model
Viewing
Transformations
M1
Model
M2
3D World
Scene
V
3D View
Scene
Model
M3
WCS
MCS
VCS
Rasterization
P
Clip
Projection
Normalize
2D/3D Device
Scene
NDCS
2D Image
DCS
SCS
●
Model coordinates to World coordinates:
Modelling transformations
Model coordinates:
1 circle (head),
2 circles (eyes),
1 line group (nose),
1 arc (mouth),
2 arcs (ears).
With their relative
coordinates and sizes
World coordinates:
All shapes with their
absolute coordinates and sizes.
circle(0,0,2)
circle(-.6,.8,.3) circle(.6,.8,.3)
lines[(-.4,0),(-.5,-.3),(.5,.3),(.4,0)]
arc(-.6,0,.6,0,1.8,180,360)
arc(-2.2,.2,-2.2,-.2,.8,45,315)
arc(2.2,.2,2.2,-.2,.8,225,135)
●
World coordinates to Viewing coordinates:
Viewing transformations
World coordinates
Viewing coordinates:
Viewers position and view
angle. i.e. rotated/translated
●
Projection: 3D to 2D. Clipping depends on
viewing frame/volume. Normalization: device
independent coordinates
Viewing coordinates:
Device Independent Coordinates:
Invisible shapes deleted, others
reduced to visible parts.
3 arcs, 1 circle, 1 line group
●
Device Independent Coordinates to Device
Coordinates. Rasterization
Device Independent
Coordinates
Unit measures
Screen Coordinates or
Device Coordinates
Pixels
2D Viewing
●
World coordinates to Viewing coordinates
●
Window to Viewport.
Window: A region of the scene selected for viewing
(also called clipping window)
Viewport: A region on display device for mapping to
window
Viewport
Window
World Coordinates
Viewing Coordinates
Clipping Window vs. Viewport
●
●
●
The clipping window selects what we want
to see in our virtual 2D world.
The viewport indicates where it is to be
viewed on the output device (or within the
display window)
By default the viewport have the same
location and dimensions of the GLUT display
window you create
–
But it can be modified so that only a part of
the display window is used for OpenGL display
The clipping window
ywmax
yv max
Rectangular
Window
yw min
yv min
xw min
xv min
xw max
y view
y world
y view
y0
yv max
Rotated
Window
yv min
x view
x0
xv max
x world
xv min
xv max
x view
World-coordinates to Viewing Coordinates
y view
y view
y world
y world
y0
y view
x view
x0
x world

x world
x viewy
T( x0 , y0 )
●
Mwc,vc= R·T
view
R ( )
x view
Normalization
●
●
Coordinate transformation:Different sizes and/or
height width ratios?
For any point:
xv  xvmin
xw  xwmin
=
xvmax  xvmin xwmax  xwmin
yv  yvmin
yw  ywmin
=
yvmax  yvmin ywmax  ywmin
should hold.
●
xv = xvmin + xw  xwmin

xvmax  xvmin 

xv  xvmin
xw  xwmin
=
xvmax  xvmin xwmax  xwmin
yv = yvmin +  yw  ywmin

yvmax  yvmin 

yv  yvmin
yw  ywmin
=
yvmax  yvmin ywmax  ywmin
xwmax  xwmin
ywmax  ywmin
This can also be accomplished in 2 steps:
1.
Scale over the fixed point:
Sxwmin , ywmin , s x , s y 
2.
Translate lower-left corner of the clipping window to
the lower-left corner of the viewport
Txvmin  xwmin , yvmin  ywmin 
OpenGL 2D Viewing Functions
●
OpenGL, GLU, and
GLUT provide
functions to specify
clipping windows,
viewports, and
display windows
within a video
screen.
