6. 3D Vectors, Geometry and Transformations

Transcription

1 6. 3D Vectors, Geometr ad Trasformatios 6.1 From 2D to 3D 6.2 Vectors ad Matrices i 3D 6.3 The Cross Product (Vector Product) 6.4 Straight Lies, Lie Segmets ad Ras 6.5 The Geometr of Plaes 6.6 More Commo Graphics Problems 6.7 3D Trasformatios 6.8 Trasformatios i OpeGL 6.9 A Virtual Trackball "It's magic, or geometr, or oe of those thigs Terr Pratchett 2008 Burkhard Wuesche Slide From 2D to 3D 3D poits ad vectors are 3-tuples Coordiate space is defied b three orthogoal uit vectors. Covetio: use upper-case letters for poits, e.g. A, Q, bold lower case letters for vectors, e.g. a, q, ad bold upper-case letters for matrices, e.g. M, R. Additio, scalig, subtractio, magitude ad ormalisatio all as for 2D, but with a etra coordiate. A cove combiatio of poits defies a cove polhedro rather tha a polgo. Products of vectors are ver importat i 3D Dot Product (Scalar Product), Cross Product (Vector Product) 2008 Burkhard Wuesche Slide 2 House3D A simple OpeGL program displaig a 3D object What the program does The code Aspects of the code Ol cosider aspects differet from the 2D eample: Represetig the 3D wireframe house 3D Orthographic Projectio Resiig the displa widow Drawig the picture Eercises Chagig the View (GluLookAt) 2008 Burkhard Wuesche Slide 3 What the program does Defies a simple house shape i wireframe form (i.e. made up of just straight lies represetig the edges) i 3-space. Displas a picture of the house usig a 3D orthographic projectio alog the ais A poit (,,) projects to a poit (,) Much more o this later Note that the -ais is UP The Situatio The Program Output 2008 Burkhard Wuesche Slide 4

2 // A basic OpeGL program displaig a 3D house shape #iclude <widows.h> #iclude <gl/gl.h> #iclude <gl/glu.h> #iclude <gl/glut.h> cost it widowwidth=400; cost it widowheight=400; // defie vertices ad edges of the house cost it umvertices=10; cost it umedges=17; cost float vertices[umvertices][3] = 0,0,0,1,0,0,0,1,0,1,1,0,0,0,2, 1,0,2,0,1,2,1,1,2,0.5f,1.5f,0,0.5f,1.5f,2; cost it edges[umedges][2] = 0,1,1,3,3,2,2,0,4,5,5,7,7,6,6,4,0,4, 1,5,3,7,2,6,2,8,8,3,6,9,9,7,8,9; void displa(void) glmatrimode(gl_modelview ); // Set the view matri... glloadidetit(); //... to idetit. glclear(gl_color_buffer_bit); // clear all piels i frame buffer glcolor3f (1.0, 0.0, 0.0); // draw subsequet objects i red glbegi(gl_lines); for(it i=0;i<umedges;i++) glverte3fv(vertices[edges[i][0]]); glverte3fv(vertices[edges[i][1]]); gled(); glflush (); 2008 Burkhard Wuesche Slide 5 void iit(void) // select clearig color (for glclear) glclearcolor (1.0, 1.0, 1.0, 0.0); // RGB-value for white // iitialie view (simple 3D orthographic projectio) glmatrimode(gl_projection); glloadidetit(); glortho(-2,2,-2,2,-3,3); void reshape(it width, it height ) it sie = mi(width, height); glviewport(0, 0, sie, sie); // Called at start, ad wheever user resies compoet // Largest possible square // create a sigle buffered colour widow it mai(it argc, char** argv) glutiit(&argc, argv); glutiitdisplamode(glut_single GLUT_RGB); glutiitwidowsie(widowwidth, widowheight); glutiitwidowpositio(100, 100); glutcreatewidow("house3d"); iit (); // iitialise view glutdisplafuc(displa); // Set fuctio to draw scee glutreshapefuc(reshape); // Set fuctio called if widow is resied glutmailoop(); retur 0; 2008 Burkhard Wuesche Slide 6 Represetig the Wireframe House Have a verte table ad a edge table cost float vertices[umvertices][3] = 0,0,0,1,0,0,0,1,0,1,1,0,0,0,2, 1,0,2,0,1,2,1,1,2,0.5f,1.5f,0,0.5f,1.5f,2; cost it edges[umedges][2] = 0,1,1,3,3,2,2,0,4,5,5,7,7,6,6,4,0,4, 1,5,3,7,2,6,2,8,8,3,6,9,9,7,8,9; Each verte table arra etr is itself a arra of 3 floats, represetig a poit i R 3 [3D-space] Edge table values are idices ito verte table 8 e.g. edge 3,7 is the edge from V 3 (1,1,0) to V (1,1,2) Coordiate sstem is right haded Burkhard Wuesche Slide 7 3D Orthographic Projectio There are tpicall at least four phases to drawig ( rederig ) a scee i OpeGL: (1) Defie required projectio (2) Defie required view (allows ou to to rotate, scale, traslate, etc. the model) (3) Set up scee lightig (4) Output scee primitives (i.e. describe/defie the scee) I this eample we have a simple orthographic projectio [i.e. (,,) (,)], a trivial view ad o lightig. (1) Defie required projectio: glmatrimode(gl_projection); // Iitialise projectio matri... Projectio volume glloadidetit(); //... to the idetit matri [more later]. glortho(-2,2,-2,2,-3,3); // Set orthographic projectio volume [more later] glortho(left, right, bottom, top, ear, far) defies the coordiates of the projectio volume, with ear ad far beig measured from the view poit i the view directio, i.e. the are depths ot -values. View poit (origi) View directio (egative of ais) (-2,-2,-3) Near plae (2,2,-3) Far plae 2008 Burkhard Wuesche Slide 8 (2,2,3)

