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Computer Graphics

Computer Graphics . Clipping Fall FCC 2006. Line Clipping. What happens when one or both endpoints of a line segment are not inside the specified drawing area? Draw just the portions of a line (or object) that fall within a given region/window/screen (usually rectangular). Drawing

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Computer Graphics

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  1. Computer Graphics Clipping Fall FCC 2006

  2. Line Clipping What happens when one or both endpoints of a line segment are not inside the specified drawing area? Draw just the portions of a line (or object) that fall within a given region/window/screen (usually rectangular) Drawing Area

  3. Line Clipping • Strategies for clipping: • Check (in inner loop) if each point is inside  Works, but slow • Clip invalid coordinate(s) to boundary  Incorrect results • Find intersection of line with boundary  Correct if (x >= xmin && x <= xmax && y >= ymin && y <= ymax) drawPoint(x,y,c); if (x < xmin) x = xmin; else if (x > xmax) x = xmax; if (y < ymin) y = ymin; else if (y > ymax) y = ymax; Clip x Output Input Clip y Clip line to intersection

  4. Line Clipping: Possible Configurations • Both endpoints are inside the region (line AB) • No clipping necessary • One endpoint in, one out (line CD) • Clip at intersection point • Both endpoints outside the region: • No intersection (lines EF, GH) • Line intersects the region (line IJ) • Clip line at both intersection points J D I A F C H B E G

  5. J D I Clip and retest A F C H B Trivially accept E G Trivially reject Line Clipping: Cohen-Sutherland • Basic algorithm: • Accept (and draw) lines that have both endpoints inside the region • Reject (and don’t draw) lines that have both endpoints less than xmin or ymin or greater than xmax or ymax • Clip the remaining lines at a region boundary and repeat steps 1 and 2 on the clipped line segments

  6. Cohen-Sutherland: Accept/Reject Tests • Assign 4-bit code to each endpoint corresponding to its position relative to region: • First bit (1000): if y > ymax • Second bit (0100): if y < ymin • Third bit (0010): if x > xmax • Fourth bit (0001): if x < xmin • Test: if code0 OR code1 = 0000 accept (draw) else if code0 AND code1 0000 reject (don’t draw) else clip and retest 1001 1000 1010 0001 0000 0010 0101 0100 0110

  7. Cohen-Sutherland: Line Clipping Intersection algorithm: if code0 0000 then code = code0 else code = code1 dx = x1 – x0; dy = y1 – y0 if codeAND 1000 then begin // ymax x = x0 + dx * (ymax – y0) / dy; y = ymax end else if codeAND 0100 then begin // ymin x = x0 + dx * (ymin – y0) / dy; y = ymin end else if codeAND 0010 then begin // xmax y = y0 + dy * (xmax – x0) / dx; x = xmax end else begin // xmin y = y0 + dy * (xmin – x0) / dx; x = xmin end if code = code0 then begin x0 = x; y0 = y; end else begin x1 = x; y1 = y; end (x1, y1) (x, y) ymax dy (x0, y0) dx ymin xmin xmax

  8. (x1, y1) (400, 300) Code (1010) ymax=200 (x0, y0) (150, 150) Code (0000) ymin=100 Xmin = 100 xmax = 300 Cohen-Sutherland: Line Clipping Example Intersection algorithm: if code0 0000 then code = code0 else code = code1 dx = x1 – x0; dy = y1 – y0 if codeAND 1000 then begin // ymax x = x0 + dx * (ymax – y0) / dy; y = ymax end else if codeAND 0100 then begin // ymin x = x0 + dx * (ymin – y0) / dy; y = ymin end else if codeAND 0010 then begin // xmax y = y0 + dy * (xmax – x0) / dx; x = xmax end else begin // xmin y = y0 + dy * (xmin – x0) / dx; x = xmin end if code = code0 then begin x0 = x; y0 = y; end else begin x1 = x; y1 = y; end Code dx dy x y

  9. (x1, y1) (400, 300) Code (1010) ymax=200 (x0, y0) (150, 150) Code (0000) ymin=100 Xmin = 100 xmax = 300 Cohen-Sutherland: Line Clipping Example Intersection algorithm: if code0 0000 then code = code0 else code = code1 dx = x1 – x0; dy = y1 – y0 if codeAND 1000 then begin // ymax x = x0 + dx * (ymax – y0) / dy; y = ymax end else if codeAND 0100 then begin // ymin x = x0 + dx * (ymin – y0) / dy; y = ymin end else if codeAND 0010 then begin // xmax y = y0 + dy * (xmax – x0) / dx; x = xmax end else begin // xmin y = y0 + dy * (xmin – x0) / dx; x = xmin end if code = code0 then begin x0 = x; y0 = y; end else begin x1 = x; y1 = y; end Code dx dy x y