Setting up a 2D Clipping-Window
●
●
●
glMatrixMode (GL_PROJECTION)
glLoadIdentity (); // reset, so that new viewing
parameters are not combined
with old ones (if any)
gluOrtho2D (xwmin, xwmax, ywmin, ywmax);
or
●
●
glOrtho (xwmin, xwmax, ywmin, ywmax, zwmin, zwmax);
Objects within the clipping window are transformed to
normalized coordinates (-1,1)
Setting up a Viewport
●
●
●
glViewport (xvmin, yvmin, vpWidth, vpHeight);
All the parameters are given in integer screen
coordinates relative to the lower-left corner of the
display window.
If we do not invoke this function, by default, a
viewport with the same size and position of the
display window is used (i.e., all of the GLUT window
is used for OpenGL display)
Creating a GLUT Display Window
●
glutInitWindowPosition (xTopLeft, yTopLeft);
–
the integer parameters are relative to the top-left
corner of the screen
●
glutInitWindowSize (dwWidth, dwHeight);
●
glutCreateWindow (“Title of Display Window”);
●
glutInitDisplayMode (GLUT_SINGLE | GLUT_RGB)
–
●
Specification of the buffer that will be used
glClearColor (red, green, blue, alpha)
–
Specify the background color
Multiple GLUT windows
●
Multiple windows may be created within an
OpenGL program
–
Need window ids to manage multiple windows
–
windowID = glutCreateWindow(“Window1”);
–
glutDestroyWindow (windowID)
// to distroy the window
●
General functions (like glutInitDisplayMode)
are applied to the current display window.
We can set the current window to a specific
window with:
–
glutSetWindow (windowID);
Other functions
●
GLUT provide functions to relocate, resize,
minimize, resize to fullscreen, change
window title, hide, show, bring to front, or
send to back, select a specific cursor for the
current display window. (pages 309-311 in
the textbook)
Display callback function
●
●
●
Each separate GLUT window can have its own
function to specify what will be drawn inside. E.g.,
–
glutSetWindow (w1);
–
glutDisplayFunction (wireframeDisplay);
–
glutSetWindow (w2);
–
glutDisplayFunction (solidDisplay);
Display callback functions are called only when GLUT
determines that the display content should be
renewed. To update the display manually call:
glutPostRedisplay();
glutIdleFunc (functionName) could be used in animations
OpenGL 2D Viewing Example
●
●
2 Viewports
One triangle is displayed
in two colors and
orientations in 2
viewports
glClear (GL_COLOR_BUFFER_BIT);
glColor3f(0.0, 0.0, 1.0);
glViewport(0, 0, 300, 300);
drawCenteredTriangle();
glColor3f(1.0, 0.0, 0.0);
glViewport(300, 0, 300, 300);
glRotatef(90.0, 0.0, 0.0, 1.0);
drawCenteredTriangle();
glutInitWindowSize (600, 300);
Clipping Algorithms
●
●
●
Clipping: identifying the parts of the objects
that will be inside of the window.
Everything outside the clipping window is
eliminated from the scene description (i.e.,
not scan converted) for efficiency.
Point clipping:
xwmin  x  xwmax
ywmin  y  ywmax
Line Clipping
●
When both end points are inside all four boundaries
(completely inside) or both end points are outside any one of
the boundaries (completely outside) no extra processing need
to be done. Otherwise:
–
For each line determine the intersection with boundaries.
Parametric line equations:
x = x1 + ux2  x1 ,
y = y1 + u y2  y1 , 0  u  1
Find u for all boundary lines. Not very efficient.
