CS 548: COMPUTER GRAPHICS REVIEW: OVERVIEW OF POLYGONS SPRING 2015 DR. MICHAEL J. REALE

Transcription

1 CS 548: COMPUTER GRPHICS REVIEW: OVERVIEW OF POLYGONS SPRING 05 DR. MICHEL J. RELE

2 NOTE: COUNTERCLOCKWISE ORDER ssuming: Right-handed sstem Vertices in counterclockwise order looking at front of polgon

3 FILL RES Fill area or filled area = area to be filled with color or pattern or both Usuall surface of object Usuall polgons

4 WHY POLYGONS? Wh? oundaries = linear equations efficient fill algorithms Can approimate most cured surfaces with polgons Shading easier especiall with triangles single plane per triangle Surface tessellation = approimating a cured surface with polgons More polgons better detail, but more ertices/polgons to process lso called fitting the surface with a polgon mesh

5 POLYGON DEFINITIONS Polgon = a figure with or more ertices connected in sequence b straight-line segments edges or sides of the polgon Most loose definition an closed-polline boundar More finick definitions contained in single plane, edges hae no common points other than their endpoints, no three successie points collinear Standard polgon or simple polgon = closed-polline with no crossing edges

6 DOES POLYGON LIE IN SINGLE PLNE? In computer graphics, polgon not alwas in same plane: Round-off error E.g., after transformations Fitting to surface makes non-planar polgons E.g., quad bent in half Thus, use triangles single plane per triangle

7 DEGENERTE POLYGONS Degenerate polgons = often used to describe polgon with: or more collinear ertices generates a line segment E.g., in etreme case, triangle with no area Repeated erte positions some edges hae length 0

8 CONVEX VS. CONCVE Interior angle = angle inside polgon boundar formed b two adjacent edges interior angle: Cone = all interior angles less than 80 Concae = one or more interior angles greater than or equal to 80 looking at ertices compared to edge lines: Cone = for all edge lines, all other ertices are on one side Concae = for one or more edge lines, some of the ertices are on one side and some are on the other lso, one or more edge lines will intersect another edge

9 CONVEX VS. CONCVE: CORNY MEMORY HOOK

10 DELING WITH DEGENERTES ND CONCVE POLYGONS OpenGL cannot deal with degenerate polgons or concae polgons programmer must detect/preprocess them Degenerate polgons detect and remoe Concae polgons detect and split into cone polgons

11 m/blog/wpcontent/uploads/0/0/irplane.j pg WHT S YOUR VECTOR, VICTOR? ector = N or N matri # of rows X # of columns In computer graphics: N usuall,, or 4 homogeneous coordinates w Use column ectors N Components of the ector: D and alues D,, and alues 4D,,, and w alues Vector interpreted as: Location w = Direction w = 0 Scalar = single alue or ector

12 LENGTH OF VECTOR ND THE ZERO VECTOR Use Euclidean distance for length: ector with a length of ero is called the ero ector:

13 SCLING VECTORS Scalar times a ector = scalar times the components of the ector Eample: multipl 5 b a D ector 5* 5* 5* 5* 5

14 NORMLIZED VECTORS Normalie a ector = diide ector b its length makes length equal to Equialent to multipling ector b / Resulting ector is called a normalied ector WRNING: This is NOT the same as a NORML ector! lthough normal ectors are often normalied.

15 DDING/SUTRCTING VECTORS dd/subtract ectors add/subtract components Geometric interpretation: dding putting head of one ector on tail of the other Subtracting gies direction from one endpoint to the other

16 DOT PRODUCT Result is a single number i.e., scalar another name for the dot product is the scalar product Dot product of ector with itself = square of length of ector: Equialent to: where θ = smallest angle between the two ectors If the ectors are normalied, then: cos cos

