Representations and Transformations. Objectives

Transcription

1 Representations and Transformations Objectives Clarify elements of geometry -Scalars -Vectors - Points Derive homogeneous coordinate transformation matrices Introduce standard transformations - Rotations - Translation - Scaling - Shear 2 1

2 Scalars, Points, Vectors Three basic elements from geometry: Scalars, Points, Vectors Scalars can be defined as members of a set which can be combined by the operations of addition and multiplication and obey the fundamental axioms: associativity, commutivity, inversion Examples include the real and complex numbers under the rules we are all familiar with Scalars alone have no geometric properties 3 Scalars, Points, Vectors A vector is a quantity with two attributes direction & magnitude and its own rules as we saw last lecture The set defines a vector space But, vectors lack position Same length and magnitude -> Vectors are insufficient to specify geometry We need points 4 2

3 Scalars, Points, Vectors Points, we know, are locations in space Certain operations translate between points and vectors - Point-point subtraction yields a vector - Leads to equivalent to point-vector addition v=p-q P=v+Q 5 Confusing Points and Vectors Consider the point P and the vector v They appear to have the similar representations p=[ ] v=[ ] which confuse the point with the vector v p But, a vector has no position v can be placed anywhere fixed 6 3

4 Confusing Points and Vectors When we learned simple geometry, most of us started with a Cartesian coordinate frame - Points were at locations in space P=(x,y,z) - We derived results by algebraic manipulations involving these Cartesian coordinates This approach is nonphysical - Physically, points exist regardless of the location of an arbitrary coordinate system - Many geometric results are independent of the coordinate system 7 Coordinate Frames A frame is determined by (P, v 1, v 2, v 3,... ) where P is the origin Within this frame: Every vector can be written as v= 1 v v n v n And every point can be written as P = P + 1 v v n v n 8 4

5 A Single Representation With these rules, we can keep track of the difference: v= 1 v v v 3 = [ ][v 1 v 2 v 3 P ] T P = P + 1 v v v 3 = [ ][v 1 v 2 v 3 P ] T Thus we obtain a four-dimensional representation for both: v = [ ] T p = [ ] T 9 A Single Representation An affine space combines point and vector as v = [ ] T p = [ ] T 1 5

6 Vector Spaces A set of vectors v 1, v 2,, v n is linearly independent if 1 v v n v n = iff 1 = 2 = = If a set of vectors is linearly independent, we cannot represent one in terms of the others If a set of vectors is linearly dependent, as least one can be written in terms of the others 11 Vector Spaces In a vector space, the maximum number of linearly independent vectors is fixed and is called the dimension of the space In an n-dimensional space, any set of n linearly independent vectors form a basis for the space Given a basis v 1, v 2,., v n, any vector v can be written as v= 1 v v n v n where the { i } are unique 12 6

7 Vector Spaces Given the basis vectors v 1, v 2,., v n A vector is written v= 1 v v n v n The list of scalars { 1, 2,. n } then is the representation of v with respect to the given basis And we write the representation as a row or column array of scalars 1 a=[ 1 2. n ] T = 2. n 13 Affine Spaces Vector spaces do not represent points Instead, we work in an affine space and add that special point, the origin, to the basis vectors, this is now our frame v 2 P v 1 v

8 Affine Spaces An affine space is both point and vector space It allows operations from vectors, points and scalars: - Vector-vector addition - Scalar-vector multiplication - Point-vector addition - All scalar-scalar operations v=p-q P=v+Q 15 Transformations Homogeneous coordinates are key to all computer graphics systems Hardware pipeline all work with 4 dimensional representations v = [ ] T p = [ ] T All standard transformations (rotation, translation, scaling) will be implemented by matrix multiplications with 4 x 4 matrices 16 8

9 Transformations A transformation maps points to other points and/or vectors to other vectors v=t(u) Q=T(P) 17 Translation Move (translate, displace) a point to a new location P P d Displacement determined by a vector d - Three degrees of freedom -P =P+d 18 9

10 Moving objects When we move a point on an object to a new location, to preserve the object, we must move all other points on the object in the same way object translation: every point displaced by the same vector, d 19 Translation Using Representations Using the homogeneous coordinate representation in some frame p=[ x y z 1] T p =[x y z 1] T d=[dx dy dz ] T Hence p = p + d or x =x+d x y =y+d y z =z+d z note that this expression is in four dimensions and expresses that point = vector + point 2 1