3 Resiig the Displa Widow The argumet of glutreshapefuc (the fuctio reshape) is called at the start ad wheever the displa widow gets resied Specifies how the scee will be redraw i the resied widow I the previous eamples the viewport was the etire OpeGL widow I this eample, we set the viewport to be the largest square possible. it sie = mi(width, height); The projectio matri maps the scee oto the viewport glviewport(0, 0, sie, sie); The rest (if a) of the widow is uused glviewport parameters are,, width ad height i piel coordiates, with (, ) beig the bottom left corer of the viewport. I OpeGL, coordiates alwas icrease upwards, so (0,0) is the bottom left corer of the widow, ot the top left as is ormal i scree coordiates. Viewport 2008 Burkhard Wuesche Slide 9 Drawig the Picture (2) Defie required view of the model. The two lies glmatrimode(gl_modelview ); glloadidetit(); iitialise the model + view matri to the idetit matri, meaig do t trasform the scee at all. Do t epect to uderstad this just et!! View directio is alog egative ais. (3) Set up scee lightig - ca ot illumiate wireframes (sice there is o surface ormal)! (4) Output scee primitives (as i the 2D eample) glclear(gl_color_buffer_bit); // clear all piels i frame buffer glcolor3f (1.0, 0.0, 0.0); // draw scee i red glbegi(gl_lines); // draw edges as lie segmets (3D vertices) for(it i=0;i<umedges;i++) glverte3fv(vertices[edges[i][0]]); glverte3fv(vertices[edges[i][1]]); gled(); glflush (); 2008 Burkhard Wuesche Slide 10 Eercises Chage the program to use glverte3f everwhere istead of glverte3fv. How could ou cetre the picture i the output widow? [Fid a solutio that ivolves ol adjustig two of the umbers i the program] How could ou icrease the sie of the picture i the output widow? What is the effect of puttig (a) the ear plae, ad (b) the far plae at = 1? What happes if the ear ad far faces of the projectio volume are rectagular rather tha square? 2008 Burkhard Wuesche Slide 11 Chagig the View We ca rotate the house to give a more useful view b chagig step (2) to glmatrimode(gl_modelview ); // Set the view matri.. glloadidetit(); //... to idetit. glrotatef(-40,1,2,-0.3f); // Rotate -40 degrees aroud a ais through the // origi i the directio (1,2,-0.3). Looks vaguel OK, but how ca we determie a suitable ais ad agle, ecept b lots of eperimetatio? Aswers: Either (a) We ca t. Yet. Need some maths! Or (b) Use a GLU fuctio to do it for us Burkhard Wuesche Slide 12

