MAP. Vectors and Transforms. Reading instructions Quick Repetition of Vector Algebra. In 3D Graphics. Repetition of the Rendering Pipeline

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1 79 MAP Skämtbil om matte å KTHanimationskurs Vectors an Transforms In 3D Grahics Reetition of the Renering Pieline Geometr er verte: Lighting (colors) Screen sace ositions Reetition of the Renering Pieline light Geometr blue re green light Geometr blue re green Rasterier Rasterier: Rasteriation Tetures Interolate colors HARDWARE Alication Geometr Rasterier HARDWARE Alication Geometr Rasterier 3 Verte shaer rogram Piel shaer rogram Disla Ulf Assarsson 7 4 Verte shaer rogram Piel shaer rogram Disla Ulf Assarsson 7 Reaing instructions VERY IMPORTANT READ HOME PAGE in connection to each lecture COURSE HOME PAGE is locate here (an nowhere else): htt:// Quick Reetition of Vector Algebra COURSE HOME PAGE

2 79 Wh transforms? We want to be able to animate objects an the camera Translations Rotations Shears An more We want to be able to use rojection transforms Tomas AkenineMőller How imlement transforms? Matrices! Can ou reall o everthing with a matri? Not everthing, but a lot! We use 33 an 44 matrices m M m m m m m m m m Tomas AkenineMőller Moving an object Demo Ul f Assarsson 7 Transformation Pieline v e r t e object ee cli normalie evice Moelview Matri Moelview Moelview Projection Matri Projection Persective Division Viewort Transform other calculations here material color shae moel (flat) olgon renering moe olgon culling cliing winow OenGL Geometr stage Mainl Ulf Assarsson on GPU 7 Moel to Worl Matri The OenGL Pieline Moel sace Wor sace camera Worl to View Matri MoelViewMt = Moel to View Matri View sace Tomas AkenineMőller From htt://eltronslair.com/glie.html Tomas AkenineMőller

3 79 How o I use transforms racticall? Sa ou have a circle with origin at (,,) an with raius unit circle gltranslatef(8,,); RenerCircle(); gltranslatef(3,,); glscalef(,,); RenerCircle(); Cont from revious slie A simle D eamle A circle in moel sace gltranslatef(3,,); glscalef(,,); gltranslatef(8,,); Tomas AkenineMőller Tomas AkenineMőller Rotation (D) Consier rotation about the origin b q egrees raius stas the same, angle increases b q ' = r cos (f + q) ' = r sin (f + q) '? '? = r cos f = r sin f?? Answer: = cos q sin q = sin q + cos q Derivation of rotation matri in D re n e if i (, ) re n ( n, n ) i r(cos f + i sinf) [rotation is mult b e ] T T n R? e i if r[(cos + i sin )(cosf + i sinf)] r(cos cosf sin sinf) + n ir(cos sin f + sin cosf) ( r cosf, r sinf) ( r(cos cosf sin sinf), r(cos sinf + sin cosf)) T T Tomas AkenineMőller Derivation D rotation, cont n n (, ) n ( n, n ) r(cos sin f + sin cosf)) n R T T ( r cosf, r sin f) cos sin sin cos R ( r(cos cosf sin sin f), what is R? T T Tomas AkenineMőller Rotations in 3D Same as in D for Zrotations, but with a 33 matri cos sin cos sin R ( ) R ( ) sin cos sin cos For X For Y R ( ) cos sin sin cos cos sin R ( ) sin cos Tomas AkenineMőller 3

4 79 Translations must be simle? Translation Rotation?????? + t R n??? Rotation is matri mult, translation is a Woul be nice if we coul onl use matri multilications Turn to homogeneous coorinates A a new comonent to each vector Tomas AkenineMőller Homogeneous notation A oint: Translation becomes: t + t t + t t + t T( t) A vector (irection): Translation of vector: T Also allows for rojections (later) ) T ) T Tomas AkenineMőller Rotations in 44 form Just a a row at the bottom, an a column at the right: R cos sin sin cos ( ) Similarl for X an Y et( R )= (for 33 matrices) Trace( R )=+cos(alha) (for an ais,33) Tomas AkenineMőller M Change of Frames M moeltoworl : a o a a a b b b c c c moeltoworl o o o worl sace E.g.: worl = M mw moel = M mw (,5,) T = 5 b (+ o) c (,5,) b moel sace More basic transforms Scaling Normal transforms Not so normal Shear Rigibo: rotation then translation X TR Concatenation of matrices Not commutative, i.e., RT TR In X TR, the rotation is one first Inverses an rotation about arbitrar ais: Rigi bo: X = X T Tomas AkenineMőller Cannot use same matri to transform normals Use: N M ) T instea of M M works for rotations an translations, though Tomas AkenineMőller 4

