CSCE 441 Computer Graphics: Clipping Lines Jinxiang Chai

OpenGL Geometric Primitives All geometric primitives are specified by vertices GL_LINE_STRIP GL_LINE_LOOP GL_POLYGON GL_LINES GL_POINTS GL_TRIANGLE_STRIP GL_QUAD_STRIP GL_TRIANGLES GL_QUADS GL_TRIANGLE_FAN

Why Clip? We do not want to waste time drawing objects that are outside of viewing window (or clipping window)

Clipping Points Given a point (x, y) and clipping window (xmin, ymin), (xmax, ymax), determine if the point should be drawn (xmax,ymax) (x,y) (xmin,ymin)

Clipping Points Given a point (x, y) and clipping window (xmin, ymin), (xmax, ymax), determine if the point should be drawn (xmax,ymax) (x,y) xmin=<x<=xmax? (xmin,ymin) ymin=<y<=ymax?

Clipping Points Given a point (x, y) and clipping window (xmin, ymin), (xmax, ymax), determine if the point should be drawn (xmax,ymax) (x2,y2) (x1,y1) xmin=<x<=xmax? (xmin,ymin) ymin=<y<=ymax?

Clipping Points Given a point (x, y) and clipping window (xmin, ymin), (xmax, ymax), determine if the point should be drawn (xmax,ymax) (x2,y2) (x1,y1) xmin=<x1<=xmax Yes (xmin,ymin) ymin=<y1<=ymax Yes

Clipping Points Given a point (x, y) and clipping window (xmin, ymin), (xmax, ymax), determine if the point should be drawn (xmax,ymax) (x2,y2) (x1,y1) xmin=<x2<=xmax No (xmin,ymin) ymin=<y2<=ymax Yes

Clipping Lines

Clipping Lines

Clipping Lines Given a line with end-points (x0, y0), (x1, y1) and clipping window (xmin, ymin), (xmax, ymax), determine if line should be drawn and clipped end-points of line to draw. (xmax,ymax) (x1, y1) (x0, y0) (xmin,ymin)

Clipping Lines

Outline Simple line clipping algorithm Cohen-Sutherland Liang-Barsky Required readings: HB 8-5, 8-6, 8-7 and 8-9

Clipping Lines

Clipping Lines – Simple Algorithm If both end-points inside rectangle, draw line If one end-point outside, intersect line with all edges of rectangle clip that point and repeat test

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Intersecting Two Lines

Intersecting Two Lines

Intersecting Two Lines

Intersecting Two Lines

Intersecting Two Lines

Intersecting Two Lines Substitute t or s back into equation to find intersection

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm

Clipping Lines – Simple Algorithm Lots of intersection tests makes algorithm expensive Complicated tests to determine if intersecting rectangle Is there a better way?

Trivial Accepts Big Optimization: trivial accepts/rejects How can we quickly decide whether line segment is entirely inside window Answer: test both endpoints

Trivial Accepts Big Optimization: trivial accepts/rejects How can we quickly decide whether line segment is entirely inside window Answer: test both endpoints

Trivial Rejects How can we know a line is outside of the window Answer:

Trivial Rejects How can we know a line is outside of the window Answer: Wrong side

Trivial Rejects How can we know a line is outside of the window Answer: both endpoints on wrong side of same edge, can trivially reject the line

Cohen-Sutherland Algorithm Classify p0, p1 using region codes c0, c1 If , trivially reject If , trivially accept Otherwise reduce to trivial cases by splitting into two segments

Cohen-Sutherland Algorithm Every end point is assigned to a four-digit binary value, i.e. Region code Each bit position indicates whether the point is inside or outside of a specific window edges bit 4 bit 3 bit 2 bit 1 top bottom right left

Cohen-Sutherland Algorithm bit 4 bit 3 bit 2 bit 1 top bottom right left top right left Region code? bottom

Cohen-Sutherland Algorithm bit 4 bit 3 bit 2 bit 1 top bottom right left top right left 0010 bottom ?

