Note 2: Transformation (modeling and viewing)

Transcription

1 Note : Tranformation (modeling and viewing Reading: tetbook chapter 4 (geometric tranformation and chapter 5 (viewing.. Introduction (model tranformation modeling coordinate modeling tranformation world coordinate viewing tranformation viewing coordinate (ee coordinate projection tranformation projection coordinate Thi lecture dicue the modeling iue. With given gl primitive, we have two option to put object on the creen: - hard code the coordinate---man man time (BAD - hard code it once, and ue (intance tranformation to get man ame object. OpenGL provide the necear matrice; we are tring to undertand the inide of OpenGL matrice.. Space - calar pace: field, +. a+(-a, a* a. - vector pace: calar pace with vector, vector-vector addition (+ and calar-vector multiplication (*. - Linear combination: u a*u + + an*un - Linear independent: a*u + + an*un, iff a an. - Dimenion: greatet number of linearl independent vector. - Bai: n linearl independent vector - No location, no ditance, no geometric concept. - Affine pace: vector pace with point (location in pace CS57 Computer Graphic Note : Tranformation

2 - Point-point ubtraction! vector, v P Q. - Frame ( not a coordinate tem: to pecif point and vector: A frame conit of a point P and a et of vector, v,, vn, o that vav+ +anvn (a, an define v and P P + bv + + bnvn (b,, bn define P. Again, no ditance. - Euclidean pace: a vector pace with inner product, u.v v.u, (au+bv.w au.w + bv.w. If u.v, u and v are orthogonal.. Baic working unit - calar a, b: quantit/magnitude without direction (real number - point p, q: location in pace (,, -coordinate - vector u, v: magnitude with direction Right hand rule: we ue thi with -ai pointing out of the creen Homogeneou coordinate: So far, there i no ditinction between point and vector. P(,,, mean Pv+v+v+P and, for vector, v(,, mean vv+v+v. Thu, we add a fourth coordinate. Intead of being repreented b a triple of number (,,, each point i repreented b a 4-tuple (,,, w. Two et of homogeneou coordinate (,,, w and (,,, w repreent the ame point iff one i a multiple of the other (i.e /w /w & /w /w & /w /w. Thu, (,, 4, and (4, 6, 8, are the ame point repreented b different coordinate 4-tuple. Thi new invention allow u to do tranformation b onl matri multiplication. 4. Tranformation Tranformation: a function that take a point (or vector and map that point (or vector into another point (or vector. Affine Tranformation: ha the propert of preerving parallelim of line, but not length and angle. (i tranlation: diplace point b a fied ditance in a given direction P (,, P ( +, +, + P P T(,,. P (ii caling: CS57 Computer Graphic Note : Tranformation

3 CS57 Computer Graphic Note : Tranformation S(,,. P (iii hear - we can contruct an affine tranformation from a equence of rotation, tranlation, and caling - the amount of movement in -direction (for hearing depend on how far it i from the -ai + cot ( H cot (, (, (iv rotation: rotate a point about the origin in a D plane. ( ( (, (,

4 CS57 Computer Graphic Note : Tranformation 4 - note: (a the fied point of the tranformation i the origin; (b rotation doe not change value; o can etend to D ( R R ( R R ( ( ( R R 5. Concatenation of tranformation create new affine tranformation b multipling equence of the baic tranformation. q CBAp q ( (CB A p (C (B Ap C (B (Ap etc. a matri multiplication i aociative. To tranform jut a point, better to do qc(b(ap But to tranform man point, bet to do MCBA, then do qmp We can contruct an affine tranformation from a equence of rotation, tranlation and caling. Eample (i Rotation about a Fied Point p around the -ai b (ii Rotation about an ai along the vector from the origin to ( a, b, c b degree p T( R ( p T( M

5 q p γ (a, b, c β o tan γ c / a + b tan a / b R (... vector tranformed to -plane a oq. Anwer: R ( R ( γ R ( R ( γ R ( (iii Rotation! tranlation and tranlation! Rotation ha different reult. 6. OpenGL tranformation matrice matrie: model, view and projection, hare the ame et of tranformation function. glmatrimode(glenum mode, mode can be GL_MODELVIEW, GL_PROJECTION, GL_TECTURE, or GL_COLOR. glloadidentit(; glloadmatrif(glfloat *Matri /* (4*4 organied b column, i.e. for(i;i<;i++ for (j;j<;j++ Matri[i+4*j] m[i][j]; */ glrotatef(glfloat angle, GLfloat, GLfloat, GLfloat ; gltranlatef(d,d,d; glscalef(,,; OpenGL ue potmultiplication. The tranformation pecified mot recentl i the one applied firt. For eample, MT(pR(v,v,v(thetaT(-p epreed in OpenGL: CS57 Computer Graphic Note : Tranformation 5

6 glmatrimode(gl_modelview; glloadidentit(; gltranlatef(,,; glrotate(theta, v,v,v; gltranlatef(-,-,-; Vertee pecified after thi will be multiplied b M. If ou know eactl the matri, ou can do the ame thing b ug glloadmatrif(glfloat *Matri and glmulmatri(glfloat *Matri. To retore the matri at a certain point, we uuall ue glpuhmatri(; glpopmatri(; 7. Eample Code: Solar Stem Program From: OpenGL programming for Window 95 and Window NT // draw the un glcolorf(.,.,. ; auwiresphere(. ; // draw the Earth // the following 7 line are eplained in the revered order glrotatef( (GLfloat(6.*DaOfYear/65.,.,.,. ; gltranlatef( 4.,.,. ; glpuhmatri(; // ave matri tate for ue b moon glrotatef( (GLfloat(6.*HourOfDa/4.,.,.,. ; glcolorf(.,.,. ; auwiresphere(. ; glpopmatri(; // retore matri tate // draw the moon // at here, the drawing of moon got all tranformation appl to the earth glrotatef( (GLfloat(6.*.5*DaOfYear/65.,.,.,. ; gltranlatef(.5,.,. ; glcolorf(.,.,. ; auwiresphere(.5 ; // fluh the pipeline, wap the buffer glfluh(; auswapbuffer(; CS57 Computer Graphic Note : Tranformation 6

7 8. Introduction (viewing tranformation We need to conider where to view. How to pecif the poition of the camera? either: ( move the world frame from the camera, or ( move the camera from world frame. Viewing tranformation i confug and not ea to undertand. In practice, take great care where primitive are pecified relative to the change in the mode-view matri. 9. Viewing Tranformation Eample (i imple cae Ditance are meaured from viewer to object, NOT from object to viewer. left handed camera frame. CS57 Computer Graphic Note : Tranformation 7

8 viewing along negative -ai viewing at arbitrar (a, b, c (a, b, c (,, T (,, - T( -a, -b, -c (ii Rotate the camera 9 degree along the Y ai. glmatrimode(gl_model_view; glloadidentit(; gltranlatef(.,.,-t; Rotatef(-9.,.,.,.; T(,,-T R(-9(,, (iii General cae The general equence of tranformation i. tranlate the view reference point to the origin of the world-coordinate tem. appl rotation to align the v, v, and v ae with the world w, w, and w ae, repectivel. w v v (a, b, c v T( -a, -b, -c V R w w n U CS57 Computer Graphic Note : Tranformation 8

9 Note that tep on the rotation can be implified to computing unit vector along v, v, v and putting them into R. R u v n u v n u v n u where u u u, v v v v, and n n n n are unit vector. Specifing a Camera Stem VRP: view reference point, VPN: view plane normal (n, VUP: view-up vector. CS57 Computer Graphic Note : Tranformation 9

10 #define M(row,col m[col*4+row] void APIENTRY glulookat( GLdouble ee, GLdouble ee, GLdouble ee, GLdouble center, GLdouble center, GLdouble center, GLdouble up, GLdouble up, GLdouble up { GLdouble m[6]; GLdouble [], [], []; GLdouble mag; /* Make rotation matri */ /* Z vector */ [] ee - center; [] ee - center; [] ee - center; mag qrt( []*[] + []*[] + []*[] ; if (mag { [] / mag; [] / mag; [] / mag; } /* Y vector */ [] up; [] up; [] up; /* X vector Y cro Z */ [][]*[]-[]*[]; []-[]*[]+[]*[]; [][]*[]- []*[]; /* Recompute Y Z cro X */ [][]*[]-[]*[]; [] -[]*[]+[]*[]; [][]*[]-[]*[]; /* cro product give area of parallelogram, which i <. for * non-perpendicular unit-length vector; o normalie, here*/ mag qrt( []*[] + []*[] + []*[] ; if (mag { [] / mag; [] / mag; [] / mag; } mag qrt( []*[] + []*[] + []*[] ; if (mag { [] / mag; [] / mag; [] / mag; } /* et up the required rotation matri */ M(, []; M(, []; M(, []; M(,.; M(, []; M(, []; M(, []; M(,.; M(, []; M(, []; M(, []; M(,.; M(,.; M(,.; M(,.; M(,.; glmultmatrid( m ; } /* Tranlate Ee to Origin */ gltranlated( -ee, -ee, -ee ; CS57 Computer Graphic Note : Tranformation