5 79 The Euler Transform Assume the view looks own the negative ais, with u in the irection, to the right E( h,, r) R ( r) R ( ) R ( h) h=hea =itch r=roll Gimbal lock can occur looses one egree of freeom Eamle: h=,=/, then the rotation is the Quaternions qˆ ( q, q ) ( q, q, q, q ) iq + jq + kq + qw Etension of imaginar numbers Avois gimbal lock that the Euler coul rouce Focus on unit quaternions: n(ˆ) q q + q + q + qw A unit quaternion is: qˆ (sinfu q,cos f) where u same as oing a revious rot aroun Tomas AkenineMőller ais Tomas AkenineMőller v w w q Unit quaternions are erfect for rotations! Comact (4 comonents) Can show that qq ˆ ˆ ˆ reresents a rotation of f raians aroun uq of qˆ (sin u,cos f) f q Projections Orthogonal (arallel) an Persective That is: a unit quaternion reresent a rotation as a rotation ais an an angle OenGL: glrotatef(u,u,u,angle); Interolation from one quaternion to another is much simler, an gives otimal results Tomas AkenineMőller Tomas AkenineMőller Orthogonal rojection Simle, just ski one coorinate Sa, we re looking along the ais Then ro, an rener M ortho M ortho image lane Orthogonal rojection Not invertible! (eterminant is ero) For Zbuffering It is not sufficient to roject to a lane Rather, we nee to roject to a bo near far Unit cube: [,,] to [,,] Tomas AkenineMőller ee Unit cube is also use for ersective roj. Simlifies cliing Tomas AkenineMőller 5

6 79 6 Tomas AkenineMőller Orthogonal rojection The unitcube rojection is invertible Simle to erive Just a translation an scale left right bottom to near far Tomas AkenineMőller What about those homogenenous coorinates? ) T w w= for vectors, an w= for oints What if w is not or? Solution is to ivie all comonents b w ) T w w w / / / Gives a oint again! Can be use for rojections, as we will see Tomas AkenineMőller Persective rojection q q q For : > / P Tomas AkenineMőller Persective rojection The arrow is the homogeniation rocess / P? P / P / / / / / / q q q Tomas AkenineMőller Persective rojection Again, the eterminant is (not invertible) To make the rest of the ieline the same as for orhogonal rojection: roject into unitcube Not much ifferent from P Do not collase coor to a lane Tomas AkenineMőller Unerstaning the rojection matri Scaling Skew Kee info / c s b s a s P c s b s a s / ) / ( ' ) / ( ' ) / ( 3 c b a q

7 79 Persective rojection matrices See Från Värl till Skärm secion 4 for more etails. BREAK... Quick Reetition of Vector Algebra Length of vector: Normaliing a vector: ˆ Normal: ) ) n v v v v (usual normalie as well) + + ) + + ) Tomas AkenineMőller Cross Prouct: Perenicular vector Area va vb sin α: sin v v va vb Dot rouct: cos v v a a b b Ra/Plane Intersections Ra: r(t)=o+t Plane: n + = ; =n Set =r(t): n (o+t) = n o+t(n ) = t = ( n o) / (n ) Put ifferentl: Line/Line intersection in D r (s) = o +s r (t) = o +t r (s) = r (t) () o +s = o +t () noting that =, [=(a,b) =(b,a)] ( o o ) s n ( o ) s = (o o ) ( ) a t n ( ) = (o o ) ( o o) t Note: for t to be the true istance, ( ) must be of unit length. Ulf Assarsson 6 Ulf Assarsson 6 Line/Line intersection in 3D r (s) = o +s r (t) = o +t r (s) = r (t) () o +s = o +t () noting that = s = (o o ) t = (o o ) s ( ) ( ) = ((o o ) ) ( ) t ( ) ( ) = ((o o ) ) ( ) et( o o,, ) ( ) = means arallel lines et( o o,, ) s t ( ) ( ) Ulf Assarsson 6 s, t correson to closest oints Area an Perimeter For olgon,... n Perimeter = omkrets = sum of length of each ege in D an 3D: O n i A Area in D: n i+ i i n i i i+ i ) + i+ i ) + i+ i ) i+ i+ i v v We can unerstan the formula from using Greens theorem: integrating over borer to get area Choose arbitrar oint to integrate from, e.g. Origin (,,) Works for nonconve olgons as well A triangle v v ) Ulf Assarsson 6 7

