Windows and Viewports Window Interface Window Viewport Information outside the viewport is clipped away
Terminology Modelling Coordinate System – Representation of an object measured in some physical units World Coordinate System (Object Space) - Representation of a set of objects, all measured in the same physical units. Window - The rectangle defining the part of the world we wish to display. Screen Coordinate System (Image Space) - The space within the image is displayed Interface Window - The visual representation of the screen coordinate system for “window” displays (coordinate system moves with interface window). Viewport - The rectangle on the raster graphics screen (or interface window for “window” displays) defining where the image will appear. (Default is usually entire screen or interface window.) Viewing Transformation - The process of going from a window in world coordinates to a viewport in screen coordinates.
The Viewing Pipeline Construct world- coordinate scene using modeling coordinate transformations Convert world coordinates to viewing coordinates Map viewing coordinates to normalized viewing coordinates using window-viewport specification Map normalized viewport to device coordinates Beginning with modelling coordinates, we go through a series of coordinate transformations which result in device coordinates that can be plotted
Viewing Transformation Choose Window in World Coordinates Clip to size of Window Translate to origin Scale to size of Viewport Translate to proper position on screen (Interface Window)
Notes on Viewing Transformation Panning - Moving the window about the world Zooming - Reducing the window size As the window increases in size, the image in the viewport decreases in size and vice versa Beware of aspect ratio.
Clipping A line is visible if both of its end points are in the window. (xr, yt) (xl, yt) (xl, yb)(xr, yb) A point is visible if xl < X < xr and yb < Y < yt
Cohen-Sutherland Algorithm Region Checks: Trivially reject or accept lines and points. Fast for large windows (everything is inside) and for small windows (everything is outside). Each vertex is assigned a four-bit outcode. Bit 1 <-- sign bit of (yt-Y) -- point is above window Bit 2 <-- sign bit of (Y-yb) -- point is below window Bit 3 <-- sign bit of (xr-X) -- point is to right of window Bit 4 <-- sign bit of (X-xl) -- point is to left of window
Cohen-Sutherland Clipping A line can be trivially accepted if both endpoints have an outcode of 0000. A line can be trivially rejected if any corresponding bits in the two outcodes are both equal to 1. (This means that both endpoints are to the right, to the left, above, or below the window.) Can be computed with a logical AND. 100110001010 000100000010 010101000110 Bit 1: Above Bit 2: Below Bit 3: Right Bit 4: Left
Clipping Lines not Accepted or Rejected In the case where a line can be neither trivially accepted nor rejected, the algorithm uses a “divide and conquer” method. Line AD: 1) Test outcodes of A and D --> can’t accept or reject. 2) Calculate intersection point B, which is conceptually on the window side of the dividing line. Form new line segment AB and discard the rest of the line because it is above the window. 3) Test outcodes of A and B. Reject. Example: A D B C E FG H
Line Intersection For intersection with the clipping boundary Use the slope-intercept form of the line equation. Intercept with a vertical boundary: y = y1 +m(x-x1) where x = xmin or x max, depending on the boundary in question and m = (y2 – y1) / (x2 – x1) And similarly for a horizontal boundary, solving for x
Other Line Clipping Algorithms Liang-Barsky Line Clipping Uses the parametric equation of a line segment x = x1 + u (x2 – x1) y = y1 = u (y2-y1) For each line, calculate values for u1 and u2 that define the part of the line that lies within the clip rectangle Comparison with Cohen-Sutherland Generally faster Does fewer intersection calculations Window intersections of a line of calculated once whereas Cohen-Sutherland can repeatedly calculate intersections along a line path, even though the line is completely outside of the window. Does the intersections calculations faster Calculates u1 and u2 by a single division Cohen-Sutherland requires division and multiplication
Polygon Clipping Polygons can be clipped against each edge of the window one edge at a time. Window/edge intersections, if any, are easy to find since the X or Y coordinates are already known. Vertices which are kept after clipping against one window edge are saved for clipping against the remaining edges. Note that the number of vertices usually changes and will often increase.
Polygon Clipping Algorithm The window boundary determines a visible and invisible region. The edge from vertex i to vertex i+1 can be one of four types: Exit visible region- save the intersection Wholly outside visible region- save nothing Enter visible region- save intersection and endpoint Wholly inside visible region- save endpoint P4 I2 P1 I1 P3 P2 P4 I2 I1 P1
Polygon clipping issues The final output, if any, is always considered a single polygon. The spurious edge may not be a problem since it always occurs on a window boundary, but it can be eliminated if necessary.
Pipelined Polygon Clipping Because clipping against one edge is independent of all others, it is possible to arrange the clipping stages in a pipeline. the input polygon is clipped against one edge and any points that are kept are passed on as input to the next stage of the pipeline. This way four polygons can be at different stages of the clipping process simultaneously. This is often implemented in hardware. Clip Top Clip Right Clip Bottom Clip Left
Further Reading See Rowe – p47 and following. Note, Rowe doesn’t describe Liang-Barsky Line Clipping – see Hearn and Baker for a description.