Liang–Barsky algorithm: Difference between revisions

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{{Short description|Line-clipping 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]].

In [[computer graphics]], the '''Liang–Barsky algorithm''' (named after [[You-Dong Liang]] and [[Brian A. Barsky]]) is a [[line clipping]] algorithm. 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:

:<math>x = x_0 + t (x_1 - x_0) = x_0 + t \Delta x,</math>

C# implementation for Liang-Barsky algorithm

:<math>y = y_0 + t (y_1 - y_0) = y_0 + t \Delta y.</math>

<source lang="csharp">

internal sealed class LiangBarskyClipping : IClippingAlgorithm {

private Vector2 _clipMin, _clipMax;

A point is in the clip window, if

public IEnumerable<Vector2> GetBoundingPolygon() {

:<math>x_\text{min} \le x_0 + t \Delta x \le x_\text{max}</math>

yield return _clipMin;

and

yield return new Vector2(_clipMax.X, _clipMin.Y);

:<math>y_\text{min} \le y_0 + t \Delta y \le y_\text{max},</math>

yield return _clipMax;

which can be expressed as the 4 inequalities

yield return new Vector2(_clipMin.X, _clipMax.Y);

:<math>t p_i \le q_i, \quad i = 1, 2, 3, 4,</math>

}

where

:<math>

public void SetBoundingRectangle(Vector2 start, Vector2 end) {

\begin{align}

_clipMin = start;

p_1 &= -\Delta x, & q_1 &= x_0 - x_\text{min}, & &\text{(left)} \\

_clipMax = end;

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{align}

</math>

To compute the final [[line segment]]:

public void SetBoundingPolygon(IEnumerable<Vector2> points) {

# A line parallel to a clipping window edge has <math>p_i = 0</math> for that boundary.

throw new NotSupportedException("see Capabilities =)");

# If for that <math>i</math>, <math>q_i < 0</math>, then the line is completely outside and can be eliminated.

}

# When <math>p_i < 0</math>, the line proceeds outside to inside the clip window, and when <math>p_i > 0</math>, the line proceeds inside to outside.

# For nonzero <math>p_i</math>, <math>u = q_i / p_i</math> gives <math>t</math> for the intersection point of the line and the window edge (possibly projected).

# The two actual intersections of the line with the window edges, if they exist, are described by <math>u_1</math> and <math>u_2</math>, calculated as follows. For <math>u_1</math>, look at boundaries for which <math>p_i < 0</math> (i.e. outside to inside). Take <math>u_1</math> to be the largest among <math>\{0, q_i / p_i\}</math>. For <math>u_2</math>, look at boundaries for which <math>p_i > 0</math> (i.e. inside to outside). Take <math>u_2</math> to be the minimum of <math>\{1, q_i / p_i\}</math>.

# If <math>u_1 > u_2</math>, the line is entirely outside the clip window. If <math>u_1 < 0 < 1 < u_2</math> it is entirely inside it.

<syntaxhighlight lang="c++">

private delegate bool ClippingHandler(float p, float q);

// Liang–Barsky line-clipping algorithm

#include<iostream>

#include<graphics.h>

#include<math.h>

using namespace std;

public bool ClipLine(ref Line line) {

Vector2 P = line.End - line.Start;

float tMinimum = 0, tMaximum = 1;

// this function gives the maximum

ClippingHandler pqClip = delegate(float directedProjection,

float maxi(float arr[], int n) {

float directedDistance) {

float m = 0;

if (directedProjection == 0) {

for (int i = 0; i < n; ++i)

if (directedDistance < 0) return false;

if (m < }arr[i])

m = else {arr[i];

return m;

float amount = directedDistance / directedProjection;

}

if (directedProjection < 0) {

if (amount > tMaximum) return false;

else if (amount > tMinimum) tMinimum = amount;

}

else {

if (amount < tMinimum) return false;

else if (amount < tMaximum) tMaximum = amount;

}

}

return true;

};

// this function gives the minimum

if (pqClip(-P.X, line.Start.X - _clipMin.X)) {

float mini(float arr[], int n) {

if (pqClip(P.X, _clipMax.X - line.Start.X)) {

float m = 1;

if (pqClip(-P.Y, line.Start.Y - _clipMin.Y)) {

for (int i = 0; i < n; ++i)

if (pqClip(P.Y, _clipMax.Y - line.Start.Y)) {

if (tMaximumm <> 1arr[i]) {

m = arr[i];

line.End.X = line.Start.X + tMaximum * P.X;

return m;

line.End.Y = line.Start.Y + tMaximum * P.Y;