8 79 Volume in 3D The same trick for comuting area in D can be use to easil comute the volume in 3D for triangulate objects Again, choose arbitrar ointofintegration, e.g. Origin (,,) With resect to ointofintegration For all backfacing triangles, a volume For all frontfacing triangles, subtract volume Works for nonconve olgons as well V tetrahero n V object 3! a 3! n i a 3! bc) et( a, b, c) bc) where a = origin b = origin c = 3 origin The sign of the eterminant will automaticall hanle ositive an negative contribution Ulf Assarsson 6 Scan Conversion of Line Segments Start with line segment in winow coorinates with integer values for enoints Assume imlementation has a write_iel function k = k + m DDA Algorithm Digital Differential Analer DDA was a mechanical evice for numerical solution of ifferential equations Line =k+ m satisfies ifferential equation / = k = / = / Along scan line = =; For(=; <=,i++) { write_iel(, roun(), line_color) +=k; } Problem DDA = for each lot iel at closest Problems for stee lines Using Smmetr Use for k For k >, swa role of an For each, lot closest The roblem with DDA is that it uses floats which was slow in the ol as Bresenhams algorithm onl uses integers 8

9 79 Bresenham s line rawing algorithm The line is rawn between two oints (, ) an (, ) ( Sloe ) k ( = k + m) ( ) Each time we ste in irection, we shoul increment with k. Otherwise the error in increases with k. If the error surasses.5, the line has become closer to the net value, so we a to simultaneousl ecreasing the error b function line(,,, ) int elta := abs( ) int elta := abs( ) real error := real eltaerr := elta / elta int := for from to lot(,) error := error + eltaerr if error.5 := + error := error. See also htt://en.wikieia.org/wiki/bresenham's_line_algorithm Ulf Assarsson 6 Bresenham s line rawing algorithm Now, convert algorithm to onl using integer comutations The trick we use is to multil all the fractional numbers above b (, ), which enables us to eress them as integers. The onl roblem remaining is the constant.5 to eal with this, we multil both sies of the inequalit b Ol float version: function line(,,, ) int elta := abs( ) int elta := abs( ) real error := real eltaerr := elta / elta int := for from to lot(,) error := error + eltaerr if error.5 := + error := error. New integer version: function line(,,, ) int elta := abs( ) int elta := abs( ) real error := real eltaerr := elta int := for from to lot(,) error := error + eltaerr if *error elta := + error := error elta Ulf Assarsson 6 Comlete Bresenham s line rawing algorithm function line(,,, ) boolean stee := abs( ) > abs( ) if stee then swa(, ) swa(, ) if > then swa(, ) swa(, ) int elta := int elta := abs( ) int error := int ste int := if < then ste := else ste := for from to if stee then lot(,) else lot(,) error := error + elta if error elta := + ste error := error elta The first case is allowing us to raw lines that still sloe ownwars, but hea in the oosite irection. I.e., swaing the initial oints if >. To raw lines that go u, we check if >= ; if so, we ste b instea of. To be able to raw lines with a sloe less than one, we take avantage of the fact that a stee line can be reflecte across the line = to obtain a line with a small sloe. The effect is to switch the an variables. Ulf Assarsson 6 ALL FOLLOWNG SLIDES ARE NOT PART OF THE COURSE, BUT BONUSSLIDES PROVIDING DIFFERENT EXPLANATIONS IF THE PREVIOUS SLIDES ARE NOT ENOUGH. Most of the following slies are from E Angel Professor of Comuter Science, Electrical an Comuter Engineering, an Meia Arts Universit of New Meico Scalars Nee three basic elements in geometr Scalars, Vectors, Points Scalars can be efine as members of sets which can be combine b two oerations (aition an multilication) obeing some funamental aioms (associativit, commutivit, inverses) Eamles inclue the real an comle number sstems uner the orinar rules with which we are familiar Scalars alone have no geometric roerties Vector Oerations Phsical efinition: a vector is a quantit with two attributes Direction Magnitue Eamles inclue Force Velocit Directe line segments Most imortant eamle for grahics Can ma to other tes Ever vector can be multilie b a scalar There is a ero vector Zero magnitue, unefine orientation The sum of an two vectors is a vector v v v v u w 9

