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

Transcription

1 CS 45: COMPUTER GRAPHICS QUATERNIONS SPRING 6 DR. MICHAEL J. REALE

2 le%3awilliam_roan_hamilton_pot ait_oal_combined.png INTRODUCTION Quatenion inented b Si William Roan Hamilton in 843 Deeloped a etenion to comple numbe Intoduced into compute gaphic b Ken Shoemake in 985 He alo appaentl ote ome pett bad poet in hi pae time Ve good fo otation tanfomation bette than Eule angle becaue: Staightfoad to conet to and fom uatenion Stable and contant intepolation of oientation

3 COMPLEX NUMBERS Comple numbe = pat eal + imagina pat A + ib Imagina numbe i = t-

4 SO WHAT IS A QUATERNION? Quatenion = ha 4 pat Fit 3 pat imagina ecto each component ue a diffeent imagina numbe Lat pat eal pat of uatenion i, j i k j k,, i j k jk kj i ki ik j ij ji k In pactice, toed a a 4D ecto of eal numbe but uuall put little hat on name

5 NOTES ABOUT QUATERNIONS i j k,, Imagina pat can be teated like an odina ecto So e can do ecto addition, dot poduct, co poduct, etc. Multipling the imagina unit i,j,k i NONCOMMUTATIVE ORDER MATTERS! Note: look like hat happen hen ou ue the co poduct e.g., ai ai = ai i j k jk kj ki ik i j ij ji k

6 MULTIPLYING QUATERNIONS Remembe: pat of each uatenion ae added togethe, BUT can t actuall add them becaue of the imagina tem, k j i k j i k j i k ji ij j ik ki i kj jk k j i Recall:

7 RULES OF QUATERNIONS Addition: Jut add imagina and eal pat epaatel Conjugate: Negate imagina pat; leae eal pat alone Nom: Imagina pat cancel out, leaing onl the eal pat Identit:,,,,, ib a ib a Conjugate fo egula imagina numbe: n, i

8 Nom: INVERSE OF QUATERNIONS n The inee mut hae the folloing popet: We kno the folloing to be tue fom the definition of the nom: n n Thu, the inee i gien b: n

9 SCALAR MULTIPLICATION To multipl a cala b a uatenion: Alo, it i commutatie:,,,,,

10 UNIT QUATERNION Unit uatenion, uch that n Anothe a to ite thi i: hee in u u,co in u i a 3D ecto uch that co u Thi ok becaue: n nin u,co in u u co in co onl ok if: u u u

11 EULER S FORMULA Tun out that ine, coine, i, and e ae all elated b Eule Fomula: co iin e i Aide: hen φ = π, get Eule Identit! i e The euialent fo uatenion i: in u co e u We ll ue thi late

12 QUATERNION TRANSFORMS Unit uatenion can epeent ANY 3D otation! Rotate point aound ai u b an angle φ Adantage: in u co Etemel compact and imple A lot eaie to intepolate beteen oientation Coneting to/fom uatenion taightfoad

13 USING QUATERNION TRANSFORMS Aume e hae a point e ant to tanfom a a 4D homogeneou ecto p Put p into a uatenion Imagina ecto = p, eal pat = The folloing otate p aound ai u b an angle φ: p Moeoe, ince e e dealing ith unit uatenion: n So, the final tanfomed point i gien b: p, Recall:,

14 EXAMPLE: ROTATE POINT AROUND Z AXIS Let otate b 9 degee aound the Z ai: Ou uatenion fo the tanfom ill be: in u co in 45,,, co 45.77,,.77,, u,, 45 and the inee ill be the ame a the conjugate, ince it a unit uatenion:,,,, Recall:,,

15 , EXAMPLE: ROTATE POINT AROUND Z AXIS Let a e ant to otate the point 5,,: p 5,,,,.77.77,, Ou tanfomed point ill be: p p So, let the do the fit multipl:,, ,,,,.77 5,, [,3.535, 3.535,, ] [3.535,3.535, ],, ,,.77,,.77 5,,

16 , EXAMPLE: ROTATE POINT AROUND Z AXIS,,.77.77,, p No, let multipl the lat to uatenion: [3.535,3.535, ] [3.535,3.535, ][,,.77.77] [3.535,3.535,,, ,3.535,,, ,3.535,,,.77] [.499,.499,.499,.499, ] [,5, ] So, ou otated point i,5,, hich i hat e epected

17 MULTIPLE ROTATIONS One impotant ule of the uatenion conjugate: So, to appl to otation, and then : p p p cp c What thi i effectiel aing i that: ANY numbe of otation = a SINGLE otation aound the coect ai Thi alo implie that e can think of a uatenion a being a paticula oientation of an object hich mean, if e intepolate beteen to uatenion = intepolating beteen to diffeent oientation!

18 CONVERT TO A MATRIX The folloing gie ou the coeponding otation mati: Whee: Once ou hae the uatenion, no tigonometic function need to be computed! M n

19 CONVERT FROM A MATRIX If i not e mall: m t M m 4 m m m 4 m 4 Otheie, hae to find laget component uing: m m m u t t m m hee t t m t Compute laget uing: m m m t M m m m m m m m m m Compute the et: m m m m m m 4 4 4

20 SPHERICAL LINEAR INTERPOLATION Spheical Linea Intepolation = gien to unit uatenion, and, and a paamete t [,], compute an intepolated uatenion Ueful fo animation Le ueful fo camea oientation can tilt camea up ecto Mathematicall epeed a:,, t t t Poe la fo uatenion: in u in t u co t co t e tu

21 SLERP Softae implementation, lep, often ue thi fom:,, t lep,, t in t in in t in To get φ: co Thee ae, hoee, fate incemental appoache that aoid the tigonometic function

22 INTERPOLATED QUATERNIONS Intepolation path i uniue PROVIDED and ae not eact oppoite Shotet ac on 4D unit phee Contant peed, eo acceleation geodeic cue NOTE: Intepolation 3 i NOT the ame path a 3 een though the detination i the ame Alo note: not mooth tanition fom to 3 Bette to ue a pline-baed intepolation ith uatenion e.g., uad

23 TRANSFORMING FROM ONE VECTOR TO ANOTHER You can alo diectl tanfom fom one DIRECTION to anothe DIRECTION t uing the hotet path poible baicall tun it into a otation aound an ai! Nomalie and t Compute nomalied otation ai uing co poduct: Compute dot poduct: e t co u t / t Get length of co poduct: t in Quatenion e ant: Uing half-angle elation and ome implification: in u, co e t, e

24 TRANSFORMING FROM ONE VECTOR TO ANOTHER The mati fo thi kind of otation can be efficientl computed ithout tigonometic function: e e h t e t h e h h h h e h h h h e t R in co co,

25 TRANSFORMING FROM ONE VECTOR TO ANOTHER NOTE: Need to detect hen and t point in: Eactl ame diection etun identit mati/uatenion Eactl oppoite diection pick an ai of otation