How To Create A Two-Dimenional Tranformation In C++ (For Windows)
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1 Computer Graphic with OpenGL 3e 5 Pearon Education
2 Chapter 5 Geometric Tranformation 5 Pearon Education
3 Baic two-dimenional geometric tranformation () Two-Dimenional tranlation One of rigid-bod tranformation, which move object without deformation Tranlate an object b Adding offet to coordinate to generate new coordinate poition Set t,t be the tranlation ditance, we have ' + t ' + t t In matri format, where T i the tranlation vector ' P' ' P t T t P ' P + T Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
4 Baic two-dimenional geometric tranformation () Two-Dimenional rotation Rotation ai and angle are pecified for rotation Convert coordinate into polar form for calculation r coφ inφ Eample, to rotation an object with angle a The new poition coordinate ' r co( φ + a) r co φ coa r in φ ina ' r co( φ + a) r coφ coa r inφ ina In matri format coa ina R ina coa P' R P Rotation about a point ( r, r ) ' ' r r + ( + ( r r )coa ( )ina + ( r r )ina )coa coa ina ina + coa Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
5 Baic two-dimenional geometric tranformation (3) Two-Dimenional caling To alter the ize of an object b multipling the coordinate with caling factor and In matri format, where S i a b caling matri ' ' P S ' P Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon 5 Pearon Education Education Chooing a fi point ( f, f ) a it centroid to perform caling ' P S ' P ) ( ' ) ( ' f f + +
6 Homogenou Coordinate Combine the geometric tranformation into a ingle matri with 3b3 matrice Epand each D coordinate to 3D coordinate with homogenou parameter Two-Dimenional tranlation matri t t ' ' Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon 5 Pearon Education Education Two-Dimenional rotation matri Two-Dimenional caling matri co in co co ' ' θ θ θ θ ' '
7 Invere tranformation Invere tranlation matri Two-Dimenional tranlation matri t t T θ θ Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon 5 Pearon Education Education Two-Dimenional tranlation matri co in in co R θ θ θ θ S
8 Matri Concatenation Propertie Aociative propertie M M M3 M (M M3 ) (M M) M3 Tranformation i not commutative (CopCD!) Order of tranformation ma affect tranformation poition Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
9 Two-dimenional compoite tranformation () Compoite tranformation A equence of tranformation Calculate compoite tranformation matri rather than appling individual tranformation ' M M P P P' M P Compoite two-dimenional tranlation Appl two ucceive tranlation, T and T T T(t,t ) T P' T T(t (T,t ) P) (T T ) P Compoite tranformation matri in coordinate form T(t,t ) T(t,t ) T(t + t,t t ) + Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
10 Two-dimenional compoite tranformation () Compoite two-dimenional rotation Two ucceive rotation, R and R into a point P P' R( θ ) {R( θ ) P} P' {R( θ ) R( θ )} P Multipl two rotation matrice to get compoite tranformation matri R( θ ) R( θ ) R( θ + θ ) Compoite two-dimenional caling S( P' P' R( θ + θ ) P, S( ) S(,, ) S( ) P, ) Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
11 Two-dimenional compoite tranformation (3) General two-dimenional Pivot-point rotation Graphic package provide onl origin rotation Perform a tranlate-rotate-tranlate equence Tranlate the object to move pivot-point poition to origin Rotate the object Tranlate the object back to the original poition Compoite matri in coordinate form T(, ) R( θ ) T(, ) R(,, θ ) r r r r r r Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
12 Two-dimenional compoite tranformation (4) Eample of pivot-point rotation Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
13 Two-dimenional compoite tranformation (5) General two-dimenional Fied-point caling Perform a tranlate-caling-tranlate equence Tranlate the object to move fied-point poition to origin Rotate the object Ue invere of tranlation in tep to return the object back to the original poition Compoite matri in coordinate form T(, ) S(, ) T(, ) S(,, f f f f f f, ) Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
14 Two-dimenional compoite tranformation (6) Eample of fied-point caling Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
15 Two-dimenional compoite tranformation (7) General two-dimenional caling direction Perform a rotate-caling-rotate equence Compoite matri in coordinate form R ( θ ) S(, ) R( θ ) co θ + in θ ( )co θ in θ ( )coθ inθ in θ + co θ θ Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
16 Other two-dimenional tranformation () Reflection image generated relative to an ai of reflection Reflection about the -ai Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon 5 Pearon Education Education Reflection about the -ai Reflection about the z-ai
17 Other two-dimenional tranformation () Reflection with ai (diagonal) Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
18 Other two-dimenional tranformation (3) Shear Ditort the hape of an object Shape appear a if the object were compoed of internal laer that had been caued to lide over each other X-ai hear, where h i the hear parameter h X-ai hear with other reference line h h ref Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
19 Other two-dimenional tranformation (4) Y-ai hear with other reference line h h ref Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
20 Tranformation between two-dimenional coordinate tem Coordinate tranfer from one reference frame to another Step Tranlate o that origin (,) of the tem i moved to the origin of (,) of the tem Rotate the ai onto the ai Eample: Reference a Carteian tem pecified with coordinate origin (,) and orientation angle θ in Carteian tem Matri tranformation of the tranlation and rotation o coθ inθ T(, ) o R( θ ) inθ coθ Compoite matri of the tranformation M,'' R( θ ) T(, ) Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
21 Rater method for geometric tranformation Rater tem tore picture information into a color pattern of in the frame buffer Tranformation done b manipulating an arra of piel value Tranlation manipulated b rater operation E.g. Block tranfer (bitblt / piblt) Arra manipulation ued to rotate a piel block b 9 o,8 o b the rotation of matri (repreenting that piel) Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
22 Geometric tranformation in threedimenional pace () Etend from two-dimenional tranformation b including conideration for the z coordinate Tranlation and caling are imilar to two-dimenion, include the three Carteian coordinate Rotation method i le traight forward Repreentation Four-element column vector for homogenou coordinate Geometric tranformation decribed 4b4 matri Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
23 Geometric tranformation in threedimenional pace () Three-dimenional tranlation A point P (,,z) in three-dimenional pace tranlate to new location with the tranlation ditance T (t, t, tz) ' + t ' + t In matri format ' t ' t z' t z z z ' z + t z P' T P Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
24 Geometric tranformation in threedimenional pace (3) Three-dimenional caling Relative to the coordinate origin, jut include the parameter for z coordinate caling in the tranformation matri P S ' P z z' ' ' z Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon 5 Pearon Education Education Relative to a fied point ( f, f z f ) Perform a tranlate-caling-tranlate compoite tranformation )z ( ) ( ) ( ) z,, T( ),, S( ),z, t( f z z f f f f f z f f f z
25 Geometric tranformation in threedimenional pace (4) Three-dimenional rotation definition Aume looking in the negative direction along the ai Poitive angle rotation produce counterclockwie rotation about a coordinate ai Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
26 Geometric tranformation in threedimenional pace (5) Three-dimenional coordinate-ai rotation Z-ai rotation equation ' coθ inθ ' coθ inθ ' inθ + coθ ' inθ coθ z' z z' z Tranformation equation for rotation about the other two coordinate ae can be obtained b a cclic permutation z Y-ai rotation equation ' z' ' coθ zinθ inθ + zcoθ Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
27 Geometric tranformation in threedimenional pace (6) Three-dimenional coordinate-ai rotation Y-ai rotation equation z' ' ' zcoθ inθ zinθ + coθ General Three-dimenional rotation Tranlate object o that the rotation ai coincide with the parallel coordinate ai Perform the pecified rotation about that ai Tranlate object back to the original poition P' T R( θ ) R T ( θ ) T P R ( θ ) T Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
28 Geometric tranformation in threedimenional pace (7) Three-dimenional coordinate-ai rotation Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
29 Geometric tranformation in threedimenional pace (8) Three-dimenional rotation along arbitrar ai It can be done b 5 tep Action Tranlate the object o that the rotation ai pae through the coordinate origin Define a rotation ai (vector V) V P P (,,z z) The unit vector u for the rotation ai V (a,b,c ), V V V Tranlation matri u T z, z V z Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
30 Geometric tranformation in threedimenional pace (9) Action Perform rotation, o that the unit vector u coincide with z-ai Action a, rotate unit vector u into z plane about -ai Define the projection of u in z plane a u and projection of u a u z Determine the rotation angle α with dot and cro product of u and u z α u' (,b,c),u coα u' u u' u (,,) + c Rotation matri reulted a c b R d d ( α) b c d d z z z b c c d u' u z in α u b d b u u' u z in Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
31 Geometric tranformation in threedimenional pace () Action b Rotate unit vector u (a,,d) (unit vector u in z plane) to z-ai about -ai Determine the rotation angel β with the imilar wa in action a Tranformation matri d a R ( β ) a d Action 3 Perform the rotation about z-ai with pecific rotation angel coθ inθ inθ coθ R z ( θ ) θ Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
32 Geometric tranformation in threedimenional pace () Action 4 Perform an invere rotation in action Action 5 Perform an invere tranlation in action Final rotation matri R ( θ ) T R ( α) R ( β ) R z ( θ ) R ( β ) R ( α) T Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
33 Geometric tranformation in threedimenional pace () Graphical epreion of the 5 tep Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
34 Other three-dimenional tranformation () Three-dimenional reflection Relative to reflection plane Eample: reflection relative to the plane M zreflect Three-dimenional Shearing Relative to coordinate ai Eample: z-ai hearing tranformation M zhear h h z z h h z z z z ref ref Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
35 Other three-dimenional tranformation () Tranformation between 3D coordinate tem Rotation matri u' u' u' 3 u' u' u' 3 R u' u' u' z z z3 Affine tranformation ' a ' a ' a z + a + a + a z + a + a + a z z zz z + b z + b z + b z Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
36 OpenGL upporting function ummar Baic OpenGL tranformation gltranlate (t, t, tz) glrotate (theta, v, v, vz) glscale (z,, z) OpenGL Matri operation Specif model-view matri glmatrimode (GL_MODELVIEW); Aign identit matri and aign matri value glloadidentit (); glloadmatri (); Matri Stack glgetintegerv (GL_MAX_MODELVIEW_STACK_DEPTH, tackize); glpuhmatri(); glpopmatri(); Computer Graphic with OpenGL, Third Edition, b Donald Hearn and M.Pauline Baler. IBSN 4 Pearon Education, Inc., Upper Saddle River, NJ. All right reerved 5 Pearon Education
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