10 79 Vectors Lack Position These vectors are ientical Same length an magnitue Vectors insufficient for geometr Nee oints Points Location in sace Oerations allowe between oints an vectors Pointoint subtraction iels a vector Equivalent to ointvector aition v=pq P=v+Q Affine Saces Point + a vector sace Oerations Vectorvector aition Scalarvector multilication Pointvector aition Scalarscalar oerations For an oint efine P = P P = (ero vector) Lines Consier all oints of the form P()=P + Set of all oints that ass through P in the irection of the vector Parametric Form This form is known as the arametric form of the line More robust an general than other forms Etens to curves an surfaces Twoimensional forms Elicit: = k + m Imlicit: a + b +c = Parametric: () = + () () = + () Ras an Line Segments If >=, then P() is the ra leaving P in the irection If we use two oints to efine v, then P( ) = Q + (RQ)=Q+v =R + ()Q For <=<= we get all the oints on the line segment joining R an Q

11 79 Conveit An object is conve iff for an two oints in the object all oints on the line segment between these oints are also in the object Q P conve P Q not conve Affine Sums Consier the sum P= P + P +..+ n P n Can show b inuction that this sum makes sense iff n = in which case we have the affine sum of the oints P,P,..P n If, in aition, i >=, we have the conve hull of P,P,..P n Conve Hull Consier the linear combination P= P + P +..+ n P n If n = (in which case we have the affine sum of the oints P,P,..P n ) an if i >=, we have the conve hull of P,P,..P n Smallest conve object containing P,P,..P n Planes A lane can be efine b a oint an two vectors or b three oints v Q R u R P(,b)=R+u+bv P(,b)=R+(QR)+b(PQ) P conve sum of P an Q Triangles conve sum of S() an R Normals Ever lane has a vector n normal (erenicular, orthogonal) to it From oint/vector form P(,b)=R+u+bv we know we can use the cross rouct to fin n = u v Plane equation: n =, where = n an is an oint in the lane for <=,b<=, we get all oints in triangle P v u

12 79 Normal for Triangle Frames lane n ( ) = n = ( ) ( ) normalie n n/ n n A coorinate sstem is insufficient to reresent oints If we work in an affine sace we can a a single oint, the origin, to the basis vectors to form a frame Note that righthan rule etermines outwar face v P v v 3 Reresenting one basis in terms of another Each of the basis vectors, u,u, u3, are vectors that can be reresente in terms of the first basis u = g v +g v +g 3 v 3 u = g v +g v +g 3 v 3 u 3 = g 3 v +g 3 v +g 33 v 3 v Matri Form The coefficients efine a 3 3 matri g M = g g an the bases can be relate b a=m T b g g g g g g 33 Translation Move (translate, islace) a oint to a new P location P How man was? Although we can move a oint to a new location in infinite was, when we move man oints there is usuall onl one wa Dislacement etermine b a vector Three egrees of freeom P =P+ object translation: ever oint islace b same vector

