Liang Barsky algorithm, Lecture notes of Computer Graphics

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Computer Graphics

Liang-Barsky

(Line Clipping Algorithm)

Covered by:-

Arvind Kumar

Assistant Professor

Vidya College of Engg.

Liang-Barsky Line Clipping Algorithm This algorithm uses the parametric equations for a line and solves four inequalities to find the range of the parameter for which the line is in the viewport. Parametric Equations are given below:

𝟏

+ t*Δ 𝒙

y = 𝒚

𝟏

+ t*Δ y , 0 ≤ t ≥ 1

Where Δ 𝑥 = 𝑥

2

• 𝑥 1

Δ y = 𝑦

2

• 𝑦 1

As we know the point clipping condition is:

𝒘𝒎𝒊𝒏

𝒘𝒎𝒂𝒙

………………(i)

𝒘𝒎𝒊𝒏

𝒘𝒎𝒂𝒙

.……………...(ii)

Liang-Barsky Line Clipping Algorithm

Following observations can be :

if 𝒑

𝒊

= 0 : Line is parallel to the clipping boundary.

if 𝒒𝒊 < 0 : Line is completely outside the clipping

boundary.

if 𝒑𝒊 < 0 : Line proceeds from outside to inside of the

clipping boundary.

if 𝒑𝒊 < 0 : Line proceeds from inside to outside of the

clipping boundary.

i.e ,

Line intersect the clipping boundary

Liang-Barsky Line Clipping Algorithm

This algorithm calculate value parameter t: 𝒕𝟏 and 𝒕𝟐.

➢ The value of 𝒕𝟏 is determined checking the edge for which

line proceed from outside to inside( 𝒑

𝒊

< 0 ). The

maximum value is taken.

➢ The value of 𝒕𝟐 is determined checking the edge for which

line proceed from inside to outside(𝒑𝒊 > 0 ). The minimum

value is taken if 𝒕 𝟏

𝒕 𝟐 , the line is completely outside the window

and it can be rejected.

Otherwise, the value of 𝒕𝟏 & 𝒕𝟐 , are substituted in the

parametric equations to get the intersection points of the

clipped line.

Liang-Barsky Line Clipping Algorithm

4. If 𝒑𝒊 = 0 , then { Line is parallel to the clipping boundary. Now , if 𝒒𝒊 < 0 then { Line is completely outside the clipping boundary. } } else Check for intersection point.
5. Initialize values 𝒕𝟏= 0 and 𝒕𝟐 = 1.
6. Calculate values for 𝒒𝒊 𝒑𝒊 for i= 1 , 2 , 3 , 4.

Liang-Barsky Line Clipping Algorithm

7. Select values of 𝒒𝒊 𝒑𝒊 , where ( 𝒑𝒊 < 0 ), assign maximum value out of them as 𝒕𝟏.
8. Select values of 𝒒𝒊 𝒑𝒊 , where ( 𝒑𝒊 > 0 ), assign minimum value out of them as 𝒕𝟐.
9. If 𝒕𝟏<𝒕𝟐 , then calculate intersection points are as : { 𝒙𝒙𝟏 = 𝒙𝟏 + 𝒕𝟏 *Δ 𝒙 𝒚𝒚𝟏 = 𝒚𝟏 + 𝒕𝟏 *Δ y 𝒙𝒙𝟐 = 𝒙𝟏 + 𝒕𝟐 *Δ 𝒙 𝒚𝒚𝟐 = 𝒚𝟏 + 𝒕𝟐 *Δ y } Draw clipped line (𝒙𝒙𝟏, 𝒚𝒚𝟏 , 𝒙𝒙𝟐 , 𝒚𝒚𝟐)
10. Stop

1. Calculate the value of parameter 𝒑𝒊 and 𝒒𝒊, for i = 1 , 2 , 3 & 4. 𝒑𝟏 = - Δ 𝒙 𝒒𝟏 = 𝒙𝟏 - 𝒙𝒘𝒎𝒊𝒏 𝒑𝟏 = - 50 𝒒𝟏 = −𝟓 𝒒𝟏 𝒑𝟏

𝒒𝟐 𝒑𝟐

𝒒𝟑 𝒑𝟑

𝒒𝟒 𝒑𝟒

➢ 𝒕𝟏= max( 0 , 0. 25 , 0. 1 ) = 0. 25 , for the values (𝒑𝒊 < 0 ) ➢ 𝒕𝟐= min( 0. 25 , 0. 1 , 1 )= 0. 3 , for the values (𝒑𝒊 > 0 )

Calculate the intersection points: 𝒙𝒙𝟏 = 𝒙𝟏 + 𝒕𝟏 *Δ 𝒙 = 10 + 0. 25 * 50 = 22. 5 𝒚𝒚𝟏 = 𝒚𝟏 + 𝒕𝟏 *Δ y = 10 + 0. 25 * 20 = 15 𝒙𝒙𝟐 = 𝒙𝟏 + 𝒕𝟐 *Δ 𝒙 = 10 + 0. 3 * 50 = 25 𝒚𝒚𝟐 = 𝒚𝟏 + 𝒕𝟐 *Δ y = 10 + 0. 3 * 20 = 16 The Clipped Line end points are ( 22.5,15) & (25, 16)