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CGMB214: Introduction to Computer Graphics CHAPTER 7 2D VIEWING Objective To be able to understand the procedures for displaying views a two dimensional picture on an output device Two-Dimensional Viewing Co-ordinate Systems. Cartesian – offsets along the x and y axis from (0.0) Polar – rotation around the angle . Graphic libraries mostly using Cartesian co-ordinates Any polar co-ordinates must be converted to Cartesian co- ordinates Four Cartesian co-ordinates systems in computer Graphics. 1. Modeling co-ordinates 2. World co-ordinates 3. Normalized device co-ordinates 4. Device co-ordinates Modeling Coordinates Also known as local coordinate. Ex: where individual object in a scene within separate coordinate reference frames. Each object has an origin (0,0) So the part of the objects are placed with reference to the object’s origin. In term of scale it is user defined, so, coordinate values can be any size. World Co-ordinates. The world coordinate system describes the relative positions and orientations of every generated objects. The scene has an origin (0,0). The object in the scene are placed with reference to the scenes origin. World co-ordinate scale may be the same as the modeling co-ordinate scale or it may be different. However, the coordinates values can be any size (similar to MC) Normalized Device Co-ordinates Output devices have their own co-ordinates. Co-ordinates values: The x and y axis range from 0 to 1 All the x and y co-ordinates are floating point numbers in the range of 0 to 1 This makes the system independent of the various devices coordinates. This is handled internally by graphic system without user awareness. Device Co-ordinates Specific co-ordinates used by a device. Pixels on a monitor Points on a laser printer. mm on a plotter. The transformation based on the individual device is handled by computer system without user concern. Two-Dimensional Viewing Example: Graphic program which draw an entire building by an architect but we only interested on the ground floor Map of sales for entire region but we only like to know from certain region of the country. Two-Dimensional Viewing When we interested to display certain portion of the drawing, enlarge the portion, windowing technique is used Technique for not showing the part of the drawing which one is not interested is called clipping An area on the device (ex. Screen) onto which the window will be mapped is called viewport. Window defines what to be displayed. A viewport defines where it is to be displayed. Most of the time, windows and viewports are usually rectangles in standard position(i.e aligned with the x and y axes). In some application, others such as general polygon shape and circles are also available However, other than rectangle will take longer time to process. Viewing Transformation Viewing transformation is the mapping of a part of a world- coordinate scene to device coordinates. In 2D (two dimensional) viewing transformation is simply referred as the window-to-viewport transformation or the windowing transformation. Mapping a window onto a viewport involves converting from one coordinate system to another. If the window and viewport are in standard position, this just involves translation and scaling. if the window and/or viewport are not in standard, then extra transformation which is rotation is required. Viewing Transformation y-world y-view window 1 window x-view 0 1 x-world world Normalised device Window-To-Viewport Coordinate Transformation Window-to-Viewport transformation Window-To-Viewport Coordinate Transformation YWmax YVmax . xw,yw xv,yv YWmin YVmin XWmin XWmax XVmin XVmax Window-To-Viewport Coordinate Transformation xv - xvmin = xw - xwmin xvmax - xvmin xwmax - xwmin yv – yvmin = yw - ywmin yvmax – yvmin ywmax - ywmin From these two equations we derived xv = xvmin + (xw – xwmin)sx yv = yvmin + (yw – ywmin)sy where the scaling factors are sx = xvmax – xvmin sy = yvmax - yvmin xwmax – xwmin ywmax – ywmin Window-To-Viewport Coordinate Transformation The sequence of transformations are: 1. Perform a scaling transformation using a fixed- point position of (xwmin,ywmin) that scales the window area to the size of the viewport. 