ClippingCyrus Beck Line Clipping(Liang and Barsky)

• Any convex region as window

Parametric line (input line AB):

)1,0(;)()( ∈∈−−++== ttABAtL

A

B

ClippingCyrus Beck Line Clipping(Liang and Barsky)

P n

• Q

Implicit line (window edge):

Tells us on which side of the line the point Q is.

nPQQl ).()( −−==

ClippingCyrus Beck Line Clipping(Liang and Barsky)

P n

• Q

nPQQl ).()( −−==Evaluate

If > 0 inside halfspace of line (plane)If < 0 outside halfspace of line (plane)If = 0 on the line

Should give indications for trivial accept and reject cases.Inside

Outside

ClippingCyrus Beck Line Clipping(Liang and Barsky)

P n

• Q

nPQQl ).()( −−==

Inside

Outside

)()( ABtAtL −−++==

Window edge

Line segment

0)( AND0)( <<<< BlAl

0)( AND0)( >>>> BlAl

Trivial Reject

Trivial Accept

ClippingCyrus Beck Line Clipping(Liang and Barsky)

P

n

ClippingCyrus Beck Line Clipping(Liang and Barsky)

P n

A B

tABAtL )()( −−++==

t for solve;0))(( ==tLl

nPQQl ).()( −−==

0).().(0).)((

0).)((

==−−++−−==−−−−++

==−−

nABtnPAnPABtA

nPtL

ClippingCyrus Beck Line Clipping(Liang and Barsky)

P n

A B

nPBnPAnPA

t

nBAnPA

t

).().().(

).().(

−−−−−−−−==

−−−−==

Clipping

B

n Which ‘t’ to select ?

(t=0)

(t=1)

Cyrus Beck Line Clipping(Liang and Barsky)

A

Clipping

A

B

n

(t=0)

(t=1)

Cyrus Beck Line Clipping(Liang and Barsky)

nABDnAB

nPAt

).().(

).(

−−==−−−−

−−==

D > 0 label t as tEEntering

Clipping

A

B

n

(t=0)

(t=1)

Cyrus Beck Line Clipping(Liang and Barsky)

nABDnAB

nPAt

).().(

).(

−−==−−−−

−−==

D < 0 label t as tLLeaving

Clipping

A

B

n

(t=0)

(t=1)

Cyrus Beck Line Clipping(Liang and Barsky)

t of interest: largest tE smallest tL

maxEtminLt

Clipping

If

Cyrus Beck Line Clipping(Liang and Barsky)

minmaxLE tt >>

Reject

Clipping

Arbitrary Convex Window

P1

V1

Cyrus Beck Line Clipping(Liang and Barsky)

V2

V3

V4

V5V6

V7

V8

P2

E1

E2

E3

E4

E5E6

E7

E8

Clipping

Arbitrary Convex Window

P1

V1

Cyrus Beck Line Clipping(Liang and Barsky)

V2

V3

V4

V5V6

V7

V8

P2

E1

E2

E3

E4

E5E6

E7

E8

Polygon is convex if for alladjacent edges the sign of cross product is same.

positivexEEpositivexEE

::

32

21

.

.

.

Clipping

Arbitrary Window

Cyrus Beck Line Clipping(Liang and Barsky)

Polygon is non-convex

negativexEEpositivexEEpositivexEE

:::

43

32

21

.

.

.

V1 V2

V3

V4

V5V6

E1

E2

E3

E4

Clipping

Arbitrary Window

Cyrus Beck Line Clipping(Liang and Barsky)

V1 V2

V3

V4

V5V6

E1

E2

E3

E4

Make the polygon convex by adding the edge V3V5

Clip against the convex polygon=> P3P4

P1

P2

P3

P4

Clipping

Arbitrary Window

Cyrus Beck Line Clipping(Liang and Barsky)

V3

V4

V5

Clip against the triangle=> P5P4

Subtract P5P4 from P3P4=> P3P5P3

P4

P5