CS420/ S-04 Intro to 3D Math 1

Transcription

1 CS420/ S-04 Intro to 3D Math : Right-Handed vs. Left-Handed Hold out our left hand (reall, do it!): Thum to the right Inde finder up Middle finger straight ahead This forms a asis for a 3D oordinate sstem 04-1: Right-Handed vs. Left-Handed Hold out our left hand (reall, do it!): Thum to the right (+ ) Inde finder up (+ ) Middle finger straight ahead (+ ) This forms a asis for a 3D oordinate sstem Left-Handed Coordinate sstem 04-2: Right-Handed vs. Left-Handed Now, Hold out our right hand (es, reall do it!): Thum to the left (+ ) Inde finder up (+ ) Middle finger straight ahead (+ ) This forms the other asis for 3D oordinate sstem Right-Handed Coordinate sstem 04-3: Right-Handed vs. Left-Handed An asis an e rotated to e either left-handed or right-handed Swap etween sstems flipping an one ais Flipping two aes leaves handedness unhanged (wh?) What aout flipping all 3? 04-4: Right-Handed vs. Left-Handed Computer Graphis tpiall uses Left-Handed oordinate sstem Book does, too Pure linear algera often uses Right-Handed oordiate sstem Ogre also uses a Right-Handed oordinate sstem Eas transformation, just invert the sign of one ais 04-5: Multiple Cooridinate Sstems OK, so we ve deided on a right-handed oordinate sstem (given Ogre), with pointing Up

2 CS420/ S-04 Intro to 3D Math 2 Pik an aritrar loation for the origin Often in the middle of the world Can plae it off in some orner Not quite done an use multiple oordinate sstems! 04-6: Multiple Cooridinate Sstems World Spae Ojet Spae Camera Spae (Speial ase of Ojet Spae) Intertial Spae 04-7: World Spae Assume that the origin of the world is the middle of the field etween SI and K Hall 2130 Fulton, the offiial Universit address is there + is East (along Fulton), + is straight up, + is North What diretion is forward from me in world spae? What is the point 5 feet in front of me in world spae? What if I rotate 15 deg. to the left? 04-8: Ojet Spae Define a new oordinate sstem Origin is at m enter + to m right + is up through m head + is straight ahead 04-9: Ojet Spae In m ojet spae, finding a point right ahead of me is trivial Given a oordinate in m ojet spae, determining where I have to look (to aim, for instane) is trivial Of ourse, we will need a wa to translate etween world spae and ojet spae Sa tuned! Define an Ojet Spae for eah ojet in our world 04-10: Camera Spae Camera Spae is a speial ase of ojet spae Ojet is the amera

3 CS420/ S-04 Intro to 3D Math 3 We ll use left-handed oordinates (+ into the sreen), swapping to right-hand is eas (invert Z) Wh is amera spae useful? : Camera Spae 04-12: Camera Spae + Is an ojet within the amera s frustum? Is ojet A in front of ojet B, or vie-versa? Is an ojet lose enough to the amera to render?... et 04-13: Intertial Spae Halfwa etwen ojet spae and world spae Aes parallel to world spae Origin same as ojet spae

4 CS420/ S-04 Intro to 3D Math Ojet : Inertial Spae + Ojet Spae 04-15: Inertial Spae World Spae Ojet Spae Inertial Spae : Inertial Spae World Spae 04-17: Nested Coordinate Spaes Eah ojet needs to e oriented in world spae That is, the aes for the loal spae of the ojet need to e oriented in world spae. We ould use a different ojet s loal spae instead of gloal spae Easiest to see with an eample 04-18: Nested Coordinate Spaes Assume that we have a dog, whih has a head and ears The head an wag ak and forth (in relation to the od) The ears an flap up and down (in relation to the head)

5 CS420/ S-04 Intro to 3D Math 5 We don t want to derie the position of the ears in world spae, or even in dog spae Head spae is muh more onvienent 04-19: Nested Coordinate Spaes Dog s ears are desried in head spae Up and down in relation to the head Dog s head is desried in dog spae Bak and forth in relation to the dog Dog s position is desried in world spae 04-20: Nested Coordinate Spaes To render the dog (with ears!) Translate the ear loation from head spae to dog spae Translate the ear loation (and the head loation) from do spae to world spae Translate all the dog from world spae to amera spae Projet the ojets from 3-spae to a plane 04-21: Nested Coordinate Spaes The Head spae is a hild of the Dog spae The Dog spae is the parent of the Head spae The Ear spae is a hild of the head spae The Head spae is the parent of the ear spae We ould also dnamiall parent and unparent ojets 04-22: Changing Coordinate Spaes Our harater is wearing a red hat The hat is at position (0,100) in ojet spae What is the position of the hat in world spae? To make life easier, we will think aout rotating the aes, instead of moving the ojets

