Liang–Barsky algorithm

In computer graphics, the Liang–Barsky algorithm (named after You-Dong Liang and Brian A. Barsky) is a line clipping algorithm first published in early 1984. The Liang–Barsky algorithm uses the parametric equation of a line and inequalities describing the range of the clipping window to determine the intersections between the line and the clip window. With these intersections, it knows which portion of the line should be drawn. So 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.

The algorithm uses the parametric form of a straight line:

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

${\displaystyle y=y_{0}+t(y_{1}-y_{0})=y_{0}+t\Delta y.}$

A point is in the clip window, if

${\displaystyle x_{\text{min}}\leq x_{0}+t\Delta x\leq x_{\text{max}}}$

and

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

which can be expressed as the 4 inequalities

${\displaystyle tp_{i}\leq q_{i},\quad i=1,2,3,4,}$

where

${\displaystyle {\begin{aligned}p_{1}&=-\Delta x,&q_{1}&=x_{0}-x_{\text{min}},&&{\text{(left)}}\\p_{2}&=\Delta x,&q_{2}&=x_{\text{max}}-x_{0},&&{\text{(right)}}\\p_{3}&=-\Delta y,&q_{3}&=y_{0}-y_{\text{min}},&&{\text{(bottom)}}\\p_{4}&=\Delta y,&q_{4}&=y_{\text{max}}-y_{0}.&&{\text{(top)}}\end{aligned}}}$

To compute the final line segment:

1. A line parallel to a clipping window edge has ${\displaystyle p_{i}=0}$ for that boundary.
2. If for that ${\displaystyle i}$, ${\displaystyle q_{i}<0}$, then the line is completely outside and can be eliminated.
3. When ${\displaystyle p_{i}<0}$, the line proceeds outside to inside the clip window, and when ${\displaystyle p_{i}>0}$, the line proceeds inside to outside.
4. For nonzero ${\displaystyle p_{i}}$, ${\displaystyle u=q_{i}/p_{i}}$ gives ${\displaystyle t}$ for the intersection point of the line and the window edge (possibly projected).
5. The two actual intersections of the line with the window edges, if they exist, are described by ${\displaystyle u_{1}}$ and ${\displaystyle u_{2}}$, calculated as follows. For ${\displaystyle u_{1}}$, look at boundaries for which ${\displaystyle p_{i}<0}$ (i.e. outside to inside). Take ${\displaystyle u_{1}}$ to be the largest among ${\displaystyle \{0,q_{i}/p_{i}\}}$. For ${\displaystyle u_{2}}$, look at boundaries for which ${\displaystyle p_{i}>0}$ (i.e. inside to outside). Take ${\displaystyle u_{2}}$ to be the minimum of ${\displaystyle \{1,q_{i}/p_{i}\}}$.
6. If ${\displaystyle u_{1}>u_{2}}$, the line is entirely outside the clip window. If ${\displaystyle u_{1}<0<1<u_{2}}$ it is entirely inside it.

// Liang–Barsky line-clipping algorithm#include<iostream>#include<graphics.h>#include<math.h>usingnamespacestd;// this function gives the maximumfloatmaxi(floatarr[],intn){floatm=0;for(inti=0;i<n;++i)if(m<arr[i])m=arr[i];returnm;}// this function gives the minimumfloatmini(floatarr[],intn){floatm=1;for(inti=0;i<n;++i)if(m>arr[i])m=arr[i];returnm;}voidliang_barsky_clipper(floatxmin,floatymin,floatxmax,floatymax,floatx1,floaty1,floatx2,floaty2){// defining variablesfloatp1=-(x2-x1);floatp2=-p1;floatp3=-(y2-y1);floatp4=-p3;floatq1=x1-xmin;floatq2=xmax-x1;floatq3=y1-ymin;floatq4=ymax-y1;floatexitParams[5],entryParams[5];intexitIndex=1,entryIndex=1;exitParams[0]=1;entryParams[0]=0;rectangle(xmin,ymin,xmax,ymax);// drawing the clipping windowif((p1==0&&q1<0)||(p2==0&&q2<0)||(p3==0&&q3<0)||(p4==0&&q4<0)){outtextxy(80,80,"Line is parallel to clipping window!");return;}if(p1!=0){floatr1=q1/p1;floatr2=q2/p2;if(p1<0){entryParams[entryIndex++]=r1;exitParams[exitIndex++]=r2;}else{entryParams[entryIndex++]=r2;exitParams[exitIndex++]=r1;}}if(p3!=0){floatr3=q3/p3;floatr4=q4/p4;if(p3<0){entryParams[entryIndex++]=r3;exitParams[exitIndex++]=r4;}else{entryParams[entryIndex++]=r4;exitParams[exitIndex++]=r3;}}floatclippedX1,clippedY1,clippedX2,clippedY2;floatu1,u2;u1=maxi(entryParams,entryIndex);// maximum of entry pointsu2=mini(exitParams,exitIndex);// minimum of exit pointsif(u1>u2){outtextxy(80,80,"Line is outside the clipping window!");return;}clippedX1=x1+(x2-x1)*u1;clippedY1=y1+(y2-y1)*u1;clippedX2=x1+(x2-x1)*u2;clippedY2=y1+(y2-y1)*u2;setcolor(CYAN);line(clippedX1,clippedY1,clippedX2,clippedY2);// draw clipped segmentsetlinestyle(1,1,0);line(x1,y1,clippedX1,clippedY1);// original start to clipped startline(x2,y2,clippedX2,clippedY2);// original end to clipped end}intmain(){cout<<"Liang-Barsky Line Clipping";cout<<"The system window layout is: (0,0) at bottom left and (631, 467) at top right";cout<<"Enter the coordinates of the window (xmin, ymin, xmax, ymax): ";floatxmin,ymin,xmax,ymax;cin>>xmin>>ymin>>xmax>>ymax;cout<<"Enter the endpoints of the line (x1, y1) and (x2, y2): ";floatx1,y1,x2,y2;cin>>x1>>y1>>x2>>y2;intgd=DETECT,gm;initgraph(&gd,&gm,"");// using winbgimliang_barsky_clipper(xmin,ymin,xmax,ymax,x1,y1,x2,y2);getch();closegraph();}