CS123 INTRODUCTION TO COMPUTER GRAPHICS. Clipping. Concepts, Algorithms for line clipping. 1 of 16. Andries van Dam. Clipping - 10/12/17
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1 Clipping Concepts, Algorithms for line clipping 1 of 16
2 Line Clipping in 2D Clipping endpoints If x min x x max and y min y y max, the point is inside the clip rectangle. Endpoint analysis for lines: if both endpoints in, do trivial acceptance if one endpoint inside, one outside, must clip if both endpoints out, don t know Brute force clip: solve simultaneous equations using four line and four clip edges slope-intercept formula handles infinite lines only doesn t handle vertical lines 2 of 16
3 Parametric Line Formulation For Clipping Parametric form for line segment Line is in clip rectangle if parametric variables t line and s edge both in [0,1] at intersection point between line and edge of clip rectangle Slow, must intersect lines with all edges 3 of 16
4 CS123 INTRODUCTION TO COMPUTER GRAPHICS Cohen-Sutherland Line Clipping in 2D Divide plane into 9 regions Compute the sign bit of 4 comparisons between a vertex and an edge y max y; y y min ; x max x; x x min Point lies inside only if all four sign bits are 0, otherwise exceeds edge 4 bit outcode records results of four bounds tests: First bit: above top edge Second bit: below bottom edge Third bit: to the right of right edge Fourth bit: to the left of left edge Clip Rectangle Compute outcodes for both vertices of each line (denoted OC 0 and OC 1 ) Lines with OC 0 = 0 (i.e., 0000) and OC 1 = 0 can be trivially accepted. Lines lying entirely in a half plane outside an edge can be trivially rejected if (OC 0 AND OC 1 ) 0 (i.e., they share an outside bit) 4 of 16
5 Cohen-Sutherland Line Clipping in 3D Very similar to 2D Divide volume into 27 regions 6-bit outcode records results of 6 bounds tests First bit: behind back plane: Z - Zmin Second bit: in front of front plane: Zmax -Z Third bit: above top plane: Ymax - Y Fourth bit: below bottom plane: Y - Ymin Fifth bit: to the right of right plane: Xmax -X Sixth bit: to the left of left plane: X - Xmin Again, lines with OC 0 = 0 and OC 1 = 0 can be trivially accepted Lines lying entirely in a volume outside of a plane can be trivially rejected: OC 0 AND OC 1 0 (i.e., they share an outside bit) Back plane (in front) (behind) Top plane (above) (below) Right plane (to left of) (to right of) Front plane (in front) (behind) Bottom plane (above) (below) Left plane (to left of) (to right of) 5 of 16
6 CS123 INTRODUCTION TO COMPUTER GRAPHICS Cohen-Sutherland Algorithm (1/3) If we can neither trivially accept/reject (T/A, T/R), divide and conquer Subdivide line into two segments; then T/A or T/R one or both segments: use a clip edge to cut line use outcodes to choose the edges that are crossed for a given clip edge, if a line s two outcodes differ in the corresponding bit, the line has one vertex on each side of the edge, thus crosses pick an order for checking edges: top bottom right left compute the intersection point the clip edge fixes either x or y can substitute into the line equation iterate for the newly shortened line, extra clips may happen (e.g., E-I at H) D C E B A Clip rectangle F G H I 6 of 16
7 Cohen-Sutherland Algorithm (2/3) ComputeOutCode(x0, y0, outcode0); ComputeOutCode(x1, y1, outcode1); repeat check for trivial reject or trivial accept pick the point that is outside the clip rectangle if TOP then x = x0 + (x1 x0) * (ymax y0) / (y1 y0); y = ymax; else if BOTTOM then x = x0 + (x1 x0) * (ymin y0) / (y1 y0); y = ymin; else if RIGHT then y = y0 + (y1 y0) * (xmax x0) / (x1 x0); x = xmax; else if LEFT then y = y0 + (y1 y0) * (xmin x0) / (x1 x0); x = xmin; if (x0, y0 is the outer point) then x0 = x; y0 = y; ComputeOutCode(x0, y0, outcode0) else x1 = x; y1 = y; ComputeOutCode(x1, y1, outcode1) until done 7 of 16