6 CS420/ S-04 Intro to 3D Math Ojet Spae : Changing Coordinate Spaes + World Spae Rotate ais lokwise 45 degrees + + Ojet Spae : Changing Coordinate Spaes World Spae 04-25: Changing Coordinate

7 CS420/ S-04 Intro to 3D Math Translate aes to the left & down Ojet Spae (now Inertial Spae) Spaes World Spae 04-27: Changing Coordinate Spaes Rotate aes to the right 45 degrees : Changing Coordinate Spaes Ojet Spae (now World Spae) + World Spae Hat rotates the the left 45 degrees, from (0,100) to (-70, 70) Translate aes to the left 150, and down 50 Hat rotates to the right 150 and up 50, to (80, 120) We ll see how to do those rotations using matries later... + (80,120) world spae + (0,100) Ojet spae + Ojet Spae : Changing Coordinate Spaes Basis A Vetor is a displaement Vetor has oth diretion and length World Spae 04-29: Bak to

8 CS420/ S-04 Intro to 3D Math 8 Can also think of a vetor as a position (just a displaement from the origin) Can e written as a row or olumn vetor Differnene an e important for multipliation 04-30: Vetor Operations Multipling a salar To multipl a vetorv a salar s, multipl eah omponent of the vetor s Effet is saling the vetor multipling 2 maintains the diretion of the vetor, ut makes the length twie as long Works the same for 2D and 3D vetors (and highter dimemsion vetors, too, for that matter) 04-31: Vetor Operations Multipling a salar Multipling a vetor -1 flips the diretion of the vetor Works for 2D and 3D Multipling a vetor -2 oth flips the diretion, and sales the vetor 04-32: Saling a Vetor V 2V -V (1/2)V 04-33: Length Vetor has oth diretion and length Length Diretion 04-34: Length

9 CS420/ S-04 Intro to 3D Math 9 Vetor has oth diretion and length Length Diretion (Two angles) 04-35: Length Vetorv = v 1,v 2,...v n ] Length of v: v = n i=1 v 2 i 2 2 a + a a a 04-36: Normaliing a Vetor Normalie a vetor setting its length to 1, ut maintining its diretion. Multipl 1/length v norm = v v Of ourse,v an t e the ero vetor Zero vetor is the onl vetor without a diretion 04-37: Vetor Addition Add two vetors adding their omponents u 1,u 2,u 3 ]+v 1,v 2,v 3 ] = u 1 +v 1,u 2 +v 2,u 3 +v +3]

10 CS420/ S-04 Intro to 3D Math 10 v2 v : Vetor Sutration v1+v2 v1 v2 Vetor sutration is the same as multipling -1 and adding v 1 v 2 is the displaement from the point at v 2 to the point atv 1 not the displaement fromv 1 to v : Vetor Sutration v2 v1 v1 v1-v2 -v : Point Distane v2 v1-v2 v1 We an use sutration and length to find the distane etween two points Represent points as vetors displaement from the origin Distane fromv to u is v u = u v Where v is the length of the vetorv : Dot Produt a = a 1,a 2,...,a n ] = 1, 2,..., n ] a = n i=1 a i i

11 CS420/ S-04 Intro to 3D Math 11 v 1 = 1, 1, 1 ],v 2 = 2, 2, 2 ] v 1 v 2 = : Dot Produt a = a osθ a θ 04-43: Dot Produt Ifaandare unit vetors: 04-44: Dot Produt ( ) a θ = aros a θ = aros(a ) If we don t need the eat angle, we an just use the sign Ifθ < 90,osθ > 0 Ifθ = 90,osθ = 0 If90 < θ < 180,osθ < 0 Sinea = a osθ: Ifa > 0,θ < 90( π 2 ) Ifa = 0,θ = 90( π 2 ) Ifa < 0,90 < θ < 180 π 2 < θ < π 04-45: Projeting Vetors

12 CS420/ S-04 Intro to 3D Math 12 v n v n v v v 04-46: Projeting Vetors Given a vetorv andn, we want to deomposev into two vetors,v (parallel ton) andv (perpendiular ton) v = n v n So all we need is v osθ = v v v = osθ v 04-47: Projeting Vetors v = n v n v = osθ v 04-48: Projeting Vetors v = n v n = n osθ v n = n osθ v n n 2 = n v n n 2 One we havev, findingv is eas, sine v = v +v

13 CS420/ S-04 Intro to 3D Math 13 v +v = v v = v v v = v n v n n : Cross Produt v 1 = 1, 1, 1 ], v 2 = 2, 2, 2 ] v 1 v 2 = , , ] Cross produt of two vetors is a new vetor perpendiular to the other two vetors 04-50: Cross Produt a a a a 04-51: Cross Produt Whih wa does the ross produt a point? It depends upon our oordinate sstem right-handed vs. left-handed For right-handed oordinate sstems, take our right hand, move our fingers fromato thum points along a For left-handed oordinate sstems, take our right hand, move our fingers from a to thum points along a 04-52: Cross Produt a Magnitude of ross produt: a = a sinθ Same as the area of the parallelogram defined aand

