CS 450: COMPUTER GRAPHICS VECTORS SPRING 2016 DR. MICHAEL J. REALE

Transcription

1 CS 45: COMPUTER GRPHICS VECTORS SPRING 216 DR. MICHEL J. RELE

2 INTRODUCTION In graphics, we are going to represent objects and shapes in some form or other. First, thogh, we need to figre ot how to represent directions and points in space this leads s to ectors

3 INTRODUCTION TO VECTORS

4 INTRODUCTION In graphics, we will hae to define a lot of directions, offsets, and points in space We will do this sing ectors

5 WHT S YOUR VECTOR, VICTOR? Let s talk abot ectors at a HIGH leel (not how the nmbers are stored or how the are sed) Vector Describes a length and a direction asicall like an arrow with length floating in space If two ectors hae the same length and direction ectors are EQUL Een if the are in different locations NOT eqal same length bt different directions These ectors are eqal NOT eqal same direction bt different lengths

6 LENGTH ND SPECIL VECTORS The length of a ector is represented b: There are two special kinds of ectors that are defined b their length: Unit ector = has a length of 1 1 Zero ector = has a length of ero Direction for the ero ector is UNDEFINED!

7 WHT CN VECTOR REPRESENT? ector can be sed to represent: direction onl Often ector is normalied so that another ector pointing in the same direction is considered eqal n offset or displacement from some (nspecified) starting point E.g., moe 3 in X and 4 in Y Howeer, where do o start? Does allow o to add offsets together + location, position, or point Use ector to store offset from agreed pon starting point (origin) Usall doesn t make sense to add to locations together Point Unless o re taking the aerage Origin

8 STORING VECTORS ector is basicall a (N 1) or (1 N) matri (# of rows X # of colmns) Can also be iewed as an arra, a list, a strct, a class, etc. In compter graphics: N is sall 2, 3, or 4 (homogeneos coordinates) Usall se colmn ectors (N 1) Components of the ector: 2D and ales 3D,, and ales 3D homogeneos,,, and w ales w Scalar = single ale (or 11 ector)

9 INTERPRETING VECTORS Man times we will se homogeneos coordinates to represent ectors 2D Homogeneos w w 3D Homogeneos asicall, add one more entr w s we ll see later, depending on the ale of w, we will interpret the ector as a: Location (w = 1) Direction (w = ) in space We ll eplain in the ftre WHY it is particlarl sefl to do this

10 COMPUTING THE LENGTH OF VECTOR We sall se Eclidean distance to get the length of a ector: The sqare of the distance (not srprisingl) is gien b:

11 ZERO VECTOR The ero ector has the form:

12 SCLING VECTORS To mltipl a scalar b a ector, o mltiple that scalar b the components of the ector asicall, o are modifing the LENGTH of the ector, bt not the direction Eample: mltipl 5 b a 3D ector 5* 5* 5* 5* 5

13 NORMLIZED VECTORS Normalie a ector = diide ector b its length makes length eqal to 1 Eqialent to mltipling ector b 1/ Reslting ector is called a normalied ector WRNING: This is NOT the same as a NORML ector! NORMLIZED ector has length of 1 NORML ector ector perpendiclar to srface (not NECESSRILY of length 1)

14 DDING/SUTRCTING VECTORS To add/sbtract ectors, o add/sbtract their respectie components: ) ( ) ( ) ( ) ( ) ( ) (

15 GEOMETRIC INTERPRETTION OF DDING VECTORS dding Ptting head of one ector on tail of the other Called parallelogram rle gies s diagonal of parallelogram formed b and Can be done in either order COMMUTTIVE (like reglar addition) a b b a gain, reall onl makes sense if we think of ectors as displacements/offsets

16 GEOMETRIC INTERPRETTION OF SUTRCTING VECTORS Sbtracting gies direction from one endpoint to the other NOT commtatie (jst like reglar sbtraction) nother wa to think of it: Scale b -1 same length, opposite direction dd (-) to Yet again, reall onl makes sense if we think of ectors as displacements/offsets

17 MULTIPLICTION? DIVISION? So far, we e seen we can add, sbtract, and scale ectors Can we mltipl two ectors? Yes, sort of mltiplication of ectors is done in two different was Dot prodct (also called scalar prodct) Cross prodct (also called ector prodct) Can we diide two ectors? Not reall, no there isn t a nice analoge for diision with ectors

18 DOT PRODUCT

19 DOT PRODUCT Gien two 3D ectors and, the dot prodct of and is gien b: asicall: Mltiple corresponding components dd them together NOTE: Reslt is a single nmber (i.e., scalar) another name for the dot prodct is the scalar prodct lso called the inner (dot) prodct

20 DOT PRODUCT WITH YOURSELF The dot prodct of a ector with itself is eqal to the sqare of its length: So, if o hae a dot prodct fnction and a sqare root fnction, o can get the length of a ector