Cohen-Sutherland Algorithm bit 4 bit 3 bit 2 bit 1 top bottom right left top right left 0010 bottom 0100

Cohen-Sutherland Algorithm

Cohen-Sutherland Algorithm

Cohen-Sutherland Algorithm Classify p0, p1 using region codes c0, c1 If , trivially reject because both points located the wrong side of one edge. If , trivially accept because both points are not in the wrong side of any edge. Otherwise reduce to trivial cases by splitting into two segments

Cohen-Sutherland Algorithm Bitwise and

Cohen-Sutherland Algorithm Bitwise and Bitwise or

Cohen-Sutherland Algorithm Bitwise and Bitwise or

Cohen-Sutherland Algorithm Classify p0, p1 using region codes c0, c1 If , trivially reject If , trivially accept Otherwise reduce to trivial cases by splitting into two segments

Cohen-Sutherland Algorithm Classify p0, p1 using region codes c0, c1 If , trivially reject If , trivially accept Otherwise reduce to trivial cases by splitting into two segments Line is outside the window! reject

Cohen-Sutherland Algorithm Classify p0, p1 using region codes c0, c1 If , trivially reject If , trivially accept Otherwise reduce to trivial cases by splitting into two segments Line is outside the window! reject

Cohen-Sutherland Algorithm Classify p0, p1 using region codes c0, c1 If , trivially reject If , trivially accept Otherwise reduce to trivial cases by splitting into two segments Line is inside the window! draw

Cohen-Sutherland Algorithm Classify p0, p1 using region codes c0, c1 If , trivially reject If , trivially accept Otherwise reduce to trivial cases by splitting into two segments

Window Intersection (x1, y1), (x2, y2) intersect with vertical edge at xright yintersect = y1 + m(xright – x1) where m=(y2-y1)/(x2-x1) (x1, y1), (x2, y2) intersect with horizontal edge at ybottom xintersect = x1 + (ybottom – y1)/m

Example 1

Example 1

Example 1

Example 1

Example 2

Example 2

Example 2

Example 2

Example 2

Example 2

Example 2

Example 3

Example 3

Example 3

Example 3

Cohen-Sutherland Algorithm Extends easily to 3D line clipping 27 regions 6 bits

Cohen-Sutherland Algorithm Use region codes to quickly eliminate/include lines Best algorithm when trivial accepts/rejects are common Must compute viewing window clipping of remaining lines Non-trivial clipping cost More efficient algorithms exist

Liang-Barsky Algorithm Parametric definition of a line: x = x1 + uΔx y = y1 + uΔy Δx = (x2-x1), Δy = (y2-y1), 0<=u<=1 Lines are oriented: classify lines as moving inside to out or outside to in Goal: find range of u for which x and y both inside the viewing window

Liang-Barsky Algorithm Current clipped line (u1=0,u2=1) (2,1) (-.5,-.5)

Liang-Barsky Algorithm Current clipped line (u1=0.333,u2=0.6) (2,1) (-.5,-.5)

Liang-Barsky Algorithm For lines starting outside of boundary, update its starting point (u1) For lines starting inside of boundary, update end point (u2) For lines paralleling the boundaries and outside window, reject it.

Liang-Barsky Algorithm For lines starting outside of boundary, update its starting point (u1) For lines starting inside of boundary, update end point (u2) For lines paralleling the boundaries and outside window, reject it.

Liang-Barsky Algorithm Mathematically: xmin <= x1 + uΔx <= xmax ymin <= y1 + uΔy <= ymax Rearranged 1: u*(-Δx) <= (x1 – xmin) 2: u*(Δx) <= (xmax – x1) 3: u*(-Δy) <= (y1 – ymin) 4: u*(Δy) <= (ymax – y1) gen: u*(pk) <= (qk), k=1,2,3,4

Liang-Barsky Algorithm Mathematically: xmin <= x1 + uΔx <= xmax ymin <= y1 + uΔy <= ymax Rearranged 1: u*(-Δx) <= (x1 – xmin) // left edge 2: u*(Δx) <= (xmax – x1) 3: u*(-Δy) <= (y1 – ymin) 4: u*(Δy) <= (ymax – y1) gen: u*(pk) <= (qk), k=1,2,3,4

