Transformations. Examples of transformations: shear. scaling

Transcription

1 Transformations Eamples of transformations: translation rotation scaling shear

2 Transformations More eamples: reflection with respect to the y-ais reflection with respect to the origin

3 Transformations Linear transformations: take straight lines to straight lines. All of the eamples are linear. Affine transformations: take paralllel lines to parallel lines. All of the eamples are affine, an eample of linear non-affine is perspective projection. Orthogonal transformations: preserve distances, move all objects as rigid bodies. rotation, translation and reflections are affine.

4 Composition of transformations Order matters! ( rotation * translation translation * rotation) Composition of transformations = matri multiplication: if T is a rotation and S is a scaling, then applying scaling first and rotation second is the same as applying transformation given by the matri TS (note the order). Reversing the order does not work in most cases

5 Transformations and matrices Any affine transformation can be written as ' y' a a a a y b b p'=ap Images of basis vectors under affine transformations: e e y (column form of writing vectors) Ae a a a a a a Ae y a a

6 Transformations and matrices Matrices of some transformations: cos sin shear sin cos s s rotation scale by factor s reflection with respect to the origin reflection with respect to y-ais

7 Homogeneous coordinates

8 Problem Even for affine transformations we cannot write them as a single matri; we need an additional vector for translations. We cannot write all linear transformations even in the form A +b where A is a matri and b is a d vector. Eample: perspective projection [,y] [,y ] = y = y/ = equations not linear!

9 Homogeneous coordinates replace d points with 3d points, last coordinate for a 3d point (,y,w) the corresponding d point is (/w,y/w) if w is not zero each d point (,y) corresponds to a line in 3d; all points on this line can be written as [k,ky,k] for some k. (,y,) does not correspond to a d point, corresponds to a direction (will discuss later) Geometric construction: 3d points are mapped to d points by projection to the plane z = from the origin

10 Homogeneous coordinates z z= [,y] line corresponding to [,y] y From homogeneous to d: [,y,w] becomes [/w,y/w] From d to homogeneous: [,y] becomes [k,ky,k] (can pick any nonzero k!)

11 Homogeneous transformations Any linear transformation can be written in matri form in homogeneous coordinates. Eample : translations z= z y [,y] [+t,y+t y ] = +t =+ t y = y+t y = y+t y w = [,y] in hom. coords is [,y,] y t t y y

12 Homogeneous transformations Eample : perspective projection w w= = y = y/ w = [,y] y Can multiply all three components by the same number -- the D point won t change! Multiply by. line = [,y ] = y = y w = y y

13 Matrices of basic transformations cos µ sin µ sin µ cos µ 5 4 t 3 rotation t y 5 translation 3 4 s 3 4 s 3 5 s y 5 scaling skew 3 a a a 3 a a a general affine transform 5

14 Homogeneous line equation Implicit line equation in D: (n (q-p)) =, n = D vector, p = D point on the line. Goal: rewrite in homogeneous coordinates. z z= N y n p D point corresponds to a 3D line through origin; D line corresponds to a plane through origin In other words, the D line is intersection of a plane through origin with the plane z=.

15 Homogeneous line equation Rewrite the line equation: ( y y, n,q p) n n y (n, p) ([n,n (n,p)],[,y, ]) (N,q) where N=[n,n y,-(n,p)] is the normal to the plane corresponding to the line, and q is the homogeneous form of q=[,y]: q=[,y,] z z= n p Homogeneous form of the line equation: ( N q) N y

16 Homogeneous p p y p z p B p y p z A p w C A A regular 3D point to homogeneous: p homogeneous point to regular 3D: C A B p y p z A p =p w p y =p w A p z =p w

17 Translation and A Similar to D; translation by a vector t = [t ; t y ; t z ] Nonuniform scaling in three directions 6 4 t t y t z s 6 s y 7 4 s z 5

18 Rotations around coord aes 6 4 angle µ, around X ais: around Y ais: 6 4 cos µ sin µ sin µ cos µ around Z ais: cos µ sin µ sin µ cos µ cos µ sin µ sin µ cos µ note where the minus is!