21 DOT PRODUCT: WHT DOES IT MEN? The dot prodct of two ectors and is in fact eqialent to: cos where θ = smallest angle between the two ectors (asicall it has to do with projecting one ector on another; we ll discss this later)

22 DOT PRODUCT OF NORMLIZED VECTORS If or ectors and are normalied (hae lengths eqal to 1), then: Dot prodct = cosine of angle between them 1 1 cos cos

23 DOT PRODUCT: CHECKING NGLES What this means: ( ) > ectors pointing in similar directions ( <= θ < 9 ) ( ) = ectors are orthogonal (i.e., perpendiclar to each other) (θ = 9 ) ( ) < ectors pointing awa from each other (9 < θ <= 18 ) For normalied ectors, dot prodct ranges from [-1, 1]: ( ) = 1 ectors pointing in the eact same direction (θ = ) ( ) = ectors are orthogonal (i.e., perpendiclar to each other) (θ = 9 ) ( ) = -1 ectors pointing in the eact opposite direction (θ = 18 ) Remember: cos cos( cos(9 cos(18 ) 1 ) ) 1 (We re going to se this trick for lighting calclations later )

24 DOT PRODUCT: CHECKING NGLES EXMPLES Non-normalied ectors pointing in a similar direction: *2 1* Non-normalied ectors pointing in different directions: *( 1) 1*( 2) Non-normalied ectors that are PERPENDICULR: *( 1) 1*(2) 2 2

25 GENERL FORM OF DOT PRODUCT general definition for the dot prodct for NY dimensionalit: n 1 i i i Some of the rles for the dot prodct: (, with if ) w w w and onl if (,,...,) ( a) a( ) (ectors are orthogonal (perpendic lar) to each other)

26 DEFINITION OF ORTHOGONL For the record: Orthogonal = perpendiclar So, if the dot prodct between two ectors is ero ectors are ORTHOGONL to each other ngle between them 9 degrees

27 VECTOR PROJECTION USING THE DOT PRODUCT The dot prodct is inoles when o want to orthogonall project one ector onto another E.g., if o orthogonall project onto splatting down on in a direction PERPENDICULR to gies o another ector w

28 VECTOR PROJECTION USING THE DOT PRODUCT Length of w: cos Remember: cosine is adjacent/hpotense length of = hpotense Direction of w normalied : The orthogonal projection (ector) w of a ector onto a ector is gien b: w cos Howeer, we can trn this into a form that ses onl dot prodcts: w cos cos 2 cos

29 VECTOR PROJECTION: FINL FORM w NOTE: Part in parentheses is a NUMER, while on the otside is a VECTOR

30 WHT IF V IS NORMLIZED? If is alread normalied w ( ) Means: ( ) lso means w = absolte ale of dot prodct between and

31 ORTHOGONL PROJECTION EXMINED FURTHER Projection gies s an orthogonal decomposition of We can describe in terms of two orthogonal ectors, w and ( w) w (-w) We ll talk abot this more when we get to orthogonal basis

32 NORM (LENGTH) REVISITED

33 NORM (LENGTH) REVISITED The norm (or length) of the ector is a non-negatie nmber that can be epressed sing the dot prodct: n 1 i 2 i (+) It too has some rles: (,,...,) a a (Cach (triangle ineqalit -Schwart ineqalit ) ) Remember: cos cos : R [ 1,1]

34 CROSS PRODUCT

35 RIGHT-HND RULE efore we go frther with coordinate spaces, etc., we need to talk abot which wa the,, and aes go relatie to each other OpenGL (and other sstems) se the right-hand rle Point right hand toward X, with palm p towards Y thmb points toward Z

36 CROSS PRODUCT Cross prodct lso called ector prodct (reslt is a ector) Gien two ectors U and V gies ector W that is orthogonal (perpendiclar) to both U and V U, V, and W form right-handed sstem! I.e., can se right-hand-rle on U and V (IN THT ORDER) to get W U V W W V Eample: (X Y) = Z ais! U

37 LENGTH OF CROSS PRODUCT The length of W (= U X V) is eqialent to: W U V U V sin where again θ = smallest angle between U and V If U and V are parallel θ = OR θ = 18 sin θ = get ero ector for W! lso happens if (UU) U U V θ = V θ = 18

38 CROSS PRODUCT: ORDER MTTERS! WRNING! ORDER MTTERS with the cross prodct! U V V U Propert of anti-commtatiit REMEMER THE RIGHT-HND-RULE!!!

