10/29/02 (c) 2002 University of Wisconsin, CS559 Today Hidden Surface Removal Exact Visibility

10/29/02 (c) 2002 University of Wisconsin, CS559 Depth Sorting (Object Precision, in view space) An example of a list-priority algorithm Sort polygons on depth of some point Render from back to front (modifying order on the fly) Rendering: For surface S with greatest depth – If no overlap in depth with other polygons, scan convert – Else, for overlaps in depth, test for overlaps in the image plane If none, scan convert and go to next polygon – If S, S’ overlap in depth and in image plane, swap order and try again – If S, S’ have been swapped already, split and reinsert

10/29/02 (c) 2002 University of Wisconsin, CS559 Depth Sorting (2) Testing for overlaps: Start drawing when first condition is met: –x-extents or y-extents do not overlap –S is behind the plane of S’ –S’ is in front of the plane of S –S and S’ do not intersect in the image plane S S’ S or z x S S’S’ z x S S’S’ S S’

10/29/02 (c) 2002 University of Wisconsin, CS559 Depth sorting Advantages: –Filter anti-aliasing works fine Composite in back to front order with a sequence of over operations –No depth quantization error Depth comparisons carried out in high-precision view space Disadvantages: –Over-rendering –Potentially very large number of splits -  (n 2 ) fragments from n polygons

10/29/02 (c) 2002 University of Wisconsin, CS559 Area Subdivision Exploits area coherence: Small areas of an image are likely to be covered by only one polygon Three easy cases for determining what’s in front in a given region: 1. a polygon is completely in front of everything else in that region 2. no surfaces project to the region 3. only one surface is completely inside the region, overlaps the region, or surrounds the region

10/29/02 (c) 2002 University of Wisconsin, CS559 Warnock’s Area Subdivision (Image Precision) Start with whole image If one of the easy cases is satisfied (previous slide), draw what’s in front Otherwise, subdivide the region and recurse If region is single pixel, choose surface with smallest depth Advantages: – No over-rendering – Anti-aliases well - just recurse deeper to get sub- pixel information Disadvantage: – Tests are quite complex and slow

10/29/02 (c) 2002 University of Wisconsin, CS559 Warnock’s Algorithm Regions labeled with case used to classify them: 1) One polygon in front 2) Empty 3) One polygon inside, surrounding or intersecting Small regions not labeled Note it’s a rendering algorithm and a HSR algorithm at the same time – Assuming you can draw squares

10/29/02 (c) 2002 University of Wisconsin, CS559 BSP-Trees (Object Precision) Construct a binary space partition tree – Tree gives a rendering order – A list-priority algorithm Tree splits 3D world with planes – The world is broken into convex cells – Each cell is the intersection of all the half- spaces of splitting planes on tree path to the cell Also used to model the shape of objects, and in other visibility algorithms – BSP visibility in games does not necessarily refer to this algorith

10/29/02 (c) 2002 University of Wisconsin, CS559 BSP-Tree Example A C B A B C

10/29/02 (c) 2002 University of Wisconsin, CS559 Building BSP-Trees Choose polygon (arbitrary) Split its cell using plane on which polygon lies – May have to chop polygons in two (Clipping!) Continue until each cell contains only one polygon fragment Splitting planes could be chosen in other ways, but there is no efficient optimal algorithm for building BSP trees – Optimal means minimum number of polygon fragments in a balanced tree

10/29/02 (c) 2002 University of Wisconsin, CS559 Building Example We will build a BSP tree, in 2D, for a 3 room building – Ignoring doors Splitting edge order is shown – “Back” side of edge is side with the number

10/29/02 (c) 2002 University of Wisconsin, CS559 Building Example (1) 1 2 3b4b a, 4a, 6 2, 3b, 4b, a 3a

10/29/02 (c) 2002 University of Wisconsin, CS559 Building Example (2) 1 2 3b4b 5a 6 1 3a, 4a, a 3a 3b, 5b 2 4b, 5a - + 5b

10/29/02 (c) 2002 University of Wisconsin, CS559 Building Example (3) 1 2 3b4b 5a a 3a 2 4b, 5a - + 5b 4a, 6 3a + 5b 3b +

10/29/02 (c) 2002 University of Wisconsin, CS559 Building Example (Done) 1 2 3b4b 5a a 3a b 3a + 3b + 4a 6 + 5b 4b 5a +

10/29/02 (c) 2002 University of Wisconsin, CS559 BSP-Tree Rendering Observation: Things on the opposite side of a splitting plane from the viewpoint cannot obscure things on the same side as the viewpoint Rendering algorithm is recursive descent of the BSP Tree At each node (for back to front rendering): –Recurse down the side of the sub-tree that does not contain the viewpoint Test viewpoint against the split plane to decide which tree –Draw the polygon in the splitting plane Paint over whatever has already been drawn –Recurse down the side of the tree containing the viewpoint