13 79 Translation Using Reresentations Using the homogeneous coorinate reresentation in some frame =[ ] T =[ ] T =[ ] T Hence = + or =+ =+ =+ note that this eression is in four imensions an eresses oint = vector + oint Translation Matri We can also eress translation using a 4 4 matri T in homogeneous coorinates =T where T = T(,, ) = This form is better for imlementation because all affine transformations can be eresse this wa an multile transformations can be concatenate together Homogeneous Coorinates The homogeneous coorinates form for a three imensional oint [ ] is given as =[ w] T =[w w w w] T We return to a three imensional oint (for w) b /w /w /w If w=, the reresentation is that of a vector Note that homogeneous coorinates relaces oints in three imensions b lines through the origin in four imensions For w=, the reresentation of a oint is [ ] Homogeneous Coorinates an Comuter Grahics Homogeneous coorinates are ke to all comuter grahics sstems All stanar transformations (rotation, translation, scaling) can be imlemente with matri multilications using 4 4 matrices Harware ieline works with 4 imensional reresentations For orthograhic viewing, we can maintain w= for vectors an w= for oints For ersective we nee a ersective ivision Rotation about the ais Rotation about ais in three imensions leaves all oints with the same Equivalent to rotation in two imensions in lanes of constant = cos q sin q = sin q + cos q = or in homogeneous coorinates =R (q) Rotation Matri cos q sin q R = R (q) = sin q cos q 3

14 79 Rotation about an aes Same argument as for rotation about ais For rotation about ais, is unchange For rotation about ais, is unchange R = R (q) = cos q R = R (q) = sin q cos q sin q sin q cos q sin q cos q s S = S(s, s, s ) = Scaling Ean or contract along each ais (fie oint of origin) =s =s =s =S s s Reflection corresons to negative scale factors s = s = original s = s = s = s = Inverses Although we coul comute inverse matrices b general formulas, we can use simle geometric observations Translation: T (,, ) = T(,, ) Rotation: R (q) = R(q) Hols for an rotation matri Note that since cos(q) = cos(q) an sin( q)=sin(q) R (q) = R T (q) Scaling: S (s, s, s ) = S(/s, /s, /s ) Concatenation Orer of Transformations We can form arbitrar affine transformation matrices b multiling together rotation, translation, an scaling matrices Because the same transformation is alie to man vertices, the cost of forming a matri M=ABCD is not significant comare to the cost of comuting M for man vertices The ifficult art is how to form a esire transformation from the secifications in the alication Note that matri on the right is the first alie Mathematicall, the following are equivalent = ABC = A(B(C)) Note man references use column matrices to reresent oints. In terms of column matrices T = T C T B T A T 4

15 79 General Rotation About the Origin Rotation About a Fie Point other than the Origin A rotation b q about an arbitrar ais can be ecomose into the concatenation of rotations about the,, an aes R(q) = R (q ) R (q ) R (q ) q q q are calle the Euler angles q v Move fie oint to origin Rotate Move fie oint back M = T( f ) R(q) T( f ) Note that rotations o not commute We can use rotations in another orer but with ifferent angles Instancing In moeling, we often start with a simle object centere at the origin, oriente with the ais, an at a stanar sie We al an instance transformation to its vertices to Scale Orient Locate Shear Helful to a one more basic transformation Equivalent to ulling faces in oosite irections Shear Matri Consier simle shear along ais = + cot q = = OenGL Transformations H(q) = cot q 5

16 79 Objectives Learn how to carr out transformations in OenGL Rotation Translation Scaling Introuce OenGL matri moes Moelview Projection OenGL Matrices In OenGL matrices are art of the state Multile tes MoelView (GL_MODELVIEW) Projection (GL_PROJECTION) Teture (GL_TEXTURE) (ignore for now) Color(GL_COLOR) (ignore for now) Single set of functions for maniulation Select which to maniulate b glmatrimoe(gl_modelview); glmatrimoe(gl_projection); Current Transformation Matri (CTM) Concetuall there is a 4 4 homogeneous coorinate matri, the current transformation matri (CTM) that is art of the state an is alie to all vertices that ass own the ieline The CTM is efine in the user rogram an loae into a transformation unit C =C vertices CTM vertices CTM oerations The CTM can be altere either b loaing a new CTM or b ostmutilication Loa an ientit matri: C I Loa an arbitrar matri: C M Loa a translation matri: C T Loa a rotation matri: C R Loa a scaling matri: C S Postmultil b an arbitrar matri: C CM Postmultil b a translation matri: C CT Postmultil b a rotation matri: C C R Postmultil b a scaling matri: C C S Rotation about a Fie Point Start with ientit matri: C I Move fie oint to origin: C CT Rotate: C CR Move fie oint back: C CT Result: C = TR T which is backwars. This result is a consequence of oing ostmultilications. Let s tr again. Reversing the Orer We want C = T R T so we must o the oerations in the following orer C I C CT C CR C CT Each oeration corresons to one function call in the rogram. Note that the last oeration secifie is the first eecute in the rogram 6

