CS Polygon Scan Conversion. Slide 1

Transcription

1 CS Polygo Sca Coversio Slide 1

2 Polygo Classificatio Covex All iterior agles are less tha 180 degrees Cocave Iterior agles ca be greater tha 180 degrees Degeerate polygos If all vertices are colliear Slide 2

3 Idetifyig Cocave Polygos Usig directio of cross products of adjacet edges If all same sig, the covex, else cocave Y V 5 E 5 V 4 E 1 X E 2 > 0 E 6 E 4 E 3 V 3 E 2 X E 3 > 0 E 3 X E 4 < 0 E 4 X E 5 > 0 E 2 E 5 X E 6 > 0 V 1 E 1 V 2 X Slide 3

4 Splittig Cocave Polygos Assume that the polygo is o XY plae Fid the cross products of adjacet edges If it has a egative z compoet Split the polygo alog the lie of the first edge vector of the cross product Slide 4

5 Idetifyig Cocave Polygos Usig directio of cross products of adjacet edges If all same sig, the covex, else cocave Y V 5 E 5 V 4 E 1 X E 2 > 0 E 6 E 4 E 3 V 3 E 2 X E 3 > 0 E 3 X E 4 < 0 E 4 X E 5 > 0 E 2 E 5 X E 6 > 0 V 1 E 1 V 2 X Slide 5

6 Example Slide 6

7 Rotatioal Method For each vertex V k Traslate so that V k coicides with origi Rotate clockwise so that V k+1 lies o the x-axis If V k+2 is below the x- axis, the cocave Split alog x-axis to get two polygos Apply iverse trasformatios Slide 7

8 Triagulatig a Covex Polygo V 5 V 4 V 3 Take ay three vertices i order Throw away the middle vertex from the list V 6 V 2 Apply to the same procedure V 1 Util oly three vertices remai Slide 8

9 Triagulatig a Covex Polygo V 5 V 4 V 3 V 1,V 2,V 3,V 4,V 5,V 6 V 1,V 2,V 3 tri 1 V 1,V 3,V 4,V 5,V 6 V 1,V 3,V 4 tri 2 V 6 V 2 V 1,V 4,V 5,V 6 V 1,V 4,V 5 tri 3 V 1 V 1,V 5,V 6 tri 4 Slide 9

10 Polygo Rasterizatio Works for both covex ad cocave polygos Basic Idea: Itersect scalie with polygo edges ad fill betwee pairs of itersectios For y = ymi to ymax 1) itersect scalie y with each edge 2) sort iteresectios by icreasig x [p0,p1,p2,p3] 3) fill pairwise (p0 - p1, p2 - p3) Slide 10

11 Fillig i Odd Parity Rule Sort the itersectios Start with eve parity, ad traverse scalie from left to right Wheever a itersectio is ecoutered, flip parity If parity is odd, draw the pixel Cotiue till the ed of the scalie Slide 11

12 Problem: Shared Boudaries Case 1: Shared vertices Case 2: Shared edges Case 3: Shared vertices ad edges Slide 12

13 How to solve this? Let the pixel be colored i the order the polygos are sca coverted Rightmost polygo is the domiat oe Problems Colorig same pixel multiple times Ca cause colorig problems depedig o what exactly the colorig fuctio does Ca create a amalgam of colors Slide 13

14 How to solve this? Defie iterior: For a give pair of itersectio poits (Xi, Y), (Xj, Y) Fill ceilig(xi) to floor(xj) Will resolve shared boudary problems Itersectio has a iteger X coordiate if Xi is iteger, we defie it to be iterior (fill) if Xj is iteger, we defie it to be exterior (do t fill) Slide 14

15 Problem: Edge edpoit Itersectio is a edge ed poit, say: (p0, p1, p2)?? (p0,p1,p1,p2), so we ca still fill pairwise I fact, if we compute the itersectio of the scalie with edge e1 ad e2 separately, we will get the itersectio poit p1 twice. Keep both of the p1. Slide 15

16 Problem: Edge edpoit But what about this case: still (p0,p1,p1,p2) Slide 16

17 Rule Rule: If the itersectio is the ymi of the edge s edpoit, cout it. Otherwise, do t. Do t cout p1 for e2 Slide 17

18 Horizotal edges Need ot be cosidered G F AB I H A itersectio of JA E Parity becomes odd at A B itersectio of BC Parity becomes eve at B Hece AB is draw J C D A B Slide 18

19 Horizotal edges CD J itersectio of IJ Not of JA Parity becomes odd at J C o chage see Parity remais odd D itersectio of ED Parity chages to eve CD is draw I J A H G C B F D E Slide 19

20 Horizotal edges IH G F I o itersectio of IJ Parity remais eve H itersectio of GH I H E Parity becomes odd IH is ot draw But right spa of IH is draw A B J C D Slide 20

21 Horizotal edges GF G F G o itersectio of GH Parity remais eve F o itersectio of FE I H E Parity becomes eve GF is ot draw J C D A B Slide 21

22 Performace Improvemet Brute force: itersect all the edges with all scalie Goal:compute the itersectios more efficietly fid the ymi ad ymax of each edge ad itersect the edge oly whe it crosses the scalie oly calculate the itersectio of the edge with the first sca lie it itersects calculate dx/dy for each additioal scalie, calculate the ew itersectio as x = x + dx/dy Slide 22

23 Data Structure Edge table: Bucket Sort A separate bucket for each scalie Each edge goes to the bucket of its y mi scalie Withi each bucket, edges are sorted by icreasig x of the y mi edpoit (x max, y max ) (x max, y max ) (x mi, y mi ) (x mi, y mi ) Slide 23

24 Edge Table Slide 24

25 Active Edge Table (AET) A list of edges active for curret scalie, sorted i icreasig x y = 9 y = 8 Slide 25

26 Algorithm Costruct the Edge Table (ET); Active Edge Table (AET) = ull; for y = Ymi to Ymax Merge-sort ET[y] ito AET by x value for each edge i AET if edge.ymax = y remove edge from AET Fill betwee pairs of x i AET for each edge i AET edge.x = edge.x + dx/dy sort AET by x value ed sca_fill Slide 26