Liang-Barsky Line Clipping
●
Use parametric equations for efficiency
xwmin  x1 + ux  xwmax
ywmin  y1 + uy  ywmax , 0  u  1
rewrite these inequalities:
upk  qk , k = 1,2,3,4
p1 =  Δx, q1 = x1  xwmin
p2 = Δx,
q2 = xwmax  x1
p3 =  Δy, q3 = y1  ywmin
p4 = Δy,
q4 = ymax  y1
qk
u=
pk
The point where the line
intersects the borders
upk  qk , 0  u  1
if pk < 0, outside to inside transitio n, candidate for u1
if pk > 0, inside to outside transitio n, candidate for u2
●
●
u 1
u2
Problem: find the clipping interval [u1,u2]
If pk<0 u1 = max(u1,pk/qk)
If pk>0 u2 = min(u2,pk/qk)
u1
u 0
Liang-Barsky Algorithm
u1 = 0 ,u2 = 1
for k = 1,2,3,4
find pk
if pk < 0 rk =
qk
pk
u1 = maxu1,rk 
else if pk > 0 rk =
u2 = minu2,rk 
else
qk
pk
parallel line, treat specially
if u1 > u2
line is outside, totally reject
x2 = x1 + u2 x, y2 = y1 + u2 y
x1 = x1 + u1x, y1 = y1 + u1y
Example
x = 8, y = 5
k =1
7
p2 = 8, q2 = 7  0 = 7, r2 =
8
7  7
u 2 = min  ,1 =
8  8
k=3
1 2  2
u1 = max ,  =
8 5  5
P2 = 8,5
y=6
1
p1 = 8, q1 = 0  1 = 1, r1 =
8
1  1
u1 = max ,0 =
8  8
k=2
p3 = 5, q3 = 0  2 = 2, r3 =
x= 7
x =1
u2 = 7 / 8
y=2
u1 = 2 / 5
P1 = 0,0
u1 = 1 / 8
xwmin  x1 + ux  xwmax
ywmin  y1 + uy  ywmax , 0  u  1
upk  qk , k = 1,2,3,4
k=4
2
5
p4 = 5, q4 = 7  0 = 7, r4 =
7
5
7 7  7
u 2 = min  ,  =
5 8  8
P1 = 0 + 2 / 5  8,0 + 2 / 5  5 = 16 / 5,2  P2 = 0 + 7 / 8  8,0 + 7 / 8  5 = 7,35 / 8
p1 =  Δx, q1 = x1  xwmin
p2 = Δx,
q2 = xwmax  x1
p3 =  Δy, q3 = y1  ywmin
p4 = Δy,
q4 = ymax  y1
u=
qk
pk
NLN
(2,6)
LT
P2
(6,7)
(6,6)
L
L
P1
(1,4)
LR
L
(2,2)
LB
(6,2)
NLN
(2,6)
LT
P2
(6,7)
(6,6)
L
P1
(1,4)
For example when P2 is in region LT if
(2,2)
Slope P1PTR < Slope P1P2 < Slope P1PTL
L
LR
L
LB
Or
(YT - Y1)/(XR – X1)) < (Y2 - Y1)/(X2 – X1) < (YT – Y1)/(XL – X1)
Ex:
●
●
(6-4)/(6-1) < (7-4)/(6-1) < (6-4)/(2-1)
2/5 < 3/5 < 2
And we clip the entire line if
(YT – Y1)(X2 – X1) < (XL – X1)(Y2 – Y1)
(6,2)
NLN
●
●
●
From the parametric equations:
x = x1 + (x2 – x1)u
y = y1 + (y2 – y1)u
An intersection position on the left
boundary is x = xL with u = (xL – x1)/(x2
– x1) will give coordinate y = y1 +((y2 –
y1)/(x2 – x1)) * (xL – x1)
An intersection position on the top
boundary has y = yT with u = (yT –
y1)/(y2 – y1) will give coordinate x = x1
+((x2 – x1)/(y2 – y1)) * (yT – y1)
NLN Line Clipping(1/4)
●
Three possible position for a line end
point P1
–
Equal position with rotation or translation
2001. 7. 13
31
NLN Line Clipping (2/4)
●
●
●
The four clipping region when P1 is inside the
clip window
The four clipping region when P1 is directly left
of the clip window
The two possible sets of clipping regions when P1
is above and to the left of the clip window
2001. 7. 13
32
NLN Line Clipping (3/4)
●
Region Determination
–
P1 is left of the clipping rectangle, then P2 is in
the region LT if
Slope P1PTR < slope P1P2 < slope P1PTL
or
–
yT  y1 y2  y1 yT  y1


xR  x1 x2  x1 xL  x1
Clipping the entire line
( yT  y1 )( x2  x1 )  ( xL  x1 )( y2  y1 )
2001. 7. 13
33
NLN Line Clipping (4/4)
●
Intersection Position Calculation
x  x1  ( x2  x1 )u
y  y1  ( y2  y1 )u
–
Left Boundary
x  xL
u  ( xL  x1 ) /( x2  x1 )
–
Top Boundary
y  yT
u  ( yT  y1 ) /( y2  y1 )
2001. 7. 13
y2  y1
y  y1 
( xL  x1 )
x2  x1
x2  x1
x  x1 
( yT  y1 )
y2  y1
34
Polygon Clipping
●
●
Find the vertices of the
new polygon(s) inside the
window.
