Geometric Transformations Hearn & Baker Chapter 5. Some slides are taken from Robert Thomsons notes.

Transcription

1 Geometric Tranformation Hearn & Baker Chapter 5 Some lie are taken from Robert Thomon note.

2 OVERVIEW Two imenional tranformation Matri repreentation Invere tranformation Three imenional tranformation OpenGL Geometric-Tranformation Function

3 Geometric tranformation function Tranlation Rotation Scaling Reflection Shearing We firt icu the 2D tranformation, then we will continue with 3D.

4 2D Tranlation Point P efine a P,, tranlate to Point P, a itance parallel to ai, parallel to ai. Define the column vector P P, Now T, P P P P T

5 2D Scaling from the origin. Point P efine a P,, Perform a cale tretchto Point an along the ai.., Define the matri S. P, b a factor along P the ai, P Now P S P or.

6 2D Rotation about the origin. P, r P, r

7 2D Rotation about the origin. P, r r P, r.co r.in

8 2D Rotation about the origin. r r P, P,.in.co r r.co.in.in.co.in.in.in.co.co.co r r r r r r

9 2D Rotation about the origin. r.co r.co.co r.in.in r.in r.co.in r.in.co Subtituting for r : r.co r.in Give u :.co.in.in.co

10 2D Rotation about the origin..co.in.in.co Rewriting in matri form give u : co in. in co Define the matri R co in in, co P R P

11 Tranformation. Tranlation. P=T + P Scale P=S P Rotation P=R P We woul like all tranformation to be multiplication o we can concatenate them epre point in homogenou coorinate.

12 Homogeneou coorinate A an etra coorinate, W, to a point. P,,W. Two et of homogeneou coorinate repreent the ame point if the are a multiple of each other. 2,5,3 an 4,,6 repreent the ame point. At leat one component mut be non-zero,, i not efine. If W, ivie b it to get Carteian coorinate of point /W,/W,. If W=, point i ai to be at infinit.

13 Homogeneou coorinate If we repreent,,w in 3-pace, all triple repreenting the ame point ecribe a line paing through the origin. If we homogenize the point, we get a point of form,, homogenie point form a plane at W=. W P W= plane X Y

14 Tranlation in homogenie coorinate Tranformation matrice for 2D tranlation are now 33..

15 Concatenation. We perform 2 tranlation on the ame point:,,, : we epect So,,,,, T T T T P T T P P T P P T P P

16 Concatenation.?. i :,, The matri prouct T T Matri prouct i varioul referre to a compouning, concatenation, or compoition

17 Concatenation. Matri prouct i varioul referre to a compouning, concatenation, or compoition. Thi ingle matri i calle the Coorinate Tranformation Matri or CTM.. i :,, The matri prouct T T

18 Propertie of tranlation.,, T 4.,,,, 3.,,, 2.,. - T T t t T t t T T t t T t t T T I T Note : 3. tranlation matrice are commutative.

19 Homogeneou form of cale. S, Recall the, form of Scale :, S In homogeneou coorinate :

20 Concatenation of cale.! multipl in the matri - ea to element iagonal Onl. i :,, The matri prouct S S

21 Reflection correpon to negative cale factor = - = original = - = - = = -

22 Homogeneou form of rotation.. co in in co i.e : are orthogonal, Rotation matrice. For rotation matrice, T R R R R i.e. the invere i the tranpoe

23 Orthogonalit of rotation matrice. co in in co, co in in co T R R co in in co co in in co R

24 Other propertie of rotation. Otherwieorer matter. the ame, rotation are the centre of if onl an R R R R R R R I R

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26 2D Compoite Tranformation Combine tranformation of ifferent tpe tranlate, rotate, tranlate tranlate, cale, tranlate tranlate, reflect, tranlate Ue to rotate or cale an object w.r.t. a point that i not the origin Implement b multiplication of the correponing homogeneou matrice

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32 reflection in an ae reflection in origin

33 a a b b Sh give the lope of a vertical line after hear, a /

34 Eample T,.4,,.4 S,,

35 Eample T,,,.4 S,,

36 Eample,.4 T,,.4 TS,.4,

37 Eample 2 T,.6,,.6 S,,

38 Eample 2.6 T,,, S,,

39 Eample 2 T,.6,.6,.6, TS,,

40 Summar Shear in : Sh h h Shear in : Sh h h.

41 Double Shear: not commutative! a b ab b a b a b a ab

42 Shear Matri efine b angle, not lope e.g. imple hear along ai = + cot = z = z cot H =

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48 3D Tranformation. Ue homogeneou coorinate, jut a in 2D cae. Tranformation are now 44 matrice. We will ue a right-hane worl coorinate tem - z out of page. Note: Convenient to think of ipla a Being left-hane!! z into the creen z out of page

49 Simple etenion to the 3D cae:

50 Scale in 3D.,, z z S Simple etenion to the 3D cae:

51 Rotation in 3D Nee to pecif which ai the rotation i about. z-ai rotation i the ame a the 2D cae. co in in co R z

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55 Rotation aroun an ai parallel to -ai

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61 H&B p27

62 Row of upper-left 33 ubmatri, when rotate b R lie on the, an z ae r r r R r r r R r r r R

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64 Scaling an object with thi tranformation will alo move it poition relative to the origin - o move it to the origin, cale it, then move it back...

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71 recall, for 2D rawing we ue glmatrimoegl_projection gluortho2d., 2.,., 2.

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