4 GLULookAt Remember: GLU is a package of utilit fuctios o top of GL. Replace start of displa() with: glmatrimode( GL_MODELVIEW ); glloadidetit(); glulookat(-1,2,5, 0.5f,0.75f,1, 0,1,0); Replace start of iit() with: glmatrimode(gl_projection); glloadidetit(); glortho(-2,2,-2,2,0,7); NB: projectio volume coords are w.r.t. camera The commad glulookat( camerax, cameray, cameraz, lookatx, lookaty, lookatz, UpX, UpY, UpZ) specifies where the virtual camera is, where it s poitig to, ad how we re orietig (rotatig) it Burkhard Wuesche Slide 13 GLULookAt (cot d) What's happeig? Camera coordiate sstem (u, v, aes) has origi at camera poit, amel (-1, 2, 5). vector from camera to lookat defies the NEGATIVE ais House image is projected oto viewplae, which is defied to be perpedicular to ais Projectio of up oto viewplae is v ais. Projectio volume is defied b glortho, i camera coordiates viewplae UpX camerax lookatx Let w = UpY, e= cameray lookaty u UpZ cameraz lookatz e w ( w ) the =, v=, ad u= v e w ( w ) projectio volume 2008 Burkhard Wuesche Slide 14 v GLULookAt (cot d) Output of program is: 6.2 Vectors ad Matrices i 3D Vector ad matri operatios as i 2D Determiat det M (also writte M ) of a 33 matri M m m m m m m m m m m m m m m m = m32 m33 m31 m33 m31 m m m m Iverse of a matri: The matri M -1 is called the iverse of M if M -1 M=MM -1 =I Each verte has bee projected i directio oto the viewplae. But to uderstad eactl what s happeig here we eed to uderstad 3D vectors, trasformatios ad other miscellaeous geometr Burkhard Wuesche Slide 15 m11 m12 m i+ j ji M = m21 m22 m23 M = A where aij = ( 1 ) A, 1 i, j 3 m31 m32 m M 33 kl ad A is the 2 2 matri formed b deletig the kth row ad lth colum of M Burkhard Wuesche Slide 16

5 The Dot Product Defiitio The dot product (or ier product, or scalar product) of two 3D vectors v = (v 1,v 2,v 3 ) ad w = (w 1,w 2,w 3 ) is: v w = v 1 w 1 + v 2 w 2 + v 3 w 3. Properties (same as i 2D) Smmetr: a b = b a Liearit: (a+b) c = a c + b c Homogeeit: (sa) b = s(a b) b 2 = b b Eample: Prove a - b 2 = a a -2a b + b b Applicatios (same as i 2D, ecept where 2D Perp Vector is used) Agle betwee two vectors The sig of a b ad perpedicularit Projectig vectors Ra reflectio 2008 Burkhard Wuesche Slide 17 Coordiate Trasformatios The compoet of a vector v i a directio represeted b a uit vector i is the perpedicular projectio of v oto the directio of i. Give b (v i) i (formula from Chapter 5, slide 12) Let P be a poit ad E be the camera locatio give i,,-coordiates The compoets of the poit P epressed i the (u,v,) coordiate sstem are: (r u), (r v) ad (r ) where r = P E 2008 Burkhard Wuesche Slide 18 E v u r P 6.3 The Cross Product Let i, j, ad k be uit vectors alog the, ad aes respectivel. Defie the cross product (or vector product) operator such that: It's a liear operator, i.e. a (b+c) = a b + a c a b = a b si θ where θ is the agle betwee a ad b i rage [0,2π] It's homogeeous, i.e. (sa) b = s (a b) i j = k, j k = i, k i = j j i = -k, k j = -i, i k = -j i i = j j = k k = 0 Ca show from this (UDOO) that if a = a 1 i + a 2 j + a 3 k ad b = b 1 i + b 2 j + b 3 k ab 2 3 ab 3 2 i j k the a b = ab 3 1 ab 1 3 = a1 a2 a3 ab 1 2 ab 2 1 b1 b2 b Burkhard Wuesche Slide 19 Properties of Cross Product Ke properties (UDOO proofs): a b is a vector perpedicular to both a ad b Directio of a b is give b "right-had rule" a b = b a a b is area of parallelogram defied b a ad b Hece area of triagle defied b a ad b is 0.5 * a b 2008 Burkhard Wuesche Slide 20 a b a b a b