Sutherland-Hodgman
Polygon Clipping:
Check each edge of the polygon
against all window boundaries.
Modify the vertices based on
transitions. Transfer the new
edges to the next clipping
boundary.
Shutherland-Hodgman Polygon Clipping
●
Traverse edges for borders; 4 cases:
–
V1 outside, V2 inside: take V1' and V2
–
V1 inside, V2 inside: take V1 and V2
V1
V1'
V2
V1
V2
–
V1 inside, V2 outside: take V1 and V2'
V1
V2
–
V1 outside, V2 outside: take none
V1
V2
V2'
●
v1
v2
v6
●
v5'
v3'
v5
v3
v4
Left border:
v1 v2 both inside
v1 v2
v2 v3 both inside
v2 v3
.....
“
“
........
v1,v2,v3,v4v5,v6,v1
Bottom Border:
v1 v2 both inside
v2 v3 v2 i, v3 o
v3 v4 both outside
v4 v5 both outside
v5 v6 v5 o, v6 i
v6 v1 both inside
v1,v2,v3',v5',v6,v1
v1 v2
v2 v3'
none
none
v5' v6
v6 v1
●
v1
●
v2'
v2
v1'
v2'''
v6
v2''
v5'
●
v3'
v5
v3
v4
v1,v2,v3',v5',v6,v1
Right border:
v1 v2
v1 i, v2 o
v2 v3'
v2 o, v3'i
v3' v5'
both inside
v5' v6
both inside
v6 v1
both inside
v1,v2',v2'',v3',v5',v6,v1
v1 v2'
v2'' v3'
v3' v5'
v5' v6
v6 v1
Top Border:
v1 v2' both outside none
v2' v2'' v2' o, v2'' i v2''' v2''
v2'' v3' both inside v2'' v3'
v3' v5' both inside v3' v5'
v5' v6 both inside v5' v6
v6 v1 v6 i, v1 o
v6 v1'
v2''',v2'',v3',v5',v6,v1'
v1
v2'
v2
v1'
v2'''
v1'
v2'''
v6
v6
v2''
v5'
v3'
v5
v3
v4
v2''
v5'
v3'
Other Issues in Clipping
●
●
●
●
Problem in Shutherland-Hodgman.
Weiler-Atherton has a solution
Clipping other shapes:
Circle, Ellipse, Curves.
Clipping a shape against another
shape
Clipping the interior.
Weiler-Atherton Polygon Clipping
●
When using Sutherland-Hodgeman,
concavities can end up linked
Remember
this?
●
A different clipping algorithm, the
Weiler-Atherton algorithm, creates
separate polygons
Weiler-Atherton Polygon Clipping
●
Example:
Out -> In
Add clip vertex
Add end vertex
In -> In
Add end vertex
In -> Out
Add clip vertex
Cache old direction
Follow clip edge until
(a) new crossing found
(b) reach vertex already
added
Weiler-Atherton Polygon Clipping
●
Example (cont’d):
Continue from
cached vertex and
direction
Out -> In
Add clip vertex
Add end vertex
Follow clip edge until
In -> Out
(a) new crossing found
Add clip vertex
(b) reach vertex already
Cache old direction
added
Weiler-Atherton Polygon Clipping
●
Example (cont’d):
Continue from
cached vertex and
direction
Nothing added
Finished
Final Result:
2 unconnected
polygons