17 79 CTM in OenGL OenGL has a moelview an a rojection matri in the ieline which are concatenate together to form the CTM Can maniulate each b first setting the correct matri moe Rotation, Translation, Scaling Loa an ientit matri: glloaientit() Multil on right: glrotatef(theta, v, v, v) theta in egrees, (v, v, v) efine ais of rotation gltranslatef(,, ) glscalef( s, s, s) Each has a float (f) an ouble () format (glscale) Eamle Rotation about ais b 3 egrees with a fie oint of (.,., 3.) glmatrimoe(gl_modelview); glloaientit(); gltranslatef(.,., 3.); glrotatef(3.,.,.,.); gltranslatef(.,., 3.); Remember that last matri secifie in the rogram is the first alie Arbitrar Matrices Can loa an multil b matrices efine in the alication rogram glloamatrif(m) glmultmatrif(m) The matri m is a one imension arra of 6 elements which are the comonents of the esire 4 4 matri store b columns In glmultmatrif, m multilies the eisting matri on the right Matri Stacks In man situations we want to save transformation matrices for use later Traversing hierarchical ata structures (Chater ) Avoiing state changes when eecuting isla lists OenGL maintains stacks for each te of matri Access resent te (as set b glmatrimoe) b glpushmatri() glpomatri() Reaing Back Matrices Can also access matrices (an other arts of the state) b quer functions glgetintegerv glgetfloatv glgetbooleanv glgetdoublev glisenable For matrices, we use as ouble m[6]; glgetfloatv(gl_modelview, m); 7

18 79 Using the Moelview Matri In OenGL the moelview matri is use to Position the camera Can be one b rotations an translations but is often easier to use glulookat Buil moels of objects The rojection matri is use to efine the view volume an to select a camera lens Quaternions Etension of imaginar numbers from two to three imensions Requires one real an three imaginar comonents i, j, k q=q +q i+q j+q 3 k Quaternions can eress rotations on shere smoothl an efficientl. Process: Moelview matri quaternion Carr out oerations with quaternions Quaternion Moelview matri Objectives Comuter Viewing E Angel Professor of Comuter Science, Electrical an Comuter Engineering, an Meia Arts Universit of New Meico Introuce the mathematics of rojection Introuce OenGL viewing functions Look at alternate viewing APIs Comuter Viewing There are three asects of the viewing rocess, all of which are imlemente in the ieline, Positioning the camera Setting the moelview matri Selecting a lens Setting the rojection matri Cliing Setting the view volume (efault is unit cube, R 3, [,]) Default Projection Default rojection is orthogonal clie out = 8

19 79 Moving the Camera Frame If we want to visualie object with both ositive an negative values we can either Move the camera in the ositive irection Translate the camera frame Move the objects in the negative irection Translate the worl frame Both of these views are equivalent an are etermine b the moelview matri Want a translation (gltranslatef(.,.,);) > Moving the Camera We can move the camera to an esire osition b a sequence of rotations an translations Eamle: sie view Rotate the camera Move it awa from origin Moelview matri C = TR OenGL coe Remember that last transformation secifie is first to be alie glmatrimoe(gl_modelview) glloaientit(); gltranslatef(.,., ); glrotatef(9.,.,.,.); The LookAt Function The GLU librar contains the function glulookat to form the require moelview matri through a simle interface Note the nee for setting an u irection Still nee to initialie Can concatenate with moeling transformations Eamle: isometric view of cube aligne with aes glmatrimoe(gl_modelview): glloaientit(); glulookat(.,.,.,.,.,.,.,...); glulookat gllookat(ee, ee, ee, at, at, at, u, u, u) Other Viewing APIs The LookAt function is onl one ossible API for ositioning the camera Others inclue View reference oint, view lane normal, view u (PHIGS, GKS3D) Yaw, itch, roll Elevation, aimuth, twist Direction angles 9