17 DOT PRODUCT: CHECKING NGLES Look at sign of dot product to check angle: > 0 ectors pointing in similar directions 0 <= θ < 90 = 0 ectors are orthogonal i.e., perpendicular to each other θ = 90 < 0 ectors pointing awa from each other 90 < θ <= 80 For normalied ectors, dot product ranges from [-, ]: = ectors pointing in the eact same direction θ = 0 = 0 ectors are orthogonal i.e., perpendicular to each other θ = 90 = - ectors pointing in the eact opposite direction θ = 80 Remember: cos cos0 cos90 cos80 0 We re going to use this trick for lighting calculations later

18 CROSS PRODUCT lso called ector product results is a ector Gien two ectors U and V gies ector W that is orthogonal perpendicular to both U and V U, V, and W form right-handed sstem! I.e., can use right-hand-rule on U and V IN THT ORDER to get W Eample: X Y = Z ais! The length of W = U X V is equialent to: where again θ = smallest angle between U and V If U and V are parallel θ = 0 sin θ = 0 get ero ector for W! sin V U V U W u u u u u u V U w w w W

19 CROSS PRODUCT: ORDER MTTERS! WRNING! ORDER MTTERS with the cross product! U V V U Propert of anti-commutatiit REMEMER THE RIGHT-HND-RULE!!!

20 COMPUTING THE CROSS PRODUCT: SRRUS S SCHEME Follow diagonal arrows for each arrow multipl elements along arrow times sign at top e = ais, e = ais, e = ais u e u e u e u e u e u e V U

21 DETECTING ND SPLITTING CONCVE POLYGONS There are was we can do this: Vector method Check interior angles using cross product Rotational method Rotate each edge in line with X ais check if erte below X ais

22 SPLITTING Y VECTOR METHOD Transform to XY plane if necessar Get edge ectors in counterclockwise order: For each pair of consecutie edge ectors, get cross product If concae negatie Z component split polgon along first ector in cross product pair Hae to intersect this line with other edges Repeat process with two new polgons NOTE: successie collinear points anwhere cross product becomes ero ector! k k k V V E E E E E E E E E E E

23 SPLITTING Y ROTTIONL METHOD Transform to XY plane if necessar For each erte V k : Moe polgon so that V k is at the origin Rotate polgon so that V k+ is on the X ais If V k+ is below X ais polgon is concae split polgon along ais Repeat concae test for each of the two new polgons Stop when we e checked all ertices

24 SPLITTING CONVEX POLYGON Eer consecutie ertices make triangle Remoe middle erte Keep going until down to last ertices

25 PLNE EQUTION Need plane equation: Collision detection, ratracing/racasting, etc. Need normal of polgon: Lighting/shading ackface culling don t draw polgon facing awa from camera General equation of a plane:,, = an point on the plane C D 0,,C,D = plane parameters ON the plane + + C + D = 0 EHIND the plane + + C + D < 0 IN FRONT OF the plane + + C + D > 0

26 CLCULTING THE PLNE EQUTION Diide the formula b D Pick an noncollinear points in the polgon Sole set of simultaneous linear plane equations to get /D, /D, and C/D use Cramer s Rule,, / / / k D C D D k k k D C D C

27 QUICK REVIEW: DETERMINNT OF MTRIX M 4 *4 * i j k M i*6 *5 j*4 *6 k*5 *

28 IF THE POLYGON IS NOT CONTINED IN PLNE Either: OR: Diide into triangles Find approimating plane Diide ertices into subsets of Get plane parameters for each subset Get aerage plane parameters

29 NORML VECTOR Normal ector = gies us orientation of plane/polgon Points towards OUTSIDE of plane From back face to front face Perpendicular to surface of plane Normal N =,,C parameters from plane equation! lthough it doesn t hae to be, the normal ector is often normalied i.e., length =

30 GETTING NORML ND PLNE EQUTION Sole for plane equation normal =,,C OR Get normal from edges using cross-product sole for D in plane equation V, V, and V = consecutie ertices in counterclockwise order: N V V V V NOTE: Counterclockwise looking from outside the polgon towards inside

31 POINT-NORML PLNE EQUTION Gien the normal N and an point on the plane, the following holds true: N P D related, alternatie equation for a plane is the point-normal form: N V P 0 where V is an D point

32 EXMPLE