  10. (x1, y1) (400, 300) Code (1010) ymax=200 (x0, y0) (150, 150) Code (0000) ymin=100 Xmin = 100 xmax = 300 Cohen-Sutherland: Line Clipping Example Intersection algorithm: if code0 0000 then code = code0 else code = code1 dx = x1 – x0; dy = y1 – y0 if codeAND 1000 then begin // ymax x = x0 + dx * (ymax – y0) / dy; y = ymax end else if codeAND 0100 then begin // ymin x = x0 + dx * (ymin – y0) / dy; y = ymin end else if codeAND 0010 then begin // xmax y = y0 + dy * (xmax – x0) / dx; x = xmax end else begin // xmin y = y0 + dy * (xmin – x0) / dx; x = xmin end if code = code0 then begin x0 = x; y0 = y; end else begin x1 = x; y1 = y; end Code 1010 dx dy x y

  11. (x1, y1) (400, 300) Code (1010) ymax=200 (x0, y0) (150, 150) Code (0000) ymin=100 Xmin = 100 xmax = 300 Cohen-Sutherland: Line Clipping Example Intersection algorithm: if code0 0000 then code = code0 else code = code1 dx = x1 – x0; dy = y1 – y0 if codeAND 1000 then begin // ymax x = x0 + dx * (ymax – y0) / dy; y = ymax end else if codeAND 0100 then begin // ymin x = x0 + dx * (ymin – y0) / dy; y = ymin end else if codeAND 0010 then begin // xmax y = y0 + dy * (xmax – x0) / dx; x = xmax end else begin // xmin y = y0 + dy * (xmin – x0) / dx; x = xmin end if code = code0 then begin x0 = x; y0 = y; end else begin x1 = x; y1 = y; end Code 1010 dx 250 dy 150 x y

  12. (x1, y1) (400, 300) Code (1010) ymax=200 (x0, y0) (150, 150) Code (0000) ymin=100 Xmin = 100 xmax = 300 Cohen-Sutherland: Line Clipping Example Intersection algorithm: if code0 0000 then code = code0 else code = code1 dx = x1 – x0; dy = y1 – y0 if codeAND 1000 then begin// ymax x = x0 + dx * (ymax – y0) / dy; y = ymax end else if codeAND 0100 then begin // ymin x = x0 + dx * (ymin – y0) / dy; y = ymin end else if codeAND 0010 then begin // xmax y = y0 + dy * (xmax – x0) / dx; x = xmax end else begin // xmin y = y0 + dy * (xmin – x0) / dx; x = xmin end if code = code0 then begin x0 = x; y0 = y; end else begin x1 = x; y1 = y; end Code 1010 dx 250 dy 150 x y

  13. (x1, y1) (400, 300) Code (1010) ymax=200 (x0, y0) (150, 150) Code (0000) ymin=100 Xmin = 100 xmax = 300 Cohen-Sutherland: Line Clipping Example Intersection algorithm: if code0 0000 then code = code0 else code = code1 dx = x1 – x0; dy = y1 – y0 if codeAND 1000 then begin// ymax x = x0 + dx * (ymax – y0) / dy; y = ymax end else if codeAND 0100 then begin // ymin x = x0 + dx * (ymin – y0) / dy; y = ymin end else if codeAND 0010 then begin // xmax y = y0 + dy * (xmax – x0) / dx; x = xmax end else begin // xmin y = y0 + dy * (xmin – x0) / dx; x = xmin end if code = code0 then begin x0 = x; y0 = y; end else begin x1 = x; y1 = y; end Code 1010 dx 250 dy 150 x 233 y 200

  14. (x1, y1) (400, 300) Code (1010) ymax=200 (x0, y0) (150, 150) Code (0000) ymin=100 Xmin = 100 xmax = 300 Cohen-Sutherland: Line Clipping Example Intersection algorithm: if code0 0000 then code = code0 else code = code1 dx = x1 – x0; dy = y1 – y0 if codeAND 1000 then begin// ymax x = x0 + dx * (ymax – y0) / dy; y = ymax end else if codeAND 0100 then begin // ymin x = x0 + dx * (ymin – y0) / dy; y = ymin end else if codeAND 0010 then begin // xmax y = y0 + dy * (xmax – x0) / dx; x = xmax end else begin // xmin y = y0 + dy * (xmin – x0) / dx; x = xmin end if code = code0 then begin x0 = x; y0 = y; end else begin x1 = x; y1 = y; end Code 1010 dx 250 dy 150 x 233 y 200

  15. (x1, y1) (400, 300) Code (1010) ymax=200 (x0, y0) (150, 150) Code (0000) ymin=100 Xmin = 100 xmax = 300 Cohen-Sutherland: Line Clipping Example Intersection algorithm: if code0 0000 then code = code0 else code = code1 dx = x1 – x0; dy = y1 – y0 if codeAND 1000 then begin// ymax x = x0 + dx * (ymax – y0) / dy; y = ymax end else if codeAND 0100 then begin // ymin x = x0 + dx * (ymin – y0) / dy; y = ymin end else if codeAND 0010 then begin // xmax y = y0 + dy * (xmax – x0) / dx; x = xmax end else begin // xmin y = y0 + dy * (xmin – x0) / dx; x = xmin end if code = code0 then begin x0 = x; y0 = y; end else begin x1 = x; y1 = y; end Code 1010 dx 250 dy 150 x 233 y 200

  16. Cohen-Sutherland: Line Clipping Summary • Choose an endpoint outside the clipping region • Using a consistent ordering (top to bottom, left to right) find a clipping border the line intersects • Discard the portion of the line from the endpoint to the intersection point • Set the new line to have as endpoints the new intersection point and the other original endpoint • You may need to run this several times on a single line (e.g., a line that crosses multiple clip boundaries)

  17. Cohen-Sutherland Line Clip Examples H 1000 1001 1010 F D G 0001 0000 C 0010 J A E 0101 I 0100 0110 B A 0001 B 0100 OR 0101 AND 0000 subdivide C 0000 D 0010 OR 0010 AND 0000 subdivide E 0000 F 0000 OR 0000 AND 0000 accept G 0000 H 1010 OR 1010 AND 0000 subdivide I 0110 J 0010 OR 0110 AND 0010 reject

  18. Cohen-Sutherland Line Clip Examples H 1000 1001 1010 G’ D G 0001 0000 C’ C 0010 A A’ 0101 0100 0110 B A 0001 A’ 0001 remove A’ 0001 B 0100 OR 0101 AND 0000 subdivide C 0000 C’ 0000 OR 0000 AND 0000 accept C’ 0000 D 1010 remove G 0000 G’ 0000 OR 0000 AND 0000 accept G’ 0000 H 1010 remove

  19. Cohen-Sutherland Line Clip Examples 1000 1001 1010 0001 0000 0010 A’ B’ 0101 0100 0110 B A’ 0001 B’ 0100 remove B’ 0100 B 0100 OR 0100 AND 0100 reject

  20. Polygon Clipping What about polygons? For concave polygons, the intersection with the clipping region may be complex

  21. Polygon Clipping: Algorithm • Clip polygon to ymin and ymax: • Create empty output vertex list (listout = empty) • Process input list (listin = (v0, v1, …, vn) where v0 = vn) in order • For each input vertex (vi where 0 in–1): • If vi is inside region  Add vi to end oflistout • If the line between viand vi+1 intersects specified boundaries  Add intersection point(s) to end oflistout • Repeat: clipping to xmin and xmax • Post-process: • Find “degenerate” sections where both sides of polygon has collapsed to region boundary • Remove those sections  Create new polygon

  22. p0 line intersect boundary: yes Polygon Clipping: Example v6 v7 Clip first to ymin and ymax v0 v8 vertex:v0 ymin inside region: no v9 v5 v1 v2 add p0 to output list ymax v3 v4 Input vertex list:(v0, v1, v2, v3, v4, v5, v6, v7, v8, v9) Output vertex list: (p0)

  23. p0 line intersect boundary: no Polygon Clipping: Example v6 v7 Clip first to ymin and ymax v0 v8 vertex:v1 ymin inside region: yes v9 v5 v1 v2 add v1 to output list ymax v3 v4 Input vertex list:(v0, v1, v2, v3, v4, v5, v6, v7, v8, v9) Output vertex list: (p0, v1)

  24. p0 line intersect boundary: yes p1 Polygon Clipping: Example v6 v7 Clip first to ymin and ymax v0 v8 vertex:v2 ymin inside region: yes v9 v5 v1 v2 add v2, p1 to output list ymax v3 v4 Input vertex list:(v0, v1, v2, v3, v4, v5, v6, v7, v8, v9) Output vertex list: (p0, v1, v2, p1)

  25. p0 line intersect boundary: no p1 Polygon Clipping: Example v6 v7 Clip first to ymin and ymax v0 v8 vertex:v3 ymin inside region: no v9 v5 v1 v2 ymax v3 v4 Input vertex list:(v0, v1, v2, v3, v4, v5, v6, v7, v8, v9) Output vertex list: (p0, v1, v2, p1)

  26. p0 line intersect boundary: yes p1 p2 Polygon Clipping: Example v6 v7 Clip first to ymin and ymax v0 v8 vertex:v4 ymin inside region: no v9 v5 v1 v2 add p2 to output list ymax v3 v4 Input vertex list:(v0, v1, v2, v3, v4, v5, v6, v7, v8, v9) Output vertex list: (p0, v1, v2, p1, p2)

  27. p3 p0 line intersect boundary: yes p1 p2 Polygon Clipping: Example v6 v7 Clip first to ymin and ymax v0 v8 vertex:v5 ymin inside region: yes v9 v5 v1 v2 add v5, p3 to output list ymax v3 v4 Input vertex list:(v0, v1, v2, v3, v4, v5, v6, v7, v8, v9) Output vertex list: (p0, v1, v2, p1, p2, v5, p3)

  28. p3 p0 line intersect boundary: no p1 p2 Polygon Clipping: Example v6 v7 Clip first to ymin and ymax v0 v8 vertex:v6 ymin inside region: no v9 v5 v1 v2 ymax v3 v4 Input vertex list:(v0, v1, v2, v3, v4, v5, v6, v7, v8, v9) Output vertex list: (p0, v1, v2, p1, p2, v5, p3)

  29. p3 p0 line intersect boundary: no p1 p2 Polygon Clipping: Example v6 v7 Clip first to ymin and ymax v0 v8 vertex:v7 ymin inside region: no v9 v5 v1 v2 ymax v3 v4 Input vertex list:(v0, v1, v2, v3, v4, v5, v6, v7, v8, v9) Output vertex list: (p0, v1, v2, p1, p2, v5, p3)

  30. p4 p3 p0 line intersect boundary: yes p1 p2 Polygon Clipping: Example v6 v7 Clip first to ymin and ymax v0 v8 vertex:v8 ymin inside region: no v9 v5 v1 v2 add p4 to output list ymax v3 v4 Input vertex list:(v0, v1, v2, v3, v4, v5, v6, v7, v8, v9) Output vertex list: (p0, v1, v2, p1, p2, v5, p3, p4)

  31. p4 p3 p5 p0 line intersect boundary: yes p1 p2 Polygon Clipping: Example v6 v7 Clip first to ymin and ymax v0 v8 vertex:v9 ymin inside region: yes v9 v5 v1 v2 add v9, p5 to output list ymax v3 v4 Input vertex list:(v0, v1, v2, v3, v4, v5, v6, v7, v8, v9) Output vertex list: (p0, v1, v2, p1, p2, v5, p3, p4, v9, p5)

  32. p4 p3 p5 p0 p1 p2 Polygon Clipping: Example This gives us a new polygon ymin v9 v5 v1 v2 ymax with vertices: (p0, v1, v2, p1, p2, v5, p3, p4, v9, p5)

  33. p4 p3 p5 p0 p1 p2 Polygon Clipping: Example (cont.) Now clip to xmin and xmax xmin xmax p9 v9 p6 v5 p7 v1 v2 p8 Input vertex list:= (p0, v1, v2, p1, p2, v5, p3, p4, v9, p5) Output vertex list:(p0, p6, p7, v2, p1, p8, p9, p3, p4, v9, p5)

  34. Polygon Clipping: Example (cont.) Now post-process xmin xmax p4 p3 p5 p0 p9 v9 p6 p7 v2 p1 v3 p8 Output vertex list:(p0, p6, p7, v2, p1, p8, p9, p3, p4, v9, p5) Post-process:(p0, p6, p9, p3,) and (p7, v2, p1, p8) and (v4, v9, p5)

  35. Polygon Orientation C D R D C L B E R L A A E B positive Negative

  36. Left o Right Let A(x1,y1) and B(x2,y2) end points of a line. A point P(x,y) will be to the left of the line segment if the expresion C= (x2-x1)*(y-y1)-(y2-y1)*(x-x1) is positive. We say the point is to the right if C is negative. If P is to the right, it is outside the Polygon If P is to the left, it is inside the Polygon

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