Notes_456_PolygonClipping2_10 (1).pdf

1/3/2023
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EI-OE-401 Open Elective IV
(Computer Graphics and CAD/CAM)
Module - 2
Polygon Clipping
Prof. Pardeep Kumar
Department of Instrumentation,
Kurukshetra University, Kurukshetra-136119
pardeepk@kuk.ac.in
• Points and lines, Line drawing algorithms, midpoint circle and ellipse
algorithms. Filled area primitives: scan line polygon fill algorithm, boundary-
fill and flood fill algorithms.
• Translation, scaling, rotation, reflection and shear transformations, matrix
representations and homogeneous coordinates, composite transforms,
transformation between coordinate systems. 2-D Viewing: The viewing
pipeline, viewing coordinate reference frame, window to viewport coordinate
transformation, viewing functions, Cohen-Sutherland and Cyrus beck line
clipping algorithms
• Reference Books:
Computer Graphics by Hearn & Donald, PHI
Computer Graphic by Plastock, McGraw Hill
Principle of Interactive graphics by Newman.W Spraul R.F., McGraw Hill
Procedural Elements of computer graphics by Rogers D.F., McGraw Hill
Module – II Syllabus
2
January 3, 2023 DOI, Kurukshetra University
There will be
continuous
evaluation in the
form of
Assignments,
quizzes etc.
After
completion
there will be a
class test

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• Polygon clipping
– Sutherland-Hodgman,
– Weiler-Atherton
• Polygon filling
– Scan filling polygons
– Flood filling polygons
• Introduction and discussion of Assignment work #2
Outline
3
Polygons
Clipping
January 3, 2023 DOI, Kurukshetra University 4

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• Ordered set of vertices (points)
– Usually counter-clockwise
• Two consecutive vertices define an
edge
• Left side of edge is inside
• Right side is outside
• Last vertex implicitly connected to first
• In 3D vertices are co-planar
Polygon
5
• Convex
–All interior angles are less than 180 degrees
• Concave
–Interior angles can be greater than 180 degrees
• Degenerate polygons
–If all vertices are collinear
Polygon Classification
6

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Clipping polygons is more complex than clipping the
individual lines
– Input: polygon
– Output: original polygon, new polygon, or nothing
When can we trivially accept/reject a polygon as opposed
to the line segments that make up the polygon?
Clipping Polygons
7
• Clipping a polygon fill area needs more than line-clipping of the
polygon edges
– would produce an unconnected set of lines
• Must generate one or more closed polylines, which can be filled
with the assigned colour or pattern
Clipping Polygons
8

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What happens to a triangle during clipping?
Possible outcomes:
How many sides can a clipped triangle have?
Clipping Polygons
9
triangle triangle triangle quad triangle 5-gon
How many sides?
10
Seven… Five…

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Polygon clipping
11
• Clipping is symmetric
Polygon clipping is complex
12
• Even when the
polygons are convex
• A really tough case:
concave polygon multiple polygons

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Polygon clipping is nasty
13
• When the polygons are concave
• Basic Algorithm
– N*m intersections
– Then must link all segment
– Not efficient and not even easy
• Lots of different cases
• Issues
– Edges of polygon need
to be tested against
clipping rectangle
– May need to add new
edges
– Edges discarded or divided
– Multiple polygons can result
from a single polygon
Polygon Clipping
14

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Sutherland-Hodgeman
Clipping
• Find the vertices of the new
polygon(s) inside the window.
• Sutherland-Hodgeman 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
Polygon Clipping
16

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Sutherland-Hodgeman Clipping
17
Basic idea:
– Consider each edge of the viewport individually
– Clip the polygon against the viewport edge’s equation
– After doing all edges, the polygon is fully clipped
Input/output for algorithm:
– Input: list of polygon vertices in order
– Output: list of clipped poygon vertices consisting of old vertices (maybe)
and new vertices (maybe)
Note: this is exactly what we expect from the clipping operation
against each edge.
18
Divide and Conquer
• Idea:
‒ Clip single polygon using
single infinite clip edge
‒ Repeat 4 times
Note the generality:
‒ 2D convex n-gons can clip
arbitrary n-gons
‒ 3D convex polyhedra can clip
arbitrary polyhedra

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• Input:
– v1, v2, … vn the vertices defining the polygon
– Single infinite clip edge w/ inside/outside info
• Output:
– v’1, v’2, … v’m, vertices of the clipped polygon
• Do this 4 (or ne) times
• Traverse vertices (edges)
• Add vertices one-at-a-time to output polygon
– Use inside/outside info
– Edge intersections
Sutherland-Hodgman Algorithm
19
Sutherland-Hodgman basic routine:
– Go around polygon one vertex at a time
– Current vertex has position p
– Previous vertex had position s, and it has been added to the output if appropriate
Edge from s to p takes one of four cases:
(Orange line can be a line or a plane)
20
inside outside
s
p
p output
inside outside
s
p
no output
inside outside
s
p
i output
inside outside
s
p
i output
p output

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• Can be done incrementally
• If first point inside add. If outside, don’t add
• Move around polygon from vi to vn and back to vi
• Check vi,vi+1 w.r.t. the clip edge
• Need vi,vi+1‘s inside/outside status
• Add vertex one at a time. There are 4 cases:
21
4 cases
22

