Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Computer GraphicsLecture 8

Dr. Marc Eduard Fr̂ıncuWest University of Timisoara

May 5, 2020

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Outline

1 Introduction

2 Point Clipping

3 Line Clipping

4 Polygon Clipping

5 Assignment

6 Recap

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Image Physics

Clipping

render only parts of the primitiveslocated inside the viewport

avoids cost of futuretransformations of the “invisible”primitive parts

Clipping methods

analytic, e.g., for lines & polygons

during the scan conversion, e.g.,for other primitives

copy pixels from one memoryregion to another

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Image Physics

Clipping

render only parts of the primitiveslocated inside the viewport

avoids cost of futuretransformations of the “invisible”primitive parts

Clipping methods

analytic, e.g., for lines & polygons

during the scan conversion, e.g.,for other primitives

copy pixels from one memoryregion to another

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Point Clipping

Point clipping

– assume window to have left bottom corner (l , b) and right top corner (r , t) ⇒P(x , y) is visible ⇔ x ∈ [l , r ] and y ∈ [b, t]

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Line Clipping

Line clipping methods

Cohen-Sutherland (encoding based) – oldest & most common

Liang-Barsky (parametric) – more efficient

Nicholl-Lee-Nicholl (encoding based) – more efficient than the first one

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Line Clipping

Cohen-Sutherland algorithm

– encode end points using 4 bit code:bit0 = 1 point is left of window

bit1 = 1 point is right of window

bit2 = 1 point is bellow of window

bit3 = 1 point is above of window

1 if C1 ∧ C2 6= 0⇒ reject2 if C1 ∨ C2 = 0⇒ accept3 otherwise subdivide segment & go

to 1):if C1 6= 0 ⇒ P1 not in viewport⇒let i = minj{bitj 6= 0} ⇒new P1 =

⋂segment with edge

i (left=0, right=1, bottom=2,top=3)and new C1 = 0 (is on the edgeof the viewport ⇔ is inside)otherwise if C2 6= 0 applyprevious logic on C2

– where C2 and C1 represent the bitcodes for the endpoints of the line seg-ment

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Line Clipping

Cohen-Sutherland algorithm

– encode end points using 4 bit code:bit0 = 1 point is left of window

bit1 = 1 point is right of window

bit2 = 1 point is bellow of window

bit3 = 1 point is above of window

1 if C1 ∧ C2 6= 0⇒ reject2 if C1 ∨ C2 = 0⇒ accept3 otherwise subdivide segment & go

to 1):if C1 6= 0 ⇒ P1 not in viewport⇒let i = minj{bitj 6= 0} ⇒new P1 =

⋂segment with edge

i (left=0, right=1, bottom=2,top=3)and new C1 = 0 (is on the edgeof the viewport ⇔ is inside)otherwise if C2 6= 0 applyprevious logic on C2

– where C2 and C1 represent the bitcodes for the endpoints of the line seg-ment

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Line Clipping

Cohen-Sutherland algorithm (cont.)

does unnecessary clipping

not efficient

clipping & testing are done in fixed order

efficient when most lines are either rejected or accepted

easy to program

parametric clipping is more efficient

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Line Clipping

Line parametric form

a line segment with endpoints (x1, y1) and (x2, y2) ⇒ (parametric form){x = x1 + u∆x

y = y1 + u∆y– with u ∈ [0, 1], ∆x = x2 − x1 and ∆y = y2 − y1

Liang-Barsky algorithm

– line inside the clipping region:

{l ≤ x1 + u∆x ≤ r

b ≤ y1 + u∆y ≤ tor:

upk ≤ qk , k = 1, 2, 3, 4 and we further need uk = qkpk

with:

p1 = −∆x q1 = x1 − l

p2 = ∆x q2 = r − x1

p3 = −∆y q3 = y1 − b

p4 = ∆y q4 = t − y1

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Line Clipping

Line parametric form

a line segment with endpoints (x1, y1) and (x2, y2) ⇒ (parametric form){x = x1 + u∆x

y = y1 + u∆y– with u ∈ [0, 1], ∆x = x2 − x1 and ∆y = y2 − y1

Liang-Barsky algorithm

– line inside the clipping region:

{l ≤ x1 + u∆x ≤ r

b ≤ y1 + u∆y ≤ tor:

upk ≤ qk , k = 1, 2, 3, 4 and we further need uk = qkpk

with:

p1 = −∆x q1 = x1 − l

p2 = ∆x q2 = r − x1

p3 = −∆y q3 = y1 − b

p4 = ∆y q4 = t − y1

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Line Clipping

Liang-Barsky algorithm (cont.)

if pk < 0 as u increases ⇒ line enters viewport

if pk > 0 as u increases ⇒ line exits viewport

if pk = 0 ⇒ line parallel to an edge of the viewport

if qk < 0 line completely outside (discard it)

for pk 6= 0 uk gives the intersection point

1 compute the uk values

2 take umin = max{0, uk} and umax = min{uk , 1}3 if (umin > umax ) reject line as being outside viewport

4 otherwise draw line from:(x1 + ∆xumin, y1 + ∆yumin)to(x1 + ∆xumax , y1 + ∆yumax )

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Line Clipping

Nicholl-Lee-Nicholl algorithm

least number of comparisons & divisionsdoes not extend well to 3D (as the previous two do)

Algorithm overview

P1 needs to be in three predetermined areas (could require translations orrotations to achieve this):

in the viewportin a “corner region”in an “edge region”

1 do a region testing like Cohen-Sutherland to see if a line segment can beaccepted/rejected:

to determine regions draw lines from P1 through all four corners of the viewport

2 determine the region with the property that no matter where in the region P2 is,the segment will have to be intersected with the same boundaries of the viewport(i.e., compare slopes of segment with the four lines through the corners)

3 determine intersection points (if existing) and draw segment

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Line Clipping

Nicholl-Lee-Nicholl algorithm

least number of comparisons & divisionsdoes not extend well to 3D (as the previous two do)

Algorithm overview

P1 needs to be in three predetermined areas (could require translations orrotations to achieve this):

in the viewportin a “corner region”in an “edge region”

1 do a region testing like Cohen-Sutherland to see if a line segment can beaccepted/rejected:

to determine regions draw lines from P1 through all four corners of the viewport

2 determine the region with the property that no matter where in the region P2 is,the segment will have to be intersected with the same boundaries of the viewport(i.e., compare slopes of segment with the four lines through the corners)

3 determine intersection points (if existing) and draw segment

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Line Clipping

Nicholl-Lee-Nicholl for the left edge region

LeftEdgeRegionCase(x1, y1, x2, y2, visible)algorithm:

Require: x1, y1 start point of the lineRequire: x2, y2 end point of the line1: {end point on the left side of the left edge of the viewport}2: if x2 < l then3: visible→ false4: else5: if y2 < b then6: LeftBottom(l, b, r, t, x1, y1, x2, y2, visible)7: else8: if y2 > t then9: {use symmetry to reduce to LeftBottom case}10: y1 = −y111: y2 = −y212: LeftBottom(l,−t, r,−b, x1, y1, x2, y2, visible)13: {reflect back}14: y1 = −y115: y2 = −y216: else17: ∆x = x2 − x118: ∆y = y2 − y119: if x2 > r then20: y2 = y1 + ∆y(r − x1)/∆x21: x2 = r22: end if23: y1 = y1 + ∆y(l − x1)/∆x24: x1 = l25: visible = true26: end if27: end if28: end if

LeftBottom(l , b, r , t, x1, y1, x2, y2, visible)algorithm:

Require: x1, y1 start point of the lineRequire: x2, y2 end point of the line1: ∆x = x2 − x12: ∆x = y2 − y13: a = (l − x1)∆y4: b = (b − y1)∆x5: {slope of P1P2 ¡ slope from P1 to (l, b)}6: if b > a then7: visible→ false8: else9: visible→ true10: if x2 < r then11: x2 = x1 + b/∆y12: y2 = b13: else14: c = (r − x1)/∆y15: if b > c then16: {P2 between line from P1 to (l, b) and

(r, b)}17: x2 = x1 + b/∆y18: y2 = b19: else20: y2 = y1 + c/∆x21: x2 = r22: end if23: end if24: end if

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Polygon Clipping

Polygon clipping methods

Suterland-Hodgeman

Weiler-Atherton

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Polygon Clipping

Sutherland-Hodgeman

– one of the earliest polygon clipping al-gorithms (1974)

1 clip the polygon against an edge ofthe viewport, i.e., the halfplanedetermined by the edge whichcontains the clip polygon

2 then clip the remaining polygonagainst the next viewport edge

3 repeat until all viewport edges havebeen considered

Four test cases for the polygon edges:

first vertex inside and second outsideviewport

both vertices inside

first vertex outside and second inside

both vertices outside

Concave polygons may be displayed prop-erly with extra lines

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Polygon Clipping

Weiler-Atherton algorithm

Based on the Suterland-Hodgeman algorithm– for ClockWise (CW) polygons example:

outside-to-inside edge ⇒ follow the polygon boundaryinside-to-outside edge ⇒ follow the window boundary in a CW direction

Clips concave polygons correctly

Algorithm overview

1 the inside of the polygon is always to the right as it gets traversed in a CW2 find intersections of the viewport with the polygon to be clipped & insert vertices

at those points into the polygons3 separate the polygon countours that are outside the clip polygon from those

inside of it4 separate intersection vertices into two lists:

entering list: consists of vertices where the polygon edges enter the clip polygonleaving list: consists of vertices where the polygon edges exit the clip polygon

5 clip(a) remove an intersection vertex from the entering list. If none we are done(b) follow the polygon edges to the next intersection(c) jump to the clip polygon vertex list(d) follow the clip polygon vertices to the next intersection(e) jump back to the polygon vertex list(f) repeat (b)-(e) until we are back to the starting point

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Polygon Clipping

Weiler-Atherton algorithm

Based on the Suterland-Hodgeman algorithm– for ClockWise (CW) polygons example:

outside-to-inside edge ⇒ follow the polygon boundaryinside-to-outside edge ⇒ follow the window boundary in a CW direction

Clips concave polygons correctly

Algorithm overview

1 the inside of the polygon is always to the right as it gets traversed in a CW2 find intersections of the viewport with the polygon to be clipped & insert vertices

at those points into the polygons3 separate the polygon countours that are outside the clip polygon from those

inside of it4 separate intersection vertices into two lists:

entering list: consists of vertices where the polygon edges enter the clip polygonleaving list: consists of vertices where the polygon edges exit the clip polygon

5 clip(a) remove an intersection vertex from the entering list. If none we are done(b) follow the polygon edges to the next intersection(c) jump to the clip polygon vertex list(d) follow the clip polygon vertices to the next intersection(e) jump back to the polygon vertex list(f) repeat (b)-(e) until we are back to the starting point

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Polygon Clipping

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Assignment

ASSIGNMENT 5:

Implement the following clipping algorithms and test them on lines and polygons (ifapplicable):

Cohen Sutherland line clipping algorithm

Liang-Barsky line clipping algorithm

Sutherland-Hodgeman polygon clipping algorithm

Hint: use Java2D for loading and manipulating pixels.

Deadline: Beginning of Lecture 10

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Introduction Point Clipping Line Clipping Polygon Clipping Assignment Recap

Lecture Recap

clipping algorithms: reduces costs by rendering only the visible parts of theprimitives

point, line & polygon clipping:Cohen-Sutherland (lines)Liang-Barsky (lines)Nicholl-Lee-Nicholl (lines)Sutherland-Hodgeman (polygon)Weiler-Atherton (polygon)

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