6 6.4 Straight Lies, Lie Segmets ad Ras Use a parametric form for lies. Straight lie through two poits P 1 ad P 2 is P( α) = (1 α) P1+ αp2 = P1+ α( P2 P1) = P + α v where v = P 2 P 1 is the displacemet vector from P 1 to P 2. If α costraied to the rage [0,1] we have a lie segmet all poits betwee P 1 ad P 2. If α costraied to the rage [0, ] we have a ra. If α is a real umber, we have a full lie i -space Burkhard Wuesche Slide The Geometr of Plaes The Poit-Normal Form of a Plae Equatio Distace of a Plae from the Origi Distace of a Poit from a Plae Iside-Outside Half-Space Test Itersectio Lie-Plae 2008 Burkhard Wuesche Slide 22 The Poit-Normal Form of a Plae Ca defie a plae b givig oe poit o it, S sa, ad its uit ormal. The for a poit P o the plae, S (P-S) is perpedicular to. s P i.e. (P-S) = 0. p ["Poit-ormal" form of plae equatio] If p ad s are the vectors to P ad S the ca write this as (p - s) = 0 i.e. p = s or p = d where d = s If is (a,b,c) ad p is (,,), the this is the familiar equatio a + b + c =d Burkhard Wuesche Slide 23 O Distace of plae from origi Let Q be a poit o the plae such that q is parallel to. The the legth of q is the "shortest distace" to the plae from the origi. We have the plae equatio p = d for a poit P o the plae. Hece q = d. But sice is parallel to q, q = q. Q Thus q = d. S q Hece, i the equatios p = d ad O a + b + c = d d is the distace to the plae from the origi provided = (a,b,c) is a uit vector. UDOO: How far is the plae = 5 from the origi? 2008 Burkhard Wuesche Slide 24

7 Distace of Poit from Plae How far is a poit Q from the plae p = d? Let P be the earest poit o the plae to Q, Q so that (Q-P) is parallel to. The the required aswer δ is δ δ = Q-P = (q - p) Sice q - p is parallel to, ca write as δ = (q - p) = q - p = q - d = aq 1 + bq 2 + cq 3 - d where = (a,b,c) is the uit ormal ad q = (q 1, q 2, q 3 ) δ is positive if Q is outside the plae, egative if Q is iside. WARNING: Alwas scale plae equatio a+b+c=d so that (a 2 +b 2 +c 2 )= Burkhard Wuesche Slide 25 P Iside-Outside Half-Space Tests (cot d) If plae defied i poit-ormal form, ca b covetio take to be the outward ormal ad poits "o that side" of the plae are said to be "outside", while poits o the other side are "iside". Hece, if S is a poit o the plae ad Q is a poit to be tested: (Q-S) >0 Q is outside (Q-S) =0 Q is o the plae Q (Q-S) <0 Q is iside (see chapter 5, slide 11) 2008 Burkhard Wuesche Slide 26 S Iside-Outside Half-Space Tests NOTE: Above plae equatios assume 3D vectors. If all vectors are 2D, the plae becomes a lie, ad the equatios give the distace of a lie from the origi, the distace of a poit from a lie, ad categorise a 2D poit as iside or outside a lie. Itersectio Lie-Plae Give is a lie p(t) = p 0 + tc ad a plae p=d where c is the lie s directio The lie itersects the plae whe d p0 t = c Q: What happes if c = 0? 2008 Burkhard Wuesche Slide Burkhard Wuesche Slide 28

8 6.6. More Commo Graphics Problems The Area of a Triagle use magitude of a cross product see "Properties of Cross Product" The Robust Normal to a Polgo 2008 Burkhard Wuesche Slide 29 The Normal to a Polgo I priciple, get ormal from the cross product of a two adjacet edge vectors, e.g. = (D-C) (B-C) E But this is o-robust gives erroeous or A urepresetative value whe: B 3 vertices co-liear C 2 adjacet vertices ver close together Magitude of cross product teds to ero ad directio is sesitive to slight movemet i either poit Polgo ot coplaar e.g. (B-A) (E-A), above, ot represetative Warig: I computer graphics, eceptioal coditios occur all the time! 2008 Burkhard Wuesche Slide 30 D A Robust Normal Algorithm A E B C D Polgo ABCDE NOTE: The orietatio of the resultig ormal is such that the vertices are listed i couterclockwise order aroud it. Just add together all the cross products of adjacet edge vectors i.e. (B A) (E A) + (C B) (A B) + (D C) (B C) + (E D) (C D) + (A-E) (D E) Normalise the result. Robust. Short edges or earl co-liear verte triples give egligible cross product cotributio Log earl-perpedicular edges give biggest cotributio 2008 Burkhard Wuesche Slide D Trasformatios Natural etesio of 2D. We will use the homogeeous coordiate form for all trasformatios. To covert betwee ordiar 3D coordiates ad homogeous 3D coordiates: 3D ordiar 3D homogeeous (,, ) T (,,, 1) T 3D homogeeous 3D ordiar (,,,w) T (/w, /w, /w) T 2008 Burkhard Wuesche Slide 32

