Transformations

CSC 341 Introduction to Computer Graphics Transformations

Objectives • Introduce standard transformations • Rotations • Translation • Scaling • Shear • Derive homogeneous coordinate transformation matrices • Learn to build arbitrary transformation matrices from simple transformations

General Transformations • A transformation is a function that maps a point (or a vector) into another point (or a vector) • All transformations operate as simple changes on vertex-coordinates (2D or 3D). • Q = T(P): P is mapped to Q by transformation T • v = R(u): vector u is mapped to vector v by R

Affine Transformations • Translation

Affine Transformations • Rotation

Affine Transformations • Uniform Scaling: stretching axis by the same constant factor

Affine Transformations • Nonuniform Scaling: stretching axis by some constant factor, each axis with a different constant

Affine Transformations • Shear: deforms squares into parallelograms

Affine Transformations • Reflection (with respect to a given axis)

Affine Transformations • Line preserving • Maps lines to lines • Translation and rotation preserve lengths and angle • Rigid body transformations • Uniform scaling preserves angles but not lengths • Nonuniform Scaling and shear do not preserve lengths or angles • Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints

Affine Transformations • They all preserve affine relationships! R = (1-a)P + aQ T(R) = (1-a)T(P) + aT(Q) • Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints • i.e. If we transform endpoints, that is sufficient to obtain linear combinations of transformed endpoints, we do not need to transform every single point R P Q T(Q) T(P)

Pipeline Implementation v T (from application program) frame buffer u T(u) transformation rasterizer T(v) T(v) T(v) v T(u) u T(u) vertices pixels vertices

Notation We will be working with both coordinate-free representations of transformations and representations within a particular frame P,Q, R: points in an affine space u, v, w: vectors in an affine space a, b, g: scalars

Translation • Move (translate, displace) a point to a new location • Displacement determined by a vector d • Three degrees of freedom • P’=P+d P’ d P

How many ways? Although we can move a point to a new location in infinite ways, when we move many points there is usually only one way object translation: every point displaced by same vector

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 0]T Hence P’ = P + d or x’=x+dx y’=y+dy z’=z+dz

How to we map P to P’ by matrix multiplication? • P’ = TP • What is T?

Translation Matrix Hence we can express translation using a 4 x 4 matrix T in homogeneous coordinates P’=TP where T = T(dx, dy, dz) = This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together

In general: 3D Transformations with Homogeneous Coordinates

Translation • How about vectors under translation transformation? • Vectors are not changed by translation! • Inversion of T? T-1 (dx , dy, dz)= T(-dx, -dy, -dz)

Scaling • Expand or contract along each axis • Performed relative to some fixed point—that is the point remains unchanged after scaling • We will assume that this point is the origin of the standard coordinated frame

Scaling: example in 2D Scale by 2 on x and y axis (2,2) (1,1) (-1,-1) (-2,-2)

Scaling: example in 2D Scale by 2 on x and y axis (4,4) (2,2) (2,2) (1,1)

Scaling • Given scaling factors for each axis, sx,sy,sz, point P is mapped to point P’ as follows • If all scaling factors are equal, that is called uniform scaling x’=sxx y’=syy z’=szz

How to we map P to P’ by matrix multiplication? • P’ = SP • What is matrix S?

Inverse of Scaling Matrix • S-1(sx, sy, sz) = S(1/sx, 1/sy, 1/sz)

Reflection corresponds to negative scale factors sx = -1 sy = 1 original sx = -1 sy = -1 sx = 1 sy = -1

Rotation (2D) • Consider rotation about the origin by q degrees • Fixed point at the origin • radius stays the same, angle increases by q x’ = r cos (f + q) y’ = r sin (f + q) x = r cos f y = r sin f

Rotation (2D) • Consider rotation about the origin by q degrees • Fixed point at the origin • radius stays the same, angle increases by q x’ = r cos (f + q) y’ = r sin (f + q) x’=r cosfcos q–r sin f sin q y’ = r cos fsin q + r sin fcos q x = r cos f y = r sin f

Rotation (2D) • Consider rotation about the origin by q degrees • Fixed point at the origin • radius stays the same, angle increases by q x’ = r cos (f + q) y’ = r sin (f + q) x’=r cosfcos q–r sin f sin q y’ = r cos fsin q + r sin fcos q x’=x cos q –y sin q y’ = x sin q + y cos q x = r cos f y = r sin f

3D Rotation • In general, rotation happens around an axis and about a fixed point • Assume origin is the fixed point • Consider rotation about z axis • leaves all points with the same z • Equivalent to rotation in 2D in planes of constant z x’=x cos q –y sin q y’ = x sin q + y cos q z’ =z If your thumb is aligned with the axis of rotation, then positive direction is indicated by your fingers: +Q z

3D Rotation Matrix • P’=Rz(q)P Recall: x’=x cos q –y sin q y’ = x sin q + y cos q z’ =z R = Rz(q) =

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 R = Rx(q) = R = Ry(q) =

Fixed Point revisited • The point that remains unchanged through rotation • Example: if the fixed point is origin (center of the square)

Fixed Point revisited • The point that remains unchanged through rotation • Example: if the fixed point is center of the square

Fixed Point revisited • The point that remains unchanged through rotation • Example: if the fixed point is the origin

Inverse Rotation • Although we could compute inverse matrices by general formulas, we can use simple geometric observations • Rotation: R-1(q) = R(-q) • Holds for any rotation matrix • Note that since cos(-q) = cos(q) and sin(-q)=-sin(q) R-1(q) = R T(q)

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 difficult part is how to form a desired transformation from the specifications in the application

Order of Transformations • Note that matrix on the right is the first applied • Mathematically, the following are equivalent P’ = ABCP = A(B(CP))

Rotation About a Fixed Point other than the Origin (e.g. center of a cube) Translate such that fixed point coincides with origin: T(-pf) Rotate: R(q) Translate fixed point back to its original position: T(pf) Concatenate: M = T(pf) R(q) T(-pf)

General Rotation About the Origin q A rotation by q about an arbitrary axis can be decomposed into the concatenation of rotations about the x, y, and z axes --is a bit complex R(q) = Rz(qz) Ry(qy) Rx(qx) y v qx qy qz are called the Euler angles Note that rotations do not commute We can use rotations in another order but with different angles x z

Instancing • 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 instance transformation to its vertices to Scale Orient Locate

Shear • Helpful to add one more basic transformation • Equivalent to pulling faces in opposite directions

Shear Matrix Consider simple shear along x axis x’ = x + y cot q y’ = y z’ = z H(q) =

Transformations