39 COMPUTING THE CROSS PRODUCT To compte the cross prodct: V U w w w W

40 COMPUTING THE CROSS PRODUCT: SRRUS S SCHEME n easier wa to remember this is with a method called Sarrs s scheme: Follow diagonal arrows for each arrow mltipl elements along arrow times sign at top e = ais, e = ais, e = ais The e s are ectors ) ( ) ( ) ( ) ( ) ( ) ( e e e e e e V U

41 LINER (IN)DEPENDENCE

42 LINER DEPENDENCE ND INDEPENDENCE Let s sa we hae two ectors, and 1 If I can jst mltipl a nmber (scalar) b to get 1, then the ectors are linearl DEPENDENT = a* 1 linearl DEPENDENT 1 Linearl DEPENDENT

43 LINER DEPENDENCE ND INDEPENDENCE a a of ales for all Linearl DEPENDENT Linearl INDEPENDENT

44 LINER DEPENDENCE ND INDEPENDENCE nother wa to look at this is to rearrange the eqation and also mltipl 1 b its own scalar: a a Linearl INDEPENDENT If the ONLY wa for this to be tre is if a and a 1 eqal ero and 1 are linearl INDEPENDENT Can t cancel ot each other Otherwise and 1 are linearl DEPENDENT With two ectors, this onl happens when ectors are PRLLEL to each other 1 Linearl DEPENDENT

45 LINER DEPENDENCE ND INDEPENDENCE: DEFINED Let s sa now we hae a n ectors (, 1,, n-1 ), each with their own scalar factor (a, a 1,, a n-1 ) a... a n 1 n1 If the ONLY wa to make the aboe statement tre is to set a = a 1 = = a n-1 = ectors (, 1,, n- 1) are linearl INDEPENDENT Otherwise, ectors are linearl DEPENDENT Some or all of the ectors can cancel each other ot, gien the proper scaling factors

46 SIZE OF SPCE How big a space is (i.e., how man dimensions a space is n) is determined b the maimm nmber of linearl independent ectors o can make Eample: R 3 can hae at most 3 ectors in a set that are linearl independent Eample set of linearl independent ectors for R 3 : (1,,) (,1,) (,,1) can t come p with another one

47 ORTHONORML SES ND COORDINTE FRMES

48 ORTHONORML SIS Orthonormal basis = basis where the ectors meet the following conditions: Eer basis ector has length eqal to 1 i = 1 The normal part Eer pair of basis ectors mst be orthogonal angle between them eqals 9 The ortho part Forces them to be linearl independent more compact wa to describe this: i j, 1, i i j j

49 2D ORTHONORML SIS Two ectors and form an orthonormal basis IF: 1 ND: 1

50 3D ORTHONORML SIS Three ectors,, and w form an orthonormal basis IF: ND: If the basis is right-handed, then: w w w w

51 STNDRD SIS Standard orthonormal basis = basis where each basis ector i has components: One for dimension i Zero elsewhere Standard basis ectors denoted e i lso called the global coordinate sstem Origin assmed to be (,,) Eample: 3D standard basis e e e

52 MKING OUR OWN ORTHONORML SIS If we re sing the global coordinate sstem don t need to store the aes Jst need the,, coordinates Sometimes, howeer, we want to describe a ector in terms of or OWN orthonormal basis (aes,, and w, with offset point p) In that case, we DO need to store the coordinates ND the aes/offset es and offset point stored in terms of global coordinate sstem a coordinates relatie to (,,w) and offset p a w a a a

53 GETTING THE COORDINTES FOR DIRECTION VECTOR Gien a DIRECTION ector a, how do we find the coordinates relatie to,, and w? nswer: project onto each ais! Remember:,, and w are normalied a a w a a a a w

54 GETTING THE COORDINTES FOR POINT VECTOR Gien a POINT ector a, how do we find the coordinates relatie to,, and w? FIRST, sbtract p: THEN, project onto each ais w b w b b a a a p a b

55 MKING SIS FROM SINGLE VECTOR What if o onl hae ONE ector? Mabe o want w to point in a certain direction, bt o re not particlar abot and E.g., basis from srface normal Can o make an orthonormal basis from jst one ector? Yes, UT: Yor answer will NOT be niqe Yo need to watch ot for a few things

56 MKING SIS FROM SINGLE VECTOR Normalie w: w w w Choose NY ector t that is NOT collinear with w, and get the cross prodct of t with w; normalied ector will be HINT: start with t = w, then set smallest component of t eqal to 1 t t w w W T U Compte as cross prodct of w and : w W V U

57 MKING SIS FROM TWO VECTORS Sometimes we hae a direction we are facing (w = a) and we want to be aligned with ector b E.g., camera points in a direction, UT o do want it to be pright its local Y is aligned with the global Y If a and b LREDY orthogonal, jst get directl: ba To be on the safe side (and especiall is a and b are NOT orthogonal) se b as or t ector from the preios procedre for one ector: b b w w

58 SIS DRIFT Sometimes becase of floating-point errors, or basis ectors ma not be orthogonal anmore: Can either: Remake bases all oer again Works, bt faors w Ma not be basis closest to starting basis SVD (Singlar Vale Decomposition) Gies basis GURNTEED to be closest to the starting basis More on this later