Topic 7: Transformations. General Transformations. Affine Transformations. Introduce standard transformations

Size: px

Start display at page:

Download "Topic 7: Transformations. General Transformations. Affine Transformations. Introduce standard transformations"

Phoebe Chambers
5 years ago
Views:

1 Tpic 7: Transfrmatins CITS33 Graphics & Animatin E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 Objectives Intrduce standard transfrmatins Rtatin Translatin Scaling Shear Derive hmgeneus crdinate transfrmatin matrices Learn t build arbitrar transfrmatin matrices frm simple transfrmatins E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 2 General Transfrmatins Affine Transfrmatins A transfrmatin maps pints t ther pints and/r vectrs t ther vectrs Q=T(P) v=t(u) Preserving parallel lines Characteristic f man phsicall imprtant transfrmatins Rigid bd transfrmatins: rtatin, translatin Scaling, shear Imprtance in graphics is that we need nl transfrm endpints f line segments and let implementatin draw line segment between the transfrmed endpints E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 3 E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 4

v Pipeline Implementatin u v u vertices (befre transfrmatin) T transfrmatin (frm applicatin prgram) T(u) T(u) T(v) vertices (after transfrmatin) T(v) rasterizer T(u) piels E. Angel and D.

frame P, Q, R: pints in an affine space p, q, r: representatins f pints - arra f 4 scalars in hmgeneus crdinates u, v, w: representatins f vectrs in an affine space - arra f 4 scalars in hmgeneus

2 v Pipeline Implementatin u v u vertices (befre transfrmatin) T transfrmatin (frm applicatin prgram) T(u) T(u) T(v) vertices (after transfrmatin) T(v) rasterizer T(u) piels E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 frame buffer T(v) 5 Ntatin We will be wrking with bth crdinate-free representatins f transfrmatins and representatins within a particular frame P, Q, R: pints in an affine space p, q, r: representatins f pints - arra f 4 scalars in hmgeneus crdinates u, v, w: representatins f vectrs in an affine space - arra f 4 scalars in hmgeneus crdinates α, β, γ: scalars E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 6 Translatin Mve (translate, displace) a pint t a new lcatin d P P Hw man was? Althugh we can mve a pint t a new lcatin in an infinite number f was, when we mve man pints (f a rigid bject) there is usuall nl ne wa Displacement determined b a vectr d Three degrees f freedm P = P + d E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 7 bject translatin: ever pint displaced b same vectr E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle

3 Translatin Using Representatins Translatin Matri Using the hmgeneus crdinate representatin in sme frame p = [ z ] T p = [ z ] T d = [d d d z ] T Hence p = p + d r We can als epress translatin using a 4 4 matri T in hmgeneus crdinates p = Tp where d d T = T(d, d, d z ) = d z = + d = + d z = z + d z nte that this epressin is in fur dimensins and epresses pint = vectr + pint This frm is better fr implementatin because all affine transfrmatins can be epressed this wa and multiple transfrmatins can be cncatenated tgether E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 9 E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 Rtatin (2D) Rtatin abut the z ais Cnsider a rtatin abut the rigin b q degrees radius stas the same, angle increases b q = r cs(φ + θ) = r sin(φ + θ) (New pint p after rtatin) = cs θ sin θ = sin θ + cs θ = r cs φ = r sin φ (Old pint p befre rtatin) E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 Rtatin in tw dimensins is equivalent t rtatin abut the z ais in three dimensins. Rtatin abut the z ais in 3D leaves the z cmpnents f all the pints unchanged: csθ sin θ R z θ = sin θ csθ In hmgeneus crdinates p = R Z (θ)p E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle

4 Rtatin abut the z ais (cnt.) Cnsider the eample = r cs φ = r sin φ z = we have z r cs φ csθ sin θ r cs φ r sin φ = R z (θ) = sin θ csθ r sin φ r cs θ cs φ r sin θ sin φ r sin θ cs φ + r cs θ sin φ = 3 Rtatin abut the z ais (cnt.) Appling the rule: cs(θ + φ) = cs θ cs φ sin θ sin φ sin θ + φ = sin θ cs φ + cs θ sin φ we get z = R z (θ) r cs φ r sin φ E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 = r cs(θ + φ) r sin(θ + φ) Thus, In general, = r cs(θ + φ) = r sin(θ + φ) z = = cs θ sin θ = sin θ + cs θ z = z 4 Rtatin abut and aes Same argument as fr rtatin abut z ais Fr rtatin abut ais, is unchanged Fr rtatin abut ais, is unchanged cs q - sin q R = R (q) = sin q cs q Nte the negative sign here R = R (q) = cs q - sin q sin q cs q E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 5 Scaling Epand r cntract alng each ais (fied pint f rigin) = s = s z = s z z p = Sp s S = S(s, s, s z ) = E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 s s z 6 4