20 79 OenGL Orthogonal Viewing OenGL Persective glortho(left,right,bottom,to,near,far) glfrustum(left,right,bottom,to,near,far ) near an far measure from camera Using Fiel of View With glfrustum it is often ifficult to get the esire view gluperective(fov, asect, near, far) often rovies a better interface front lane asect = w/h Projections elaine ifferentl Rea the following slies about orthogonal an ersective rojections b our selves The resent the same thing, but elaine ifferentl Projections an Normaliation The efault rojection in the ee (camera) frame is orthogonal For oints within the efault view volume = = = Most grahics sstems use view normaliation All other views are converte to the efault view b transformations that etermine the rojection matri Allows use of the same ieline for all views Homogeneous Coorinate Reresentation = = = w = efault orthograhic rojection M = = M In ractice, we can let M = I an set the term to ero later

21 79 Simle Persective Center of rojection at the origin Projection lane =, < Persective Equations Consier to an sie views / = / = / = Homogeneous Coorinate Form M = consier q = M where q = = / / Persective Division However w, so we must ivie b w to return from homogeneous coorinates This ersective ivision iels = / = / = the esire ersective equations We will consier the corresoning cliing volume with the OenGL functions Normaliation Rather than erive a ifferent rojection matri for each te of rojection, we can convert all rojections to orthogonal rojections with the efault view volume This strateg allows us to use stanar transformations in the ieline an makes for efficient cliing moelview transformation nonsingular Pieline View cliing against efault cube rojection transformation rojection 3D D ersective ivision 4D 3D

22 79 Notes We sta in fourimensional homogeneous coorinates through both the moelview an rojection transformations Both these transformations are nonsingular Default to ientit matrices (orthogonal view) Normaliation lets us cli against simle cube regarless of te of rojection Dela final rojection until en Imortant for hiensurface removal to retain eth information as long as ossible Orthogonal Normaliation glortho(left,right,bottom,to,near,far) normaliation fin transformation to convert secifie cliing volume to efault Orthogonal Matri Two stes Move center to origin T((left+right)/, (bottom+to)/,(near+far)/)) Scale to have sies of length S(/(leftright),/(tobottom),/(nearfar)) right left P = ST = to bottom near far right left right left to + bottom to bottom far + near far near Final Projection Set = Equivalent to the homogeneous coorinate transformation M orth = Hence, general orthogonal rojection in 4D is P = M orth ST General Shear Shear Matri shear ( values unchange) to view sie view H(q,f) = Projection matri General case: cot θ cot φ P = M orth H(q,f) P = M orth STH(q,f)

23 79 Effect on Cliing The rojection matri P = STH transforms the original cliing volume to the efault cliing volume object to view = Simle Persective Consier a simle ersective with the COP at the origin, the near cliing lane at =, an a 9 egree fiel of view etermine b the lanes =, = DOP cliing volume near lane far lane = = DOP = istorte object (rojects correctl) Persective Matrices Simle rojection matri in homogeneous coorinates M = Note that this matri is ineenent of the far cliing lane Generaliation N = α β after ersective ivision, the oint (,,, ) goes to = / = / Z = (+b/) which rojects orthogonall to the esire oint regarless of an b If we ick near + far = far near near far b = near far Picking an b Normaliation Transformation istorte object rojects correctl the near lane is mae to = the far lane is mae to = an the sies are mae to =, = Hence the new cliing volume is the efault cliing volume original cliing volume original object new cliing volume 3

24 79 Normaliation an HienSurface Removal Although our selection of the form of the ersective matrices ma aear somewhat arbitrar, it was chosen so that if > in the original cliing volume then the for the transforme oints > Thus hien surface removal works if we first al the normaliation transformation However, the formula = (+b/) imlies that the istances are istorte b the normaliation which can cause numerical roblems eseciall if the near istance is small OenGL Persective glfrustum allows for an unsmmetric viewing frustum (although glupersective oes not) OenGL Persective Matri The normaliation in glfrustum requires an initial shear to form a right viewing rami, followe b a scaling to get the normalie ersective volume. Finall, the ersective matri results in neeing onl a final orthogonal transformation P = NSH our reviousl efine ersective matri shear an scale Wh o we o it this wa? Normaliation allows for a single ieline for both ersective an orthogonal viewing We sta in four imensional homogeneous coorinates as long as ossible to retain threeimensional information neee for hiensurface removal an shaing We simlif cliing 4