Liang–Barsky algorithm

In computer graphics, the Liang–Barsky algorithm is a line clipping algorithm. The Liang–Barsky algorithm uses the parametric equation of a line and inequalities describing the range of the clipping box to determine the intersections between the line and the clipping box. With these intersections it knows which portion of the line should be drawn. This algorithm is significantly more efficient than Cohen–Sutherland.

The idea of the Liang-Barsky clipping algorithm is to do as much testing as possible before computing line intersections. Consider first the usual parametric form of a straight line:

${\displaystyle x=x_{0}+u(x_{1}-x_{0})=x_{0}+u\Delta x\,\!}$

${\displaystyle y=y_{0}+u(y_{1}-y_{0})=y_{0}+u\Delta y\,\!}$

A point is in the clip window, if

${\displaystyle x_{\text{min}}\leq x_{0}+u\Delta x\leq x_{\text{max}}\,\!}$

and

${\displaystyle y_{\text{min}}\leq y_{0}+u\Delta y\leq y_{\text{max}}\,\!}$,

which can be expressed as the 4 inequalities

${\displaystyle up_{k}\leq q_{k},\quad k=1,2,3,4\,\!}$,

where

${\displaystyle p_{1}=-\Delta x,q_{1}=x_{0}-x_{\text{min}}\,\!}$ (left)

${\displaystyle p_{2}=\Delta x,q_{2}=x_{\text{max}}-x_{0}\,\!}$ (right)

${\displaystyle p_{3}=-\Delta y,q_{3}=y_{0}-y_{\text{min}}\,\!}$ (bottom)

${\displaystyle p_{4}=\Delta y,q_{4}=y_{\text{max}}-y_{0}\,\!}$ (top)

To compute the final line segment:

1. A line parallel to a clipping window edge has ${\displaystyle p_{k}=0}$ for that boundary.
2. If for that ${\displaystyle k}$, ${\displaystyle q_{k}>0}$, the line is completely outside and can be eliminated.
3. When ${\displaystyle p_{k}<0}$ the line proceeds outside to inside the clip window and when ${\displaystyle p_{k}>0}$, the line proceeds inside to outside.
4. For nonzero ${\displaystyle p_{k}}$, ${\displaystyle u={\frac {q_{k}}{p_{k}}}}$ gives the intersection point.
5. For each line, calculate ${\displaystyle u_{1}}$ and ${\displaystyle u_{2}}$. For ${\displaystyle u_{1}}$, look at boundaries for which ${\displaystyle p_{k}<0}$ (outside -> in). Take ${\displaystyle u_{1}}$ to be the largest among ${\displaystyle \left(0,{\frac {q_{k}}{p_{k}}}\right)}$. For ${\displaystyle u_{2}}$, look at boundaries for which ${\displaystyle p_{k}>0}$ (inside -> out). Take ${\displaystyle u_{2}}$ to be the minimum of ${\displaystyle \left(1,{\frac {q_{k}}{p_{k}}}\right)}$. If ${\displaystyle u_{1}>u_{2}}$, the line is outside and therefore rejected.

See also

Algorithms used for the same purpose:

• Cyrus–Beck
• Nicholl–Lee–Nicholl
• Fast-clipping