2. Translate the scaled window area to the position of the viewport. Window-To-Viewport Coordinate Transformation Relative proportions of objects are maintained if the scaling factors are the same (sx = sy). Otherwise, world objects will be stretched or contracted in either x or y direction when displayed on output device. How about character strings when map to viewport? maintains a constant character size (apply when standard character fonts cannot be changed). If character size can be changed, then windowed will be applied like other primitives. For characters formed with line segments, the mapping to viewport is carried through sequence of line transformations . Viewport-to-Normalized Device Coordinate Transformation From normalized coordinates, object descriptions can be mapped to the various display devices When mapping window-to-viewport transformation is done to different devices from one normalized space, it is called workstation transformation. The Viewing Pipeline OpenGL 2D Viewing Functions To transform from world coordinate to screen coordinates, the appropriate matrix mode must be chosen glMatrixMode (GL_PROJECTION); glLoadIdentity( ); To define a 2D clipping window, we use OpenGL Utility function: gluOrtho2D( xwmin, xwmax, ywmin, ywmax); This function also perform normalization (NDC) OpenGL 2D Viewing Functions To specify the viewport parameters in OpenGL, we use function glViewport(xvmin, yvmin, vpWidth, vpHeight); Clipping Operations Procedures that identifies those portions of a picture that are either inside or outside a specified region of a space is referred to as clipping algorithm or clipping The region against which an object is to clipped is called clip window before after Clipping Operation Note: different graphic elements may require different clipping techniques. Character, need to include all or completely omit depending on whether or not its center lies within the window. Clipping Application of clipping include: Extracting part of defined scene for viewing. Identifying visible surfaces in three-dimensional views. Anti-aliasing line segments or object boundaries. Creating objects using solid-modeling procedures. Displaying a multiwindow environment. Drawing and painting operations that allow parts of picture to be selected for copying, moving, erasing, or duplication. Clipping Clipping algorithm can be applied in both world coordinate and normalized device coordinates In world coordinate: only the contents of the window interior are mapped to device coordinates world coordinate clipping removes those primitives outside the window from further consideration. Eliminating the unnecessary processing to transform In normalized device coordinate: mapping is fully done from world to normalized device. clipping will reduce calculations by allowing concatenation of viewing and geometric transformation matrices. Clipping Algorithms Algorithms have been developed for clipping for the following primitives types: 1. Point Clipping. 2. Line Clipping (straight-line segments) 3. Area Clipping (polygons) 4. Curve Clipping 5. Text Clipping. Point Clipping Assuming clip window is standard rectangle position, point P-(x,y) will be save for display if its satisfied: xwmin <= x <= xwmax ywmin <= y <= ywmax where the edges of the clip window (xwmin, xwmax, ywmin, ywmax) can be either the world coordinate window boundaries or viewport boundaries. Line Clipping Cohen-Sutherland Outcode Algorithm This algorithm quickly removes lines which entirely to one side of the clipping where both endpoints above, below, right or left. The advantage of the algorithm is, it uses bit operations for testing. Cohen Sutherland I E K 1001 B 1000 G 1010 L H S M A 0001 0010 0000 F T J D 0110 0101 C 0100 Cohen Sutherland Bit convention 1 First bit y > ymax Second bit y < ymin Third bit x > xmax Fourth bit x < xmin Bit convention 2 (1 through 4 from right to left) bit 1 : left bit 2 : right bit 3 : below bit 4 : above Cohen Sutherland Segment endpoints are each given 4-bit binary codes. The high order bit is set to 1 if the point above window The next bit set to 1 if point is below window The third and fourth indicate right or left of window. Testing result: 0000 if both endpoints are inside window. if the line segment lies entirely on one side of the window, then both endpoints will have a 1 in the outcode bit position for that side. How?: By taking AND operation. If NONZERO then line segment may be rejected. Ex: AB and CD Cohen Sutherland (Problem?) Certain lines cannot be identified as completely inside or outside a clip window by doing AND test. Lines which completely outside the clip window can also produce AND 0000. So, process of discarding is required and intersection with boundary must be calculated (divide and conquer). Difficulty is when a line crossed one or more of the lines which contain the clipping boundary. Ex: EF and DJ Cohen Sutherland (Solution) The point of intersection between the line segment and clipping lines may be used to break up the line segment. Intersection points with a clipping boundary can be calculated using the slope-intercept form of the line equation. Cohen Sutherland (Solution) For a line with endpoint (x1,y1) to (x2,y2) Y coordinate of the intersection point with vertical boundary is obtained with y = y1 + m(x-x1) where x value is set either xwmin or xwmax