9 Traslatio The matri for a traslatio b a vector t = (t, t, t ) is: t t T = t Burkhard Wuesche Slide 33 Scalig The matri for a scalig b factors of S, S, S i, ad respectivel is: S S 0 0 S = 0 0 S A egative S gives a reflectio about the = 0 plae. Similarl for egative S or S Burkhard Wuesche Slide 34 Shearig The geeral shear matri is: 1 h h 0 h 1 h 0 H = h h D shearig i its most geeral form is ver rare. Occasioall meet horiotal shearig e.g. a pile of paper pushed to oe side so that the sides of the pile are still straight but ot vertical. UDOO: What would the matri for that look like? Rotatio Rotatios are b far the most cofusig of the trasformatios Most tets cover rotatios aroud the three coordiate aes ad tr to build all other rotatios from those But some cases ver difficult We will cosider three differet rotatio situatios: Rotatio aroud the three coordiate aes Rotatio to alig a object with a ew coordiate sstem Rotatio aroud a arbitrar ais 2008 Burkhard Wuesche Slide Burkhard Wuesche Slide 36

10 Rotatio aroud Coordiate Aes Have three aes to rotate about, so three differet matrices. Let C = cos θ ad S = si θ. The the three matrices for positive (right haded) rotatio are: Rotatio about the -ais: C S 0 R = 0 S C Rotatio about the ais: C 0 S R = S 0 C 0 R : UDOO Note o 3 3 rotatio matrices: Row ad colum correspodig to ais of rotatio are as for idetit I Other elemets are C o diagoal, ± S off diagoal, so that R I if θ 0. Sig of S ca be iferred from the fact that rotatio aroud,, b θ=90 ο trasforms,,, respectivel Burkhard Wuesche Slide 37 Rotatig to Alig with New Coordiate Aes Ofte we have some object ad wat it at a ew positio ad with a ew orietatio Geerall ivolves both rotatio ad traslatio Traslatio trivial -- focus ol o rotatio here Problem: what is the rotatio matri R that rotates a coordiate sstem (,,) to alig with a ew coordiate sstem (a,b,c) with the same origi, where a,b,c are uit vectors alog the ew aes. (0,1,0) (0,0,1) (1,0,0) R*(,,)= 2008 Burkhard Wuesche Slide 38 c b a Rotatig to Alig with New Coordiate Aes (cot'd) To get R: we have R (1 0 0) T = a R (0 1 0) T = b R (0 0 1) T = c Above 3 eqs equivalet to: a b c = a b c a b c R a b c = a b c R a b c or R HC.. a b c 0 a b c 0 = a b c SO IMPORTANT GENERAL RESULT: Colums of a 3 3 rotatio matri are uit vectors alog the rotated coordiate ais directios UDOO derive R, R, R from this rule Burkhard Wuesche Slide 39 Rotatio about a arbitrar ais Ofte, whe buildig a 3D scee or object, eed to rotate a compoet about some arbitrar ais through a referece poit o it. [e.g. forearm of robot rotatig aroud a ais through the elbow]. Ivolves three steps: (1) Traslate referece poit to origi (2) Do the rotatio (3) Traslate referece poit back agai Three approaches for step (2) [et 3 slides]: Tetbook method Decompose rotatio ito primitive rotatios about,, aes Nice eercise, but hard to get right i practice Coordiate sstem aligmet method Geeralised rotatio matri A aside: Quaterios provide a elegat wa of maipulatig (ais, agle) rotatios directl Burkhard Wuesche Slide 40

