Computer Graphics I, Fall 2010 Implementation II

Computer Graphics I, Fall 2010 Objectives Introduce clipping algorithms for polygons Survey hidden-surface algorithms

Computer Graphics I, Fall 2010 Polygon Clipping Not as simple as line segment clipping ­Clipping line segment yields at most one line segment ­Clipping polygon can yield multiple polygons However, clipping convex polygon can yield at most one other polygon

Computer Graphics I, Fall 2010 Tessellation and Convexity One strategy: replace nonconvex (concave) polygons with set of triangular polygons (= tessellation) Also makes fill easier Tessellation code in GLU library

Computer Graphics I, Fall 2010 Clipping as a Black Box Can consider line segment clipping as process that takes in two vertices and produces either no vertices or vertices of clipped line segment

Computer Graphics I, Fall 2010 Pipeline Clipping of Line Segments Clipping against each side of window independent of other sides ­Can use four independent clippers in pipeline

Computer Graphics I, Fall 2010 Pipeline Clipping of Polygons Three dimensions: add front and back clippers Strategy used in SGI Geometry Engine Small increase in latency

Computer Graphics I, Fall 2010 Bounding Boxes Rather than doing clipping on complex polygon can use axis-aligned bounding box or extent ­Smallest rectangle aligned with axes that encloses polygon ­Simple to compute: max & min of x and y

Computer Graphics I, Fall 2010 Bounding boxes Can usually determine accept/reject based only on bounding box reject accept requires detailed clipping

Computer Graphics I, Fall 2010 Clipping and Visibility Clipping has much in common with hidden-surface removal In both cases, try to remove objects not visible to camera Often can use visibility or occlusion testing early in process to eliminate as many polygons as possible before going through entire pipeline

Computer Graphics I, Fall 2010 Hidden Surface Removal Object-space approach: use pairwise testing between polygons (objects) Worst case complexity O(n 2 ) for n polygons partially obscuringcan draw independently

Computer Graphics I, Fall 2010 Painter’s Algorithm Render polygons back-to-front ==> polygons behind others painted over B behind A as seen by viewer Fill B then A

Computer Graphics I, Fall 2010 Depth Sort Requires ordering of polygons first ­O(n log n) calculation for ordering ­Not every polygon is either in front or behind all other polygons Order polygons and deal with easy cases first, harder later Polygons sorted by distance from COP

Computer Graphics I, Fall 2010 Easy Cases A lies behind all other polygons ­Can render Polygons overlap in z but not in either x or y ­Can render independently

Computer Graphics I, Fall 2010 Hard Cases Overlap in all directions but one fully on one side of other cyclic overlap penetration

Computer Graphics I, Fall 2010 Back-Face Removal (Culling)  face visible iff 90    -90 equivalently cos   0 or v n  0 plane of face has form ax + by + cz + d = 0 ==> n = [a b c d] T but after normalization n = [ ] T need only test sign of c In OpenGL can simply enable culling but may not work correctly if have nonconvex objects

Computer Graphics I, Fall 2010 Image Space Approach Look at each projector ( nm for n x m frame buffer) find closest of k polygons Complexity O(nmk) Ray tracing z -buffer

Computer Graphics I, Fall 2010 z-Buffer Algorithm Use buffer “z or depth buffer” to store depth of closest object at each pixel found so far As render each polygon, compare depth of each pixel to depth in z buffer If less place shade of pixel in color buffer update z buffer

Computer Graphics I, Fall 2010 Efficiency If work scan line by scan line as move across scan line ==> depth changes satisfy a  x + b  y + c  z = 0 Along scan line  y = 0  z = -  x In screen space  x = 1

Computer Graphics I, Fall 2010 Scan-Line Algorithm Can combine shading and HSR through scan line algorithm scan line i: no need for depth information, can only be in no or one polygon scan line j: need depth information only when in more than one polygon

Computer Graphics I, Fall 2010 Implementation Need data structure to store ­Flag for each polygon (inside/outside) ­Incremental structure for scan lines stores which edges encountered ­Parameters for planes

Computer Graphics I, Fall 2010 Visibility Testing In many realtime applications, e.g., games, want to eliminate as many objects as possible within application ­Reduce burden on pipeline ­Reduce traffic on bus Partition space with Binary Spatial Partition (BSP) Tree

Computer Graphics I, Fall 2010 Simple Example consider 6 parallel polygons top view plane of A separates B and C from D, E and F

Computer Graphics I, Fall 2010 BSP Tree Can continue recursively ­Plane of C separates B from A ­Plane of D separates E and F Can put this information in BSP tree ­Use for visibility and occlusion testing