Computer Graphics : Viewing In 2D

Computer Graphics :Viewing In 2D UNIT-3

Contents • Windowing Concepts • Clipping • Introduction • Brute Force • Cohen-Sutherland Clipping Algorithm • Area Clipping • Sutherland-Hodgman Area Clipping Algorithm

Windowing I • A scene is made up of a collection of objects specified in world coordinates World Coordinates

Windowing II • When we display a scene only those objects within a particular window are displayed Window wymax wymin wxmax wxmin World Coordinates

Windowing III • Because drawing things to a display takes time we clip everything outside the window Window wymax wymin wxmax wxmin World Coordinates

Clipping • For the image below consider which lines and points should be kept and which ones should be clipped P4 Window P2 wymax P6 P3 P1 P5 P7 P9 P8 wymin P10 wxmin wxmax

Point Clipping • Easy - a point (x,y) is not clipped if: • wxmin ≤ x ≤ wxmax AND wymin ≤ y ≤ wymax • otherwise it is clipped P4 Clipped Clipped Window P2 wymax Clipped P5 P1 P7 Points Within the Window are Not Clipped P9 P8 wymin P10 Clipped wxmin wxmax

Line Clipping • Harder - examine the end-points of each line to see if they are in the window or not

Cohen-Sutherland Algorithm • Idea: eliminate as many cases as possible without computing intersections • Start with four lines that determine the sides of the clipping window y = ymax x = xmin x = xmax y = ymin

The Cases • Case 1: both endpoints of line segment inside all four lines • Draw (accept) line segment as is • Case 2: both endpoints outside all lines and on same side of a line • Discard (reject) the line segment y = ymax x = xmin x = xmax y = ymin

The Cases • Case 3: One endpoint inside, one outside • Must do at least one intersection • Case 4: Both outside • May have part inside • Must do at least one intersection y = ymax x = xmin x = xmax

Defining Outcodes • For each endpoint, define an outcode • Outcodes divide space into 9 regions • Computation of outcode requires at most 4 subtractions b0b1b2b3 b0 = 1 if y > ymax, 0 otherwise b1 = 1 if y < ymin, 0 otherwise b2 = 1 if x > xmax, 0 otherwise b3 = 1 if x < xmin, 0 otherwise

Using Outcodes • Consider the 5 cases below • AB: outcode(A) = outcode(B) = 0 • Accept line segment

Using Outcodes • CD: outcode (C) = 0, outcode(D)  0 • Compute intersection • Location of 1 in outcode(D) determines which edge to intersect with • Note if there were a segment from A to a point in a region with 2 ones in outcode, we might have to do two interesections

Using Outcodes • EF: outcode(E) logically ANDed with outcode(F) (bitwise)  0 • Both outcodes have a 1 bit in the same place • Line segment is outside of corresponding side of clipping window • reject

Using Outcodes • GH and IJ: same outcodes, neither zero but logical AND yields zero • Shorten line segment by intersecting with one of sides of window • Compute outcode of intersection (new endpoint of shortened line segment) • Reexecute algorithm

Efficiency • In many applications, the clipping window is small relative to the size of the entire data base • Most line segments are outside one or more side of the window and can be eliminated based on their outcodes • Inefficiency when code has to be reexecuted for line segments that must be shortened in more than one step

Liang – Barsky clipping • 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 clipping window. With these intersections it knows which portion of the line should be drawn. • 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. Consider first the usual parametric form of a straight line: • A point is in the clip window, if

To compute the final line segment: • A line parallel to a clipping window edge has for that boundary. • If for that the line is completely outside and can be eliminated. • When the line proceeds outside to inside the clip window and when , the line proceeds inside to outside. • For nonzero , gives the intersection point.

Here

Liang-Barsky Leave Enter Leave Leave Leave Enter Enter Enter

Liang-Barsky - Algorithm • Compute entering uvalues, which are qk/pk for each pk<0 • Compute leaving uvalues, which are qk/pk for each pk>0 • Parameter value for small u end of line is:usmall= max(0, entering u’s) • parameter value for large t end of line is:ularge=min(1, leaving u’s) • Ifusmall<ularge, there is a line segment - compute endpoints by substituting t values • Improvement (and actual Liang-Barsky): • compute u’sfor each edge in turn (some rejects occur earlier like this.)

Area Clipping • Similarly to lines, areas must be clipped to a window boundary • Consideration must be taken as to which portions of the area must be clipped

Sutherland-Hodgman Area Clipping Algorithm Sutherland turns up again. This time with Gary Hodgman with whom he worked at the first ever graphics company Evans & Sutherland • A technique for clipping areas developed by Sutherland & Hodgman • Put simply the polygon is clipped by comparing it against each boundary in turn Original Area Clip Left Clip Right Clip Top Clip Bottom

The Sutherland-Hodgeman Polygon-Clipping Algorithms clips a given polygon successively against the edges of the given clip-rectangle. • These clip edges are denoted with e1, e2, e3, and e4, here. • The closed polygon is represented by a list of its vertices (v1 to vn; Since we got 15 vertices in the example shown above, vn = v15.). • Clipping is computed successively for each edge. The output list of the previous clipping run is used as the input list for the next clipping run.

Sutherland-Hodgman Area Clipping Algorithm (cont…) • To clip an area against an individual boundary: • Consider each vertex in turn against the boundary • Vertices inside the boundary are saved for clipping against the next boundary • Vertices outside the boundary are clipped • If we proceed from a point inside the boundary to one outside, the intersection of the line with the boundary is saved • If we cross from the outside to the inside intersection point and the vertex are saved

S S P I P Save Point I Save Point P S P P I S Save Points I & P No Points Saved Sutherland-Hodgman Example • Each example shows the point being processed (P) and the previous point (S) • Saved points define area clipped to the boundary in question

Window Window Window Window Other Area Clipping Concerns • Clipping concave areas can be a little more tricky as often superfluous lines must be removed • Clipping curves requires more work • For circles we must find the two intersection points on the window boundary

Computer Graphics : Viewing In 2D