11 Tetbook Method Rotate the object so that the required ais of rotatio r lies alog the ais [R aligz ] Do the rotatio about ais Udo origial rotatio [R aligz -1 ] How to get R aligz? Measure aimuth, θ, as a right haded rotatio about the ais, startig at the ais. Measure elevatio, ø (or "latitude") as agle above plae =0. R aligz = R (φ) R ( θ) 2008 Burkhard Wuesche Slide 41 φ = ta 1 u u 2 + u 2 θ r φ U = (u, u, u) θ = ata2 (u, u ) i.e. a 4 - quadrat ta 1 u u Coordiate Sstem Aligmet Method Assume we have a coordiate sstem (a,b,c) attached to the object ad wat to rotate it to a ew kow orietatio (a',b',c') Slightl differet problem from previous oe Etesio of "Rotate to alig" problem Solutio: Traslate object to origi Rotate (a,b,c) to alig with world coord aes The iverse of the "rotate to alig" case Rotate coord aes to alig with (a',b',c') Same as "rotate to alig" case Traslate back agai TR R T Full matri is: 1 O abc ' ' ' abc O 2008 Burkhard Wuesche Slide 42 O c b a b' O c' a' The Iverse of a Rotatio Matri [eeded o previous slide] Remember: colums of a rotatio matri are uit vectors alog the rotated coordiate ais directios So colums are orthogoal, i.e dot products = 0 So: a a a a b c b b b a b c = c c c a b c i.e. T RR= I ad hece R 1 T = R So the iverse of a rotatio matri is its traspose (Note: a matri with this propert is called orthogoal.) 2008 Burkhard Wuesche Slide 43 Geeralised Rotatio Matri Matri for a arbitrar rotatio is: 2 t + c t s t + s 2 R= t+ s t + c t s 2 t s t + s t + c where the ais of rotatio (ormalised) is (,,), c ad s are resp. the cosie ad sie of the agle of rotatio, ad t=(1-c). Proof outlie [for ethusiasts ol] see Maillot, Graphics Gems I, P498 for details Rotatio ais, P (poit to rotate) (ormalised) P (= P rotated θ) O (origi) 2008 Burkhard Wuesche Slide 44 H Not eam relevat! θ V (= P rotated 90º)

12 6.8 Trasformatios i OpeGL OpeGL rederig has two 4 4 trasformatio matrices: The Projectio matri, P The Model-View matri, M All vertices (i.e. poits, polgo vertices, etc) are multiplied b M the P before the (,,) (,) projectio is doe Trasformatios i OpeGL (cot d) The P matri hadles perspective projectios (see later) ad scalig from world coordiates to scree coordiates. The M matri hadles both modellig operatios i.e. the trasformatios that are part of the process of specifig a scee, e.g. positioig some geeric chair to a certai poit i the scee) the viewig trasformatio i.e. the rotatio ad traslatio required to allow us to view the scee from somewhere other tha alog the -ais Burkhard Wuesche Slide Burkhard Wuesche Slide 46 Trasformatios i OpeGL (cot d) To set the value of oe of the two matrices: Select the oe of iterest, e.g. glmatrimode( GL_MODELVIEW ) Set it to the idetit, or load it with a specific matri glloadidetit(), or glloadmatrif(cost GLfloat *m ) Multipl it o the right b oe or more primitive matrices, e.g. gltraslatef(glfloat d, GLfloat d, GLfloat d) glscalef(glfloat Factor, GLfloat Factor, GLfloat Factor) glrotatef(glfloat agleidegrees, GLfloat aisx, GLfloat aisy, GLfloat aisz) glmultmatrif(cost GLfloat *m) // geeral purpose matri // Matri is 16 floats, columwise, i.e. m 00, m 10, m 20, m 30,m 01, m 11,... m 33 Note: sice matrices are multiplied o the right the last matri multiplied i is the first to be applied to the vertices Sice (P Q R)v = P (Q (R v)) 2008 Burkhard Wuesche Slide 47 Eample MODEL_VIEW Trasformatios I the displa method of the House3D program... glloadidetit(); glloadidetit(); gltraslatef(-0.5f,-0.5f,-1.0f); glrotatef(90,0,0,1); glloadidetit(); glrotatef(90,0,0,1); glloadidetit(); glrotatef(90,0,0,1); gltraslatef(-0.5f,-0.5f,-1.0f); glloadidetit(); gltraslatef(-0.5f,-0.5f,-1.0f); glloadidetit(); glscalef(2,0.5f,1); 2008 Burkhard Wuesche Slide 48