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Four cases:
1. s inside plane and p inside plane
– Add p to output
– Note: s has already been added
2. s inside plane and p outside plane
– Find intersection point i
– Add i to output
3. s outside plane and p outside plane
– Add nothing
4. s outside plane and p inside plane
– Find intersection point i
– Add i to output, followed by p
23
inside outside
s
p
p output
inside outside
s
p
i output
inside outside
s
p
no output
inside outside
s
p
i output
p output
• A very general test to determine if a point p is “inside” a plane P,
defined by q and n:
(p - q) • n < 0: p inside P
(p - q) • n = 0: p on P
(p - q) • n > 0: p outside P
Remember: p • n = |p| |n| cos (q)
q = angle between p and n
24
P
n
p
q
P
n
p
q
P
n
p
q

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for each polygon P P’ = P
for each clipping edge (there are 4)
{ Clip polygon P’ to clipping edge
for each edge in polygon P’
Check clipping cases (there are 4)
Case 1 : Output vi+1
Case 2 : Output intersection point
Case 3 : No output
Case 4 : Output intersection point & vi+1
}
Sutherland-Hodgman Algorithm
25
Implementation
26

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27
28

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• The algorithm correctly clips convex polygons, but may display extraneous
lines for concave polygons
• When we have concave polygon then the above algorithm might produce
polygons with coincident edges (Connected edges)
• This is fine for rendering but maybe not for other applications (e.g shadows)
• The Weiler-Atherton algorithm produces separate polygons for each visible
fragment
Issues with Sutherland-Hodgman Algorithm
29
Weiler-Atherton
Algorithm

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• Problem in Sutherland-Hodges.
Weiler-Artherton has a solution
• Clipping other shapes: Circle,
Ellipse, curves
• Clipping a shape against other
shapes
• Clipping the exteriors.
Other Issues in clipping
31
• General clipping algorithm for concave polygons with
holes
– Produces multiple polygons (with holes)
– Make linked list data structure
– Traverse to make new polygon(s)
Weiler-Atherton Algorithm
32

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• Given polygons A and B as linked list of vertices (counter-
clockwise order)
• Find all edge intersections & place in list
• Insert as “intersection” nodes
• Nodes point to A & B
• Determine in/out status of vertices
33
• If “intersecting” edges are parallel, ignore
• Intersection point is a vertex
– Vertex of A lies on a vertex or edge of B
– Edge of A runs through a vertex of B
• Replace vertex with an intersection node
Intersection Special Cases
34

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If polygons don’t intersect
– Union
• If one inside the other, return polygon that surrounds the other
• Else, return both polygons
– Intersection
• If one inside the other, return polygon inside the other
• Else, return no polygons
Boolean Special Cases
35
• Find a vertex of A outside of B
• Traverse linked list
• At each intersection point switch to other polygon
• Do until return to starting vertex
• All visited vertices and nodes define union’ed polygon
Weiler-Atherton Algorithm: Union
36

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• Start at intersection point
– If connected to an “inside” vertex, go there
– Else step to an intersection point
– If neither, stop
• Traverse linked list
• At each intersection point switch to other polygon and remove
intersection point from list
• Do until return to starting intersection point
• If intersection list not empty, pick another one
• All visited vertices and nodes define and’ed polygon
Weiler-Atherton Algorithm: Intersection
37
38
1
2
3
5
4
6
8
7
9
0
a
b
c
d
a b
c
d
0
1
2
3
4
5
6
7
8
9
A
B
i j
k
l
clip region
polygon
loop of polygon
vertices
loop of region
vertices

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• Find the intersection vertices and connect them in the two lists
39
1
2
3
5
4
6
8
7
9
0
a
b
c
d
a b
c
d
0
1
2
3
4
5
6
7
8
9
A
B
i j
k
l
clip region
polygon
i
l
k
j
Add vertex i:
40
a b
c
d
0
1
2
3
4
5
6
7
8
9
A
B
i j
k
l
clip region
polygon
1
2
3
5
4
6
8
7
9
0
a
b
c
d
i
l
k
j
Add vertex l:

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41
a b
c
d
0
1
2
3
4
5
6
7
8
9
A
B
i j
k
l
clip region
polygon
1
2
3
5
4
6
8
7
9
0
a
b
c
d
i
l
k
j
Add vertex k:
• Computed loop
42
a b
c
d
0
1
2
3
4
5
6
7
8
9
A
B
i j
k
l
clip region
polygon
1
2
3
5
4
6
8
7
9
0
a
b
c
d
i
l
k
j
Add vertex j:

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• Classify each intersection vertex as Entering or Leaving
43
a b
c
d
0
1
2
3
4
5
6
7
8
9
A
B
i j
k
l
clip region
polygon
1
2
3
5
4
6
8
7
9
0
a
b
c
d
i
l
k
j
Entering
Leaving
• Start at an entering vertex
• If you encounter a leaving vertex
swap to right hand (clip polygon)
loop
• If you encounter an entering
vertex swap to left hand
(polygon) loop
• A loop is finished when you arrive
back at start
• Repeat whilst there are entering
vertices
Capture clipped polygons
44
1
2
3
5
4
6
8
7
9
0
a
b
c
d
i
l
k
j
Entering
Leaving
Loop 1:
L, 4, 5, K
Loop 2:
J, 9, 0, i

1/3/2023
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January 3, 2023
DOI, Kurukshetra University 45
Any Question?
Thanks!

Notes_456_PolygonClipping2_10 (1).pdf

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