10/29/02 (c) 2002 University of Wisconsin, CS559 BSP-Tree Rendering Example A C B A B C st View 2nd3rd4th

10/29/02 (c) 2002 University of Wisconsin, CS559 BSP-Tree Rendering (2) Advantages: –One tree works for any viewing point –Filter anti-aliasing and transparency work Have back to front ordering for compositing –Can also render front to back, and avoid drawing back polygons that cannot contribute to the view User two trees – an extra one that subdivides the window Disadvantages: –Can be many small pieces of polygon –Over-rendering

10/29/02 (c) 2002 University of Wisconsin, CS559 Exact Visibility An exact visibility algorithm tells you what is visible and only what is visible –No over-rendering –Warnock’s algorithm is an example Difficult to achieve efficiently in practice –Small detail objects in an environment make it particularly difficult But, in mazes and other simple environments, exact visibility is extremely efficient

10/29/02 (c) 2002 University of Wisconsin, CS559 Cells and Portals Assume the world can be broken into cells –Simple shapes –Rooms in a building, for instance Define portals to be the transparent boundaries between cells –Doorways between rooms, windows, etc In a world like this, can determine exactly which parts of which rooms are visible –Then render visible rooms plus contents

10/29/02 (c) 2002 University of Wisconsin, CS559 Cell and Portal Visibility Start in the cell containing the viewer, with the full viewing frustum Render the walls of that room and its contents Recursively clip the viewing frustum to each portal out of the cell, and call the algorithm on the cell beyond the portal

10/29/02 (c) 2002 University of Wisconsin, CS559 Cell-Portal Example (1) View

10/29/02 (c) 2002 University of Wisconsin, CS559 Cell-Portal Example (2) View

10/29/02 (c) 2002 University of Wisconsin, CS559 Cell-Portal Example (3) View

10/29/02 (c) 2002 University of Wisconsin, CS559 Cell-Portal Example (4) View

10/29/02 (c) 2002 University of Wisconsin, CS559 Cell-Portal Example (5) View

10/29/02 (c) 2002 University of Wisconsin, CS559 Cell-Portal Example (6) View

10/29/02 (c) 2002 University of Wisconsin, CS559 Cell-Portal Operations Must clip polygons to the current view frustum (not the original one) –Can be done with additional hardware clipping planes, if you have them Must clip the view frustum to the portal –Easiest to clip portal to frustum, then set frustum to exactly contain clipped portal In Project 2, you implement these things in software, for a 2.5d environment

10/29/02 (c) 2002 University of Wisconsin, CS559 Cell-Portal Properties Advantages –Extremely efficient - only looks at cells that are actually visible: visibility culling –Easy to modify for approximate visibility - render all of partially visible cells, let depth buffer clean up –Can handle mirrors as well - flip world about the mirror and pretend mirror is a portal Disadvantages –Restricted to environments with good cell/portal structure

10/29/02 (c) 2002 University of Wisconsin, CS559 Rendering a 2.5D Maze (1) Assume you are given the following: –Rooms, defined in 2D by the edges that surround the room –The height of the ceiling –Each edge is marked opaque or clear –For each clear edge, there is a pointer to the thing on the other side You know where the viewer is and what the field of view is –The viewer is given as (c x,c y,c z ) position –The view frustum is given as a direction vector (d x,d y,d z ) and an angle for the field of view (c x,c y,c z ) (d x,d y,d z )

10/29/02 (c) 2002 University of Wisconsin, CS559 Rendering a 2.5D Maze (2) Work in 2D for the visibility Represent the frustum as a left and right clipping line –You don’t have to worry about the top and bottom –Each clip line starts at the viewer’s position and goes to infinity in the viewing direction Write a procedure that clips an edge to the view frustum –This takes a frustum and returns the endpoints of the clipped edge, or a flag to indicate that the edge is not visible

10/29/02 (c) 2002 University of Wisconsin, CS559 Rendering a 2.5D Maze (3) Write a procedure that takes a room and a frustum, and draws the room –Clip each edge to the frustum –If the edge is visible, draw the wall that the edge represents Create the 3D wall from the 2d piece of edge Project the vertices Draw the polygon in 2D –If the edge is clear, recurse to draw the room through the edge Use the clipped edge to create a new, clipped frustum Call the same procedure with the neighboring room and the clipped frustum Draw the floor and ceiling first, because they will be behind everything

10/29/02 (c) 2002 University of Wisconsin, CS559 Where We Stand So far we know how to: –Transform between spaces –Draw polygons –Decide what’s in front Next –Deciding a pixel’s intensity and color

10/29/02 (c) 2002 University of Wisconsin, CS559