and the slope of the line is m = (y2 – y1) / (x2 – x1) X coordinate of the intersection point with horizontal boundary is calculated by x = x1 + (y-y1)/m where y is set either to ywmin or to ywmax Cohen Sutherland (Pseudocode) Brief outline of the algorithm. First compute the outcodes for the two endpoints (p1 and p2). Enter the loop, check whether both outcodes are zero Both zero, then enter segment in display list. If not both zero, perform the AND function. If nonzero, reject the line. If zero, subdivide the line segment. Repeat the loop. Liang Barsky Line Clipping Parametric equation of a line segment (x1,y1) and (x2,y2) x = x1 + u∆x y = y1 + u∆y Where 0 <= u <=1 ∆x = x2 – x1 ∆y = y2 –y1 From point clipping, we have xwmin < = x <= xwmax Ywmin <= y <= ywmax For line clipping, we derived xwmin < = x1 + u∆x <= xwmax ywmin <= y1 + u∆y <= ywmax Liang-Barsky Line Clipping Based on these four equations, the following rules are introduced upk <= qk where k = 1 (left boundary) 2 (right boundary) 3 (bottom boundary) 4 (top boundary) parameters p and q are defined as: p1 = - ∆x, q1 = x1 - xwmin p2 = ∆x, q2 = xwmax – x1 p3= - ∆y, q3 = y1 - ywmin p4= ∆y, q4 = ywmax – y1 Liang-Barsky Line Clipping If any line that is parallel to one of the clipping boundaries, pk = 0. If qk < 0 then line is completely outside the boundary. When pk < 0, the infinite extension of the line proceeds from the outside to the inside. If pk > 0, the line proceeds from the inside to the outside. For nonzero value of pk , we can calculate the value of u corresponds to the point where infinitely extended line intersects the extension boundary k as u = qk/ pk u1(outside to inside) where its value ranges from 0 to r u2 (inside to outside) where its value ranges from 1 to r. If u1 > u2, the line is completely outside the clip window and can be rejected Liang Barsky vs Cohen Sutherland Liang-Barsky algorithm is more efficient than Cohen since intersection calculations are reduced. Liang-Barsky require only one division and window intersections of the line are computed only once. Cohen required repeated calculation even though the line may be completely outside the clip window. Nicholl-Lee-Nicholl Line Clipping (NLN) Creating more regions around the clip window to avoid multiple clipping of an individual line segment. Different to Cohen where Cohen require multiple calculation along the path of a single line before an intersection on the clipping rectangle is locate or completely rejected. NLN eliminate extra calculations through new region created. Nicholl-Lee-Nicholl Line Clipping (NLN) Compared to Cohen and Liang; NLN perform fever comparisons and division. The trade-off is that NLN can only be applied two two-dimensional clipping only. Nicholl-Lee-Nicholl Line Clipping (NLN) How its work? First , need to determine the endpoints P1 and P2 from the possible nine regions. Out of nine, NLN only interested on three regions. If P1 lies in any one of other six regions, symmetry transformation will be applied first. Then, is to determine where P2 is. If P1 is inside and P2 is outside we will follow the first case where intersection with appropriate window boundary is carried out. Four regions of L,T,R,B which contains P2 are introduced. If P1 and P2 are inside the clipping rectangle, we save the entire line. Nicholl-Lee-Nicholl Line Clipping (NLN) P2 (6,7) (2,6) LT (6,6) L L P1 LR L (1,4) (2,2) LB (6,2) Nicholl-Lee-Nicholl Line Clipping (NLN) For example when P2 is in region LT if Slope P1PTR < Slope P1P2 < Slope P1PTL Or (YT - Y1)/(XR – X1)) < (Y2 - Y1)/(X2 – X1) < (YT – Y1)/(XL – X1) Ex: (6-4)/(6-1) < (7-4)/(6-1) < (6-4)/(2-1) 2/5 < 3/5 < 2 And we clip the entire line if (YT – Y1)(X2 – X1) < (XL – X1)(Y2 – Y1) Nicholl-Lee-Nicholl Line Clipping (NLN) From the parametric equations: x = x1 + (x2 – x1)u y = y1 + (y2 – y1)u An intersection position on the left boundary is x = xL with u = (xL – x1)/(x2 – x1) will give coordinate y = y1 +((y2 – y1)/(x2 – x1)) * (xL – x1) An intersection position on the top boundary has y = yT with u = (yT – y1)/(y2 – y1) will give coordinate x = x1 +((x2 – x1)/(y2 – y1)) * (yT – y1) Line Clipping Using Nonrectangular Clip Windows Some applications clip lines against arbitrarily shaped polygons. Liang and Cyrus-Beck algorithm can be extended to convex polygon windows. For concave polygon-clipping regions, we need to split the concave polygon into set of convex polygons. Circles or other curved-boundary clipping regions are also possible but slower because intersection calculations involve nonlinear curve equation. Line can be identified as completely inside if the distance for both endpoints of a line is less than or equal to the radius squared. If not, then intersection calculation is performed.
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