11 Translation Matrix We can also express translation using a 4 x 4 matrix T in homogeneous coordinates p =Tp where 1 dx 1 d T = T(d x, d y, d z ) = y 1 dz 1 This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together 21 Scaling Expand or contract along each axis (fixed point of origin) x =s x x y =s y y z =s z z p =Sp S = S(s x, s y, s z ) = sx s y s z

12 Reflection corresponds to negative scale factors s x = -1 s y = 1 original s x = -1 s y = -1 s x = 1 s y = Rotation (2D) Consider rotation about the origin by degrees - radius stays the same, angle increases by What is this rotation about the z axis? 24 12

13 Rotation (2D) Consider rotation about the origin by degrees - radius stays the same, angle increases by x = r cos ( y = r sin ( x =x cos y sin y = x sin + y cos x = r cos y = r sin 25 Rotation about the z axis Rotation about z axis in three dimensions leaves all points with the same z - Equivalent to rotation in two dimensions in planes of constant z x =x cos y sin y = x sin + y cos z =z - or in matrix notation (with p as a column) p =R z ()p 26 13

14 Rotation Matrix Homogeneous Coordinates: R = R z () = cos sin sin cos Rotation about x and y axes Same argument as for rotation about z axis - For rotation about x axis, x is unchanged - For rotation about y axis, y is unchanged 1 cos - sin R = R x () = sin cos 1 R = R y () = cos - sin 1 sin cos

15 Basic transforms in OpenGL 29 Affine Transformations Line preserving Characteristic of many physically important transformations - Rigid body transformations: translation, rotation - Non-rigid: Scaling, shear Importance in graphics is that we need only transform vertices (points) of line segments and polygons, then system draws between the transformed points 3 15

16 Inverses Although we could compute inverse matrices by general formulas, we can also use simple geometric observations, for example: - Translation: T -1 (d x, d y, d z ) = T(-d x, -d y, -d z ) - Rotation: R -1 () = R(-) Holds for any rotation matrix Note that since cos(-) = cos() and sin(-)= -sin() R -1 () = R T () - Scaling: S -1 (s x, s y, s z ) = S(1/s x, 1/s y, 1/s z ) 31 Concatenation We can form arbitrary affine transformation matrices by multiplying together rotation, translation, and scaling matrices Because the same transformation is applied to many vertices, the cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p The combination of transformations must be managed with care, b/c order matters 32 16

17 Order of Transformations Note that matrix on the right is the first applied Mathematically, the following are equivalent p = ABCp = A(B(Cp)) (but does not = CBA p) Some references use row matrices to present points. In terms of rows, we get p T = p T C T B T A T 33 Order of Transformations In modeling, we often start with a simple object centered at the origin, oriented with the axis, and at a standard size We apply an composite transformation to its vertices to Scale Orient Locate 34 17

18 Composite Transformations Scaling about a fixed point - Applying the scale transformation also moves the object being scaled. Q Q' P P' 35 Composite Transformations Exception: Scaling about origin -> no movement Origin is a fixed point for the scale transformation We use composite transformations to create scale transformations with different fixed points Q Q' P P' 36 18

19 Composite Transformations Fixed point scale transformation Move fixed point (px,py,pz) to origin Scale by desired amount Move fixed point back to original position M = T(px, py, pz) S(s x, s y, s z ) T(-px, -py, -pz) 37 Composite Transformations Rotating about a fixed point - basic rotation alone will rotate about origin but we want: 38 19

20 Composite Transformations Rotating about a fixed point Move fixed point (px,py,pz) to origin Rotate by desired amount Move fixed point back to original position M = T(px, py, pz) R x () T(-px, -py, -pz) 39 Composite Transformations 4 2

21 Rotation about an arbitrary axis Rotating about an axis by theta degrees Rotate about x to bring axis to xz plane Rotate about y to align axis with z -axis Rotate theta degrees about z Unrotate about y, unrotate about x M = Rx -1 Ry -1 Rz() Ry Rx Can you determine the values of Rx and Ry? 41 Composite transformations A series of transformations on an object can be applied as a series of matrix multiplications : position in the global coordinate : position in the local coordinate 42 21