14 CS420/ S-04 Intro to 3D Math : Cross Produt a h θ w Area of parallelogram = w h w = sinθ = h/ a,h = a sinθ w h = a sinθ = a 04-54: Matries A 43 matrim: h w 04-55: Matries M = m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 m 41 m 42 m 43 A Square matri has the same width and height M = m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 A diagonal matri is a square matri with non-diagonal elements equal to ero 04-56: Matries M = m m m 33 The Identit Matri is a diagonal matri with all diagonal elements = : Matries I 3 = Matries and vetors

15 CS420/ S-04 Intro to 3D Math : Matries Vetors are a speial ase of matries Row vetors (as we ve seen so far),,] Column vetors : Transpose Written M T Ehange rows and olums 04-59: Transpose a d e f g h i j k l T = a d g j e h k f i l The transpose of a row vetor is a olumn vetor For an matrim,(m T ) T = M For a diagonal matrid, D T =? 04-60: Transpose The transpose of a row vetor is a olumn vetor For an matrim,(m T ) T = M For a diagonal matrid, D T = D True for an matri that is smmetri along the diagonal 04-61: Matri Multipliation Multipling a Matri a salar Multipl eah element in the Matri the salar Just like multipling a vetor a salar 04-62: Matri Multipliation km = k Multipling two matriesaandb m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 m 41 m 42 m 43 A dimensionsn m,b dimensionsm p = km 11 km 12 km 13 km 21 km 22 km 23 km 31 km 32 km 33 km 41 km 42 km 43

16 CS420/ S-04 Intro to 3D Math 16 C = AB C dimensionsn p 04-63: Matri Multipliation m ij = a ik kj k=1 a a a a a a = : Matri Multipliation a a a a a a = = a a a : Matri Multipliation a a a a a a = = a a a 23 31

17 CS420/ S-04 Intro to 3D Math : Matri Multipliation a a a a a a = = a a a : Matri Multipliation a a a a a a = = a a a : Matri Multipliation Vetors are speial ases of matries Multipling a vetor and a matri is just like multipling two matries ] m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 = m 11 + m 21 + m 31 m 12 + m 22 + m 32 m 13 + m 23 + m 33 ] 04-69: Matri Multipliation Vetors are speial ases of matries Multipling a vetor and a matri is just like multipling two matries 04-70: Matri Multipliation m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 = m 11 + m 12 + m 13 m 21 + m 22 + m 23 m 31 + m 32 + m 33

18 CS420/ S-04 Intro to 3D Math 18 Note that the following multipliations are not legal: m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 ] 04-71: Matri Multipliation Matri Multipliaton is not ommutative: AB BA (at least not for all A and B is it true for at least one A andb?) Matri Multipliation is assoiative: (AB)C = A(BC) Transposing produt is the same as the produt of the transpose, in reverse order: (AB) T = B T A T m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 ] m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m : Matri Multipliation Matri Multipliaton is not ommutative: AB BA (at least not for all A and B is it true for at least one A andb?) Matri Multipliation is assoiative: (AB)C = A(BC) Transposing produt is the same as the produt of the transpose, in reverse order: (AB) T = B T A T m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 T = ] m 11 m 21 m 31 m 12 m 22 m 32 m 13 m 23 m : Matri Multipliation Identit Matri I: AI = A (for appropriatei) IA = A (for appropriatei) 04-74: Matri Multipliation m 11 m 21 m 31 m 12 m 22 m 32 m 13 m 23 m m 11 m 21 m 31 m 12 m 22 m 32 m 13 m 23 m 33 Identit Matri I: AI = A (for appropriatei) IA = A (for appropriatei) 04-75: Matri Multipliation Identit Matri I: ]

19 CS420/ S-04 Intro to 3D Math 19 AI = A (for appropriatei) IA = A (for appropriatei) : Matri Multipliation Identit Matri I: AI = A (for appropriatei) IA = A (for appropriatei) 1 ] ] 1 ] 04-77: Row vs. Column Vetors A vetor an e reresented as a row vetor or a olumn vetor This makes a differene when using matries Row: va, ColumnAv It gets even more fun when using matries to do several transformations of a vetor: Row vabc, Column CBAv (note that to get the same transformation, ou need to take the transpose of A, B, andc when swapping etween row and olumn vetors 04-78: Row vs. Column Vetors a d e f g h i ] = a + + d + e + f g + h + i a d g e h f i = a + + d + e + f g + h + i ] 04-79: Row vs. Column Vetors ] a d ] e f g h a + + d ] e f g h ] = ] = (a + )e + ( + d)g (a + )f + ( + d)h ] ] ] e g a f h d e f g h ] a + + d ] = e(a + ) + g(a + ) f(a + ) + h(a + d) ] = ] 04-80: Row vs. Column Vetors DiretX and the tet use row vetors OpenGL and Ogre use olumn vetors