Liang-Barsky Algorithm Mathematically: xmin <= x1 + uΔx <= xmax ymin <= y1 + uΔy <= ymax Rearranged 1: u*(-Δx) <= (x1 – xmin) 2: u*(Δx) <= (xmax – x1) // right edge 3: u*(-Δy) <= (y1 – ymin) 4: u*(Δy) <= (ymax – y1) gen: u*(pk) <= (qk), k=1,2,3,4

Liang-Barsky Algorithm Mathematically: xmin <= x1 + uΔx <= xmax ymin <= y1 + uΔy <= ymax Rearranged 1: u*(-Δx) <= (x1 – xmin) 2: u*(Δx) <= (xmax – x1) 3: u*(-Δy) <= (y1 – ymin) // bottom edge 4: u*(Δy) <= (ymax – y1) gen: u*(pk) <= (qk), k=1,2,3,4

Liang-Barsky Algorithm Mathematically: xmin <= x1 + uΔx <= xmax ymin <= y1 + uΔy <= ymax Rearranged 1: u*(-Δx) <= (x1 – xmin) 2: u*(Δx) <= (xmax – x1) 3: u*(-Δy) <= (y1 – ymin) 4: u*(Δy) <= (ymax – y1) // top edge gen: u*(pk) <= (qk), k=1,2,3,4

Liang-Barsky Algorithm Mathematically: xmin <= x1 + uΔx <= xmax ymin <= y1 + uΔy <= ymax Rearranged 1: u*(-Δx) <= (x1 – xmin) 2: u*(Δx) <= (xmax – x1) 3: u*(-Δy) <= (y1 – ymin) 4: u*(Δy) <= (ymax – y1) Gen: u*(pk) <= (qk), k=1,2,3,4

Liang-Barsky Algorithm Rules: pk = 0: the line is parallel to boundaries If for that same k, qk < 0, it’s outside Otherwise it’s inside pk < 0: the line starts outside this boundary rk = qk/pk u1 = max(0, rk, u1) /**update starting point**/ pk > 0: the line starts inside the boundary u2 = min(1, rk, u2) /** update end point**/ If u1 > u2, the line is completely outside

Liang-Barsky Algorithm Current clipped line (u1=0,u2=1) (2,1) (-.5,-.5)

Liang-Barsky Algorithm Current clipped line (u1=0,u2=1) (2,1) Δx = (x2-x1)=2.5 Δy = (y2-y1)=1.5 p1 = -Δx =-2.5<0 (-.5,-.5) q1 = (x1-xmin)=-.5 r1 = q1/p1=0.2

Liang-Barsky Algorithm Current clipped line (u1=0,u2=1) (2,1) Δx = (x2-x1)=2.5 Δy = (y2-y1)=1.5 p1 = -Δx =-2.5<0 (-.5,-.5) q1 = (x1-xmin)=-.5 line starts outside this boundary r1 = q1/p1=0.2

Liang-Barsky Algorithm Current clipped line (u1=0,u2=1) (2,1) Δx = (x2-x1)=2.5 Δy = (y2-y1)=1.5 p1 = -Δx =-2.5<0 (-.5,-.5) q1 = (x1-xmin)=-.5 r1 = q1/p1=0.2

Liang-Barsky Algorithm Current clipped line (u1=0,u2=1) (2,1) Δx = (x2-x1)=2.5 Δy = (y2-y1)=1.5 p1 = -Δx =-2.5<0 (-.5,-.5) q1 = (x1-xmin)=-.5 r1 = q1/p1=0.2

Liang-Barsky Algorithm Current clipped line (u1=0,u2=1) (2,1) Δx = (x2-x1)=2.5 Δy = (y2-y1)=1.5 p1 = -Δx =-2.5<0 (-.5,-.5) q1 = (x1-xmin)=-.5 u1 = max(0, r1, u1) r1 = q1/p1=0.2

Liang-Barsky Algorithm Current clipped line (u1=0.2,u2=1) (2,1) u1<u2? yes: continue no: the line is completely outside window (-.5,-.5)

Liang-Barsky Algorithm Current clipped line (u1=0.2,u2=1) (2,1) (-.5,-.5)