13 6.9 A Virtual Trackball (from Agel, sectio ) A wa of usig the mouse to rotate the scee Imagie the scee is ecased i a freel rotateable trasparet sphere. Half of sphere is "stickig out" of scree widow Clickig ad draggig the mouse over the widow is like rotatig the sphere to a ew positio. To compute rotatio: 1. Map mouse drag ed-poits (e 1 ad e 2 ) from 2D widow coordiates to 3D coordiates (p 1 ad p 2 ) o surface of virtual sphere. If widow coords p 2 (,) are i rage -1 to +1, ad sphere is uit p 1 sphere, mappig is widow 2 2 ( ) (, ),, 1 2. Compute the rotatio reqd to move p 1 to p 2 [et slide] 2008 Burkhard Wuesche Slide 49 e 1 e 2 A Virtual Trackball (cot'd) Sphere rotatio is about a ais θ 1 = cos v1 v2 = v v v v where v 1 ad v 2 are positio vectors of p 1 ad p 2 [NB: the have uit legth. Wh?] Rotatio agle is 2008 Burkhard Wuesche Slide 50 v 2 v 1 θ Trackball.h class CTrackball public: CTrackball(); Code ot eam relevat! virtual ~CTrackball(); void tbiit(gluit butto); void tbmatri(); void tbreshape(it width, it height); void tbmouse(it butto, it state, it, it ); void tbkeboard(it ke); void tbmotio(it, it ); private: GLuit tb_lasttime; GLfloat tb_lastpositio[3]; GLfloat tb_agle; GLfloat tb_ais[3]; // rotatio ais ad agle GLfloat tb_trasform[4][4]; // curret rotatio matri for GL_MODEL_VIEW GLuit tb_width; GLuit tb_height; // width ad height of widow GLit tb_butto; GLboolea tb_trackig; void _tbpoittovector(it, it, it width, it height, float v[3]); void _tbstartmotio(it, it, it butto, it time); void _tbstopmotio(it butto, usiged time); ; 2008 Burkhard Wuesche Slide 51 #iclude <math.h> #iclude "Trackball.h" #iclude <gl/glut.h> CTrackball::CTrackball() tb_agle = 0.0; tb_ais[0]=0.0;tb_ais[1]=0.0;tb_ais[2]=0.0; tb_trackig = GL_FALSE; CTrackball::~CTrackball() Trackball.cpp void CTrackball::_tbPoitToVector(it, it, it width, it height, float v[3]) float d, a; // project, oto a hemi-sphere cetered withi width, height. v[0] = (float) ((2.0 * - width) / width); v[1] = (float) ((height * ) / height); d = (float) (sqrt(v[0] * v[0] + v[1] * v[1])); v[2] = (float) (cos(( / 2.0) * ((d < 1.0)? d : 1.0))); a = (float) (1.0 / sqrt(v[0] * v[0] + v[1] * v[1] + v[2] * v[2])); v[0] *= a; v[1] *= a; v[2] *= a; 2008 Burkhard Wuesche Slide 52

14 void CTrackball::_tbStartMotio(it, it, it butto, it time) tb_trackig = GL_TRUE; tb_lasttime = time; _tbpoittovector(,, tb_width, tb_height, tb_lastpositio); void CTrackball::_tbStopMotio(it butto, usiged time) tb_trackig = GL_FALSE; tb_agle=0.0; void CTrackball::tbIit(GLuit butto) tb_butto = butto; tb_agle = 0.0; // put the idetit i the trackball trasform for(it i=0;i<4;i++) for(it j=0;j<4;j++) tb_trasform[i][j]=0.0; tb_trasform[i][i]=1.0; Trackball.cpp (cot d) 2008 Burkhard Wuesche Slide 53 Trackball.cpp (cot d) void CTrackball::tbMatri() glpushmatri(); glloadidetit(); glrotatef(tb_agle, tb_ais[0], tb_ais[1], tb_ais[2]); glmultmatrif((glfloat *)tb_trasform); glgetfloatv(gl_modelview_matrix, (GLfloat *)tb_trasform); glpopmatri(); glmultmatrif((glfloat *)tb_trasform); void CTrackball::tbReshape(it width, it height) tb_width = width; tb_height = height; void CTrackball::tbMouse(it butto, it state, it, it ) if (state == GLUT_DOWN && butto == tb_butto) _tbstartmotio(,, butto, glutget(glut_elapsed_time)); else if (state == GLUT_UP && butto == tb_butto) _tbstopmotio(butto, glutget(glut_elapsed_time)); 2008 Burkhard Wuesche Slide 54 Trackball.cpp (cot d) void CTrackball::tbKeboard(it ke) it i,j; for(i=0;i<4;i++) for(j=0;j<4;j++) tb_trasform[i][j]=0.0; tb_trasform[3][3]=1.0; switch (ke) case (it) '': tb_trasform[0][0]=tb_trasform[1][1]=tb_trasform[2][2]=1.0; break; case (it) '': tb_trasform[0][1]=tb_trasform[1][2]=tb_trasform[2][0]=1.0; break; case (it) '': tb_trasform[0][2]=tb_trasform[1][0]=tb_trasform[2][1]=1.0; break; default:; // remember to draw ew positio glutpostredispla(); 2008 Burkhard Wuesche Slide 55 void CTrackball::tbMotio(it, it ) GLfloat curret_positio[3], d, d, d; if (tb_trackig == GL_FALSE) retur; _tbpoittovector(,, tb_width, tb_height, curret_positio); // calculate the agle to rotate b (directl proportioal to the // legth of the mouse movemet d = curret_positio[0] - tb_lastpositio[0]; d = curret_positio[1] - tb_lastpositio[1]; d = curret_positio[2] - tb_lastpositio[2]; tb_agle = (float) (90.0 * sqrt(d * d + d * d + d * d)); Trackball.cpp (cot d) // calculate the ais of rotatio (cross product) tb_ais[0] = tb_lastpositio[1] * curret_positio[2] - tb_lastpositio[2] * curret_positio[1]; tb_ais[1] = tb_lastpositio[2] * curret_positio[0] - tb_lastpositio[0] * curret_positio[2]; tb_ais[2] = tb_lastpositio[0] * curret_positio[1] - tb_lastpositio[1] * curret_positio[0]; // reset for et time tb_lasttime = glutget(glut_elapsed_time); tb_lastpositio[0] = curret_positio[0]; tb_lastpositio[1] = curret_positio[1]; tb_lastpositio[2] = curret_positio[2]; // remember to draw ew positio glutpostredispla(); 2008 Burkhard Wuesche Slide 56