20 CS420/ S-04 Intro to 3D Math 20 Ogre has a ak end for oth OpenGL and Diret3D Ogre transposes matries efore sending them to D3D liraries Leture will use oth This is on purpose I want ou to reall understand what s going on, not just memorie formulas 04-81: More Matries Consider the vetor,, ] Rewrite as: V = V = = = : More Matries letp,q,r e unit vetors for+,+ and+ v = p+q+r We have definedv as a linear omination ofp,q andr. p, q, andrare asis vetors 04-83: Basis Vetors Vetors 04-84: Basis p,q, andrare unit vetors along thex,y andz aes we re used to seeing vetors deomposed this wa

21 CS420/ S-04 Intro to 3D Math 21 Tehniall, an 3 linearl-independent vetors ould e used as asis vetors Tpiall, mutuall perpendiular verties are used as asis vetors Basis vetors not aligned with aes: Ojet spae rotated from world spae v v = 2p - q p 04-85: Non-Perpendiular Basis q 04-86: v v = 3/5p + 6/5q Perpendiular Basis & Basis q p 04-87: Maries Look ak at our asis vetors p, q and r. Create a 33 matrim usingp,q andras rows: M = p q = p p p q q q r r r r

22 CS420/ S-04 Intro to 3D Math 22 Multipl a vetor this matri: ] p p p q q q r r r = 04-88: Maries & Basis ] p p p q q q r r r = p + q + r p + q + r p + q + r ] = 04-89: Maries & Basis This is reall ool. Wh? p + q + r Take a loal spae, defined 3 asis vetors Rotation onl (no translation) Create a matri with these vetors as rows (or ols) Matri transforms from loal spae into gloal spae Coordinates in loal spae ( l, l ) v l Coordinates in gloal spae l p + l q Coordinates in gloal spae q l p ] l l p q 04-90: Matries & Basis 04-91: Matries as Transforms A 33 matri is a transform Transforms a vetor Sine a 3D model is just a series of points, an also transform a model Transforming eah point in the model What does the transformation look like? Can ou look at the matri, and see what the transformation will e? 04-92: Matries as Transforms Let s look at what happens when we multipl the asis vetors 1,0,0],0,1,0] and 0,0,1] an aritrar matri:

23 CS420/ S-04 Intro to 3D Math : Matries as Transforms 04-94: Matries as Transforms ] ] ] m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 = m 11 m 12 m 13 ] = m 21 m 22 m 23 ] = m 31 m 32 m 33 ] Eah row of the matri is a asis vetor after transformation (Or eah olumn of the matri, if we re using olumn vetors) Let s look at an eample in 2D: ] What happens when we transform a vetor (or a 2D polgon) using this matri? Assume row vetors for the moment : Matries as Transforms ] ] = 2 1 ] 1 2 ] ] = 1 2 ] 1 2 ] ] = 1 3 ] : Matries as Transforms

24 CS420/ S-04 Intro to 3D Math : Matries as Transforms 04-98: Matries as Transforms 04-99: Matries as Transforms The matri: ] oth saled and rotated a 2D image It is possile, of ourse for a matri to just sale, or just rotate an image as well

25 CS420/ S-04 Intro to 3D Math : Matries as Transforms : Matries as Transforms : Matries as Transforms Can a matri do something other than sale and rotate? : Matries as Transforms Can a matri do something other than sale and rotate? Yes! What would a matri that did something other than sale or rotate look like? (sta 2D, for the moment) : Matries as Transforms Can a matri do something other than sale and rotate? Yes! What would a matri that did something other than sale or rotate look like? (sta 2D, for the moment) Basis vetors in matri non-orthogonal

26 CS420/ S-04 Intro to 3D Math : Matries as Transforms : Matries as Transforms : Matries as Transforms This translates (reasonal) easil into 3D Instead of strething, rotating, or skewing part of a plane, streth, rotate, or skew a ue No transformation (or identit transformation) : Matries as Transforms This translates (reasonal) easil into 3D Instead of strething, rotating, or skewing part of a plane, streth, rotate, or skew a ue

27 CS420/ S-04 Intro to 3D Math 27 What is this? : Matries as Transforms : Matries as Transforms : Matries as Transforms This translates (reasonal) easil into 3D Instead of strething, rotating, or skewing part of a plane, streth, rotate, or skew a ue Rotation aout they ais, π 2 (90 degrees)