2d Transforms

Computer Graphics MSIT 3rd Miniterm, Nov-Dec 2004 Instructor: Kiran Varanasi [email protected] Lecture Slides: Prof P.J. Narayanan IIIT - Hyderabad

Week 1 2D Transformations 3D Transformations Graphics Pipeline

Coordinates • A point is represented by a pair of real numbers (x, y) • There are two axes which serve as a reference for measuring distance of each point from the origin • Conventionally the X and Y axes are perpendicular to each other. But need not be. • The coordinates give the length of projections of the point onto the two axes

Coordinate Geometry • The coordinates of the point depend upon the location of the origin and the direction of the axes • Coordinates change not only when the point moves to a new position, but also when there is a change in the origin/axes

Y

P

y

O

x

X

Types of Transformation • Translation • Rotation • Scaling • Shearing In each case, it can be either the point or the reference (origin + axes)

Translation of a point • Translate the point by (a, b) • (x + a, y + b) are the coordinates of the new point in the old (same) coordinate system.

Y

P'

y' b y P

a b a

x

x'

p' = p + t

X

Translation of Axes • Translate the axes by (-a, -b). Y' New coordinates are • (x + a, y + b). • (x + a, y + b) are the y' coordinates of the same point in new coordinate system.

Y

y

P x

a b x'

p' = p + t

X X'

Transformations in reverse • Translating the axes by (a, b) is the same as translating each point by (-a, -b) • Similarly scaling the axes (zooming in) is the same as zooming out of each point • Rotating the axes by an angle θ is the same as rotating each point by the reverse angle -θ.

Coordinates under Scaling • x' = s x, y' = s y. • s is the scale factor. • Point undergoes shift when scaling with respect to the origin! • To scale shapes “in place”, translate to origin, scale, and translate back.

Y

X

p' = s p

Rotation of a point about Origin Y y'

y

P'

θ

α

P

r x'

x

θ

x = r cos α , y=r sin α x' = r cos ( α + θ ) = r cos α cos θ − r sin α sin y' = r sin ( α + θ ) = r sin α cos θ + r cos α sin

θX

Rotation: Matrix Notation Y y'

y

R(θ )

P'

θ

P α

x'

x

cos θ -sin θ x' = y' sin θ cos θ

x y

X

NewCoords = Matrix * OldCoord

p' = M p

Rotation of Coordinate Axes Y'

Y

P

y

x' = r cos (α − θ ) = r cos α cos θ + r sin α sin θ y' = r sin (α − θ ) = − r cos α sin θ + r sin α cos θ

P

X'

y'

x'

α θ

x

R(-θ ) X

cos θ x' = y' -sin θ

sin θ cos θ

x y

Opposite Rotation of Axes Y

x' = r cos α cos θ - r sin α sin θ y' = r cos α sin θ + r sin α cos θ

Y' y'

P

X y

x

α θ

x'

X'

R(θ ) cos θ -sin θ x' = y' sin θ cos θ

x y

Equivalent Matrices Y

Y

Y'

y'

y

P'

θ

y' y

P α

x'

x

X

cos θ -sin θ x' = y' sin θ cos θ

P θ

x y

α

X

x x' X'

Multiple Transformations • Translation followed by rotation of a point p΄ = R(p + t) • Rotation followed by translation of a point p΄ = t + Rp They are not identical • Rotation followed by translation followed by scaling p΄ = s(t + Rp) Cumbersome jumble of additions and multiplications !!

Homogeneous Coordinates • Represent 2D coordinates using 3-vectors. • The last number w is sort of a “scaling constant.” • Multiple representations for each point. Multiplying x, y, and w by a non-zero k doesn’t change the point.

p=

x/w x y = y/w 1 w

Transformations: Uniform Form p=

x/w x y = y/w 1 w

cos θ -sin θ R(θ ) = sin θ cos θ 0 0

1

0

a

s