Liang-Barsky Algorithm Current clipped line (u1=0.2,u2=1) (2,1) Δx = (x2-x1)=2.5 Δy = (y2-y1)=1.5 p2 = Δx =2.5>0 (-.5,-.5) q2 = (xmax-x1)=1.5 r2 = q2/p2=0.6

Liang-Barsky Algorithm Current clipped line (u1=0.2,u2=1) (2,1) Δx = (x2-x1)=2.5 Δy = (y2-y1)=1.5 p2 = Δx =2.5>0 (-.5,-.5) q2 = (xmax-x1)=1.5 u2 = min(1, r2, u2) r2 = q2/p2=0.6

Liang-Barsky Algorithm Current clipped line (u1=0.2,u2=0.6) (2,1) Δx = (x2-x1)=2.5 Δy = (y2-y1)=1.5 p2 = Δx =2.5>0 (-.5,-.5) q2 = (xmax-x1)=1.5 r2 = q2/p2=0.6

Liang-Barsky Algorithm Current clipped line (u1=0.2,u2=0.6) (2,1) u1<u2? yes: continue no: the line is completely outside window (-.5,-.5)

Liang-Barsky Algorithm Current clipped line (u1=0.2,u2=0.6) (2,1) Δx = (x2-x1)=2.5 Δy = (y2-y1)=1.5 p3 = -Δy =-1.5<0 (-.5,-.5) q3 = (y1-ymin)=0.5 r3 = q3/p3=0.333

Liang-Barsky Algorithm Current clipped line (u1=0.2,u2=0.6) (2,1) Δx = (x2-x1)=2.5 Δy = (y2-y1)=1.5 p3 = -Δy =-1.5<0 (-.5,-.5) q3 = (y1-ymin)=0.5 u1 = max(0, r3, u1) r3 = q3/p3=0.333

Liang-Barsky Algorithm Current clipped line (u1=0.333,u2=0.6) (2,1) Δx = (x2-x1)=2.5 Δy = (y2-y1)=1.5 p3 = -Δy =-1.5<0 (-.5,-.5) q3 = (y1-ymin)=0.5 r3 = q3/p3=0.333

Liang-Barsky Algorithm Current clipped line (u1=0.333,u2=0.6) (2,1) (-.5,-.5)

Liang-Barsky Algorithm Current clipped line (u1=0.333,u2=0.6) (2,1) Δx = (x2-x1)=2.5 Δy = (y2-y1)=1.5 p3 = -Δy =-1.5<0 (-.5,-.5) q3 = (y1-ymin)=0.5 r3 = q3/p3=0.333

Liang-Barsky Algorithm Current clipped line (u1=0.333,u2=0.6) (2,1) Δx = (x2-x1)=2.5 Δy = (y2-y1)=1.5 p4 = Δy =1.5>0 (-.5,-.5) q4 = (ymax-y1)=1.5 r4 = q4/p4=1

Liang-Barsky Algorithm Current clipped line (u1=0.333,u2=0.6) (2,1) Δx = (x2-x1)=2.5 Δy = (y2-y1)=1.5 p4 = Δy =1.5>0 (-.5,-.5) q4 = (ymax-y1)=1.5 u2 = min(1, r4, u2) r4 = q4/p4=1

Liang-Barsky Algorithm Check the left edge (u1=0.333,u2=0.6) (2,1) u1<u2 (-.5,-.5) xnew1=x1+Δx*u1 xnew2=x2+Δx*u2 ynew1=y1+Δy*u1 ynew2=y2+Δy*u2

Liang-Barsky Algorithm Faster than Cohen-Sutherland Extension to 3D is easy - Parametric representation for 3D lines - Compute u1,u2 based on the intersection between line and plane

Comparison Cohen-Sutherland Liang-Barsky Repeated clipping is expensive Best used when trivial acceptance and rejection is possible for most lines Liang-Barsky Computation of t-intersections is cheap (only one division) Computation of (x,y) clip points is only done once Algorithm doesn’t consider trivial accepts/rejects Best when many lines must be clipped

Curve Clipping

Curve Clipping

Curve Clipping Approximate a curve using a set of straight-line segments Apply line clipping for curve clipping

Next Lecture Polygon fill-area clipping

Next Lecture Polygon fill-area clipping