15 #iclude <widows.h> #iclude <gl/gl.h> #iclude <gl/glu.h> #iclude <gl/glut.h> #iclude <iostream> usig amespace std; #iclude "Trackball.h" cost it widowwidth=400; House3DWithTrackball cost it widowheight=400; House3DWithTrackball (cot d) void displa(void) glclear(gl_color_buffer_bit); glcolor3f (1.0, 0.0, 0.0); // clear all piels i frame buffer // (red,gree,blue) colour compoets glmatrimode( GL_MODELVIEW ); // Set the view matri... glloadidetit(); //... to idetit trackball.tbmatri(); // defie vertices ad edges of the house cost it umvertices=10; cost it umedges=18; cost float vertices[umvertices][3] = 0,0,0,1,0,0,0,1,0,1,1,0,0,0,2, 1,0,2,0,1,2,1,1,2,0.5f,1.5f,0,0.5f,1.5f,2; cost it edges[umedges][2] = 0,1,1,3,3,2,2,0,4,5,5,7,7,6,6,4,0,4, 1,5,3,7,2,6,2,8,8,3,6,9,9,7,8,9; CTrackball trackball; // Add a trackball to our OpeGL program void hadlemousemotio(it, it ) trackball.tbmotio(, ); void hadlemouseclick(it butto, it state, it, it ) trackball.tbmouse(butto, state,, ); void hadlekeboardevet(usiged char ke, it, it ) trackball.tbkeboard(ke); 2008 Burkhard Wuesche Slide 57 // Rest is the same as i House3D void iit(void) // Rest of iitialisatio same as i House3D trackball.tbiit(glut_left_button); void reshape(it width, it height ) // Rest of reshape is the same as i House3D trackball.tbreshape(width, height); 2008 Burkhard Wuesche Slide 58 House3DWithTrackball (cot d) it mai(it argc, char** argv) glutiit(&argc, argv); glutiitdisplamode(glut_single GLUT_RGB); glutiitwidowsie(widowwidth, widowheight); glutiitwidowpositio(100, 100); glutcreatewidow("house3d"); iit (); // iitialise view glutmousefuc(hadlemouseclick); // Set fuctio to hadle mouse clicks glutmotiofuc(hadlemousemotio); // Set fuctio to hadle mouse motio glutkeboardfuc(hadlekeboardevet); // Set fuctio to hadle keboard iput Notes o House3DWithTrackball The mai class is almost idetical to House3D ecept: Add a global trackball variable Pass callback fuctios to GLUT for mouse click evets, mouse motio evets ad keboard evets. I more comple programs we have to decide which evets appl to the trackball ad which evets are related to other parts of the program. Iitialise trackball ad specif the associated mouse butto. Update trackball if the widow is reshaped. Add trackball rotatio matri to the MODEL_VIEW matri stack. glutdisplafuc(displa); glutreshapefuc(reshape); glutmailoop(); retur 0; // Set fuctio to draw scee // Set fuctio called if widow gets resied The CTrackball class cotais fuctios for hadlig trackball evets. Mouse positios are trasformed ito rotatios. Trackball accumulates rotatios Use glutpostredispla() to redraw the widow Burkhard Wuesche Slide Burkhard Wuesche Slide 60