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Arrangements

# Arrangements - Arrangements and Duality Supersampling in...

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Arrangements and Duality Supersampling in Ray Tracing Khurram Hassan Shafique

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Duality The concept: we can map between different ways of interpreting 2D values. Points (x,y) can be mapped in a one-to-one manner to lines (slope,intercept) in a different space. There are different ways to do this, called duality transforms .
Duality Transforms A duality transform is a mapping which takes an element e in the primal plane to element e* in the dual plane . One possible duality transform: point p : ( p x , p y ) line p* : y = p x x p y line l : y = mx + b point l* : ( m , - b )

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p Duality Transforms This duality transform takes points to lines, lines to points line segments to double wedges This duality transform preserves order Point p lies above line l point l* lies above line p* l p* l*
Duality The dualized version of a problem is no easier or harder to compute than the original problem. But the dualized version may be easier to think about.

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New Concept: Arrangements of Lines L is a set of n lines in the plane. L induces a subdivision of the plane that consists of vertices, edges, and faces. This is called the arrangement induced by L , denoted A ( L ) The complexity of an arrangement is the total number of vertices, edges, and faces.
Arrangments Number of vertices of A ( L ) – Vertices of A ( L ) are intersections of l i , l j L Number of edges of A ( L ) n 2 Number of edges on a single line in A ( L ) is one more than number of vertices on that line. Number of faces of A ( L ) Inductive reasoning: add lines one by one Each edge of new line splits a face. Total complexity of an arrangement is O ( n 2 ) 2 n 1 2 2 2 + + n n = + n i i 1 1

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How Do We Store an Arrangement? Data Type: doubly-connected edge-list (DCEL) Vertex: Coordinates, Incident Edge Face: an Edge Half-Edges Origin Vertex Twin Edge Incident Face Next Edge, Prev Edge
Building the Arrangement Iterative algorithm: put one line in at a time. • Start with the first edge e that l i intersects. Split that edge, and move to Twin ( e )

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ConstructArrangement Algorithm Input : A set L of n lines in the plane Output : DCEL for the subdivision induced by the part of A ( L ) inside a bounding box 1. Compute a bounding box B ( L ) that contains all vertices of A ( L ) in its interior 2. Construct the DCEL for the subdivision induced by B ( L ) 3. for i =1 to n do 4. Find the edge e on B ( L ) that contains the leftmost intersection point of l i and A i 5. f = the bounded face incident to e 6. while f is not the face outside B ( L ) do 7. Split f , and set f to be the next intersected face
ConstructArrangement Algorithm -Running Time- We need to insert n lines.

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Arrangements - Arrangements and Duality Supersampling in...

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