}

}

if (tMinimum > 0) {

line.Start.X += tMinimum * P.X;

line.Start.Y += tMinimum * P.Y;

}

return true;

}

}

}

}

return false;

}

void liang_barsky_clipper(float xmin, float ymin, float xmax, float ymax,

public ClippingCapabilities Capabilities {

float x1, float y1, float x2, float y2) {

get {

// defining variables

return ClippingCapabilities.RectangleWindow;

float p1 = -(x2 - }x1);

float p2 }= -p1;

float p3 = -(y2 - y1);

float p4 = -p3;

float q1 = x1 - xmin;

public override string ToString() {

float q2 = xmax - x1;

return "Liang-Barsky algorithm";

float q3 = y1 - ymin;

float q4 = ymax - y1;

float exitParams[5], entryParams[5];

int exitIndex = 1, entryIndex = 1;

exitParams[0] = 1;

entryParams[0] = 0;

rectangle(xmin, ymin, xmax, ymax); // drawing the clipping window

if ((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) {

float r1 = q1 / p1;

float r2 = q2 / p2;

if (p1 < 0) {

entryParams[entryIndex++] = r1;

exitParams[exitIndex++] = r2;

} else {

entryParams[entryIndex++] = r2;

exitParams[exitIndex++] = r1;

}

}

if (p3 != 0) {

float r3 = q3 / p3;

float r4 = q4 / p4;

if (p3 < 0) {

entryParams[entryIndex++] = r3;

exitParams[exitIndex++] = r4;

} else {

entryParams[entryIndex++] = r4;

exitParams[exitIndex++] = r3;

}

}

float clippedX1, clippedY1, clippedX2, clippedY2;

float u1, u2;

u1 = maxi(entryParams, entryIndex); // maximum of entry points

u2 = mini(exitParams, exitIndex); // minimum of exit points

if (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 segment

setlinestyle(1, 1, 0);

line(x1, y1, clippedX1, clippedY1); // original start to clipped start

line(x2, y2, clippedX2, clippedY2); // original end to clipped end

}

// This code was implemented by Grishul Eugeny as part of preparation

// to exam in ITMO university

</source>

int main() {

==See also==

cout << "\nLiang-Barsky Line Clipping";

cout << "\nThe system window layout is: (0,0) at bottom left and (631, 467) at top right";

cout << "\nEnter the coordinates of the window (xmin, ymin, xmax, ymax): ";

float xmin, ymin, xmax, ymax;

cin >> xmin >> ymin >> xmax >> ymax;

cout << "\nEnter the endpoints of the line (x1, y1) and (x2, y2): ";

float x1, y1, x2, y2;

cin >> x1 >> y1 >> x2 >> y2;

int gd = DETECT, gm;

initgraph(&gd, &gm, ""); // using winbgim

liang_barsky_clipper(xmin, ymin, xmax, ymax, x1, y1, x2, y2);

getch();

closegraph();

}

</syntaxhighlight>

==See also==

Algorithms used for the same purpose:

* [[Cyrus-BeckCyrus–Beck algorithm]]

* [[Nicholl–Lee–Nicholl algorithm]]

* [[Nicholl-Lee-Nicholl]]

* [[Fast-Clipping clipping]]

==References==

* Liang, Y. D., and Barsky, B., "[https://dl.acm.org/doi/pdf/10.1145/357332.357333 A New Concept and Method for Line Clipping]", ''ACM Transactions on Graphics'', 3(1):1–22, January 1984.

* Liang, Y. D., B. A., Barsky, and M. Slater, ''[https://www2.eecs.berkeley.edu/Pubs/TechRpts/1992/CSD-92-688.pdf Some Improvements to a Parametric Line Clipping Algorithm]'', CSD-92-688, Computer Science Division, University of California, Berkeley, 1992.

* James D. Foley. ''[https://books.google.com/books/about/Computer_graphics.html?id=-4ngT05gmAQC Computer graphics: principles and practice]''. Addison-Wesley Professional, 1996. p.&nbsp;117.

==External links==

* http://hinjang.com/articles/04.html#eight

* [http://www.skytopia.com/project/articles/compsci/clipping.html Skytopia: The Liang-Barsky line clipping algorithm in a nutshell!]

{{DEFAULTSORT:Liang-}}

[[Category:Computer graphics]]

[[Category:Line clipping algorithms]]