8 Cohen-Sutherland Algorithm (3/3) Similar algorithm for using 3D outcodes to clip against canonical parallel view volume: xmin = ymin = -1; xmax = ymax = 1; zmin = -1; zmax = 0; ComputeOutCode(x0, y0, z0, outcode0); ComputeOutCode(x1, y1, z1, outcode1); repeat check for trivial reject or trivial accept pick the point that is outside the clip rectangle if TOP then x = x0 + (x1 x0) * (ymax y0) / (y1 y0); z = z0 + (z1 z0) * (ymax y0) / (y1 y0); y = ymax; else if BOTTOM then x = x0 + (x1 x0) * (ymin y0) / (y1 y0); z = z0 + (z1 z0) * (ymin y0) / (y1 y0); y = ymin; else if RIGHT then y = y0 + (y1 y0) * (xmax x0) / (x1 x0); z = z0 + (z1 z0) * (xmax x0) / (x1 x0); x = xmax; else if LEFT then y = y0 + (y1 y0) * (xmin x0) / (x1 x0); z = z0 + (z1 z0) * (xmin x0) / (x1 x0); x = xmin; else if NEAR then x = x0 + (x1 x0) * (zmax z0) / (z1 z0); y = y0 + (y1 y0) * (zmax z0) / (z1 z0); z = zmax; else if FAR then x = x0 + (x1 x0) * (zmin z0) / (z1 z0); y = y0 + (y1 y0) * (zmin z0) / (z1 z0); z = zmin; if (x0, y0, z0 is the outer point) then x0 = x; y0 = y; z0 = z; ComputeOutCode(x0, y0, z0, outcode0) else x1 = x; y1 = y; z1 = z; ComputeOutCode(x1, y1, z1, outcode1) until done 8 of 16
9 Scan Conversion after Clipping Clip rectangle Don t round and then scan convert, because the line will have the wrong slope: calculate decision variable based on pixel chosen on left edge (remember: y = mx + B) Horizontal edge problem: Clipping/rounding produces pixel A; to get pixel B, round up x of the intersection of line with y = y min ½ and pick pixel above: x = x min y = y min y = y min 1/2 y = y min 1 B A 9 of 16
10 Sutherland-Hodgman Polygon Clipping The 2D Sutherland-Hodgman algorithm generalizes to higher dimensions We can use it to clip polygons to the 3D view volume one plane at a time Section 36.5 in textbook BALSA demonstration: 10 of 16
11 Cyrus-Beck/Liang-Barsky Parametric Line Clipping (1/3) 11 of 16
12 Cyrus-Beck/Liang-Barsky Parametric Line Clipping (2/3) Now solve for the value of t at the intersection of P 0 P 1 with the edge E i : First, substitute for P(t) with P 0 + (P 1 P 0 )t: Next, group terms and distribute dot product: Let D be the vector from P 0 to P 1 = (P 1 P 0 ), and solve for t: note that this gives a valid value of t only if the denominator of the expression is nonzero. For this to be true, it must be the case that: N i 0 (that is, the normal should not be 0; this could occur only as a mistake) D 0 (that is, P 1 P 0 ) N i D 0 (edge E i and line D are not parallel; if they are, no intersection). The algorithm checks these conditions. 12 of 16
13 Cyrus-Beck/Liang-Barsky Parametric Line Clipping (3/3) Eliminate all t s outside [0, 1] on the line Of remaining t s which produce interior intersections? Can t just take the innermost t values! Move from P 0 to P 1 ; for a given edge, just before crossing: If N i D < 0 Potentially Entering (PE); If N i D > 0 Potentially Leaving (PL) Pick inner PE/PL pair: t E for P PE with max t, t L for P PL with min t, and t E > 0, t L < 1 If t L < t E, no intersection 13 of 16
14 Cyrus-Beck/Liang-Barsky Line Clipping Algorithm Pre-calculate N i and select P Ei for each edge; for each line segment to be clipped if P 1 = P 0 then line is degenerate so clip as a point; else begin t E = 0; t L = 1; for each candidate intersection with a clip edge if Ni D 0 then {Ignore edges parallel to line} begin calculate t; {of line and clip edge intersection} use sign of N i D to categorize as PE or PL; if PE then t E = max(t E,t); if PL then t L = min(t L,t); end if t E > t L then return nil else return P(t E ) and P(t L ) as true clip intersections end 14 of 16
15 Parametric Line Clipping for Upright Clip Rectangle (1/2) D = P 1 P 0 = (x 1 x 0, y 1 y 0 ) Leave P Ei as an arbitrary point on clip edge: it s a free variable and drops out Calculations for Parametric Line Clipping Algorithm Clip Edge i Normal N i P Ei P 0 -P Ei t N i ( P P ) 0 Ei N D i left: x = x min (-1,0) (x min, y) (x 0 - x min,y 0 -y) ( x x ) 0 min ( x x ) 1 0 right: x = x max (1,0) (x max,y) (x 0 - x max,y 0 -y) ( x x ) 0 max ( x x ) 1 0 bottom: y = y min (0,-1) (x, y min ) (x 0 -x,y 0 - y min ) ( y y ) 0 min ( y y ) 1 0 top: y = y max (0,1) (x, y max ) (x 0 -x,y 0 - y max ) ( y y ) 0 max ( y y ) of 16
16 Parametric Line Clipping for Upright Clip Rectangle (2/2) Examine t: Numerator is just the directed distance to an edge; sign corresponds to OC Denominator is just the horizontal or vertical projection of the line, dx or dy; sign determines PE or PL for a given edge Ratio is constant of proportionality: how far over from P 0 to P 1 intersection is relative to dx or dy 16 of 16
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