Reflectin Inverses crrespnds t negative scale factrs s = - s = s = - s = - riginal s = s = - Althugh we culd cmpute inverse matrices b general frmulas, we can use simple gemetric bservatins

5 Reflectin Inverses crrespnds t negative scale factrs s = - s = s = - s = - riginal s = s = - Althugh we culd cmpute inverse matrices b general frmulas, we can use simple gemetric bservatins Translatin: T (d, d, d z ) = T( d, d, d z ) Rtatin: R (θ) = R( θ) Hlds fr an rtatin matri Nte that since cs( θ) = cs(θ) and sin( θ) = sin(θ) R (θ) = R T (θ) Scaling: S (s, s, s z ) = S(/s, /s, /s z ) E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 7 E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 8 Cncatenatin Order f Transfrmatins We can frm arbitrar affine transfrmatin matrices b multipling tgether rtatin, translatin, and scaling matrices Because the same transfrmatin is applied t man vertices, the cst f frming a matri M = ABCD is nt significant cmpared t the cst f cmputing Mp fr man vertices p The difficult part is hw t frm a desired transfrmatin frm the specificatins in the applicatin Nte that matri n the right is the first applied Mathematicall, the fllwing are equivalent p = ABCp = A(B(Cp)) Nte man references use rw vectrs t represent pints. Fr such references: p T = p T C T B T A T E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 9 E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle

6 General Rtatin Abut the Origin A rtatin angle f q abut an arbitrar ais can be decmpsed int the cncatenatin f rtatins abut the,, and z aes R(θ) = R z (θ z ) R (θ ) R (θ ) θ, θ, and θ z are called the Euler angles q v Rtatin Abut a Fied Pint ther than the Origin. Mve the rigin t the fied pint 2. Rtate 3. Mve the rigin back t fied pint back => M = T(p f ) R(q) T(-p f ) Nte that rtatins d nt cmmute We can use rtatins in anther rder but with different angles z E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle 22 2 After Step After Step 2 After Step 3 E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle Rtatin Abut a Fied Pint ther than the Origin (cnt.) A 2D eample: Objective: want t rtate a square 45 degrees abut its wn centre, p. p Rtatin Abut a Fied Pint ther than the Origin (cnt.) Our aim is t cnstruct a matri M s that when the fur vertices f the square are pre-multiplied b we get the desired utput. Step : appl a translatin s that the rigin is at p. Befre rtatin Output wanted after rtatin p This is the same as rtating abut the z-ais (pinting ut f the page) in 3D. M = T( p) 6

Rtatin Abut a Fied Pint ther than the Origin (cnt.) Step 2: appl a 45 degree rtatin abut the z-ais at the rigin. Rtatin Abut a Fied Pint ther than the Origin (cnt.

7 Rtatin Abut a Fied Pint ther than the Origin (cnt.) Step 2: appl a 45 degree rtatin abut the z-ais at the rigin. Rtatin Abut a Fied Pint ther than the Origin (cnt.) Step 3: mve the rigin back t where it was befre. M = R z θ T( p) M = T p R z θ T( p) Instancing In mdeling, we ften start with a simple bject centered at the rigin, riented with the ais, and at a standard size We appl an instance transfrmatin t its vertices t Scale Orient Lcate Shear It is helpful t add ne mre basic transfrmatin, the shearing transfrmatin, t the cllectin f transfrmatin we have learnt Shearing is equivalent t pulling faces in ppsite directins E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle

8 Shear Matri Cnsider a simple shear alng the ais = + ct θ = z = z => H(q) = ct q E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle References Interactive Cmputer Graphics A Tp-Dwn Apprach with Shader-Based OpenGL b Edward Angel and Dave Shreiner, 6 th Ed, 22 Sec 3.7 Affine Transfrmatins (all subsectins) Sec 3.8 Translatin, Rtatin, and Scaling Sec 3.9 Transfrmatins in Hmgeneus Crdinates Sec 3. Cncatenatin f Transfrmatins E. Angel and D. Shreiner: Interactive Cmputer Graphics 6E Addisn-Wesle

Transformations. Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico

Transformations. Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico Transformations Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico Angel: Interactive Computer Graphics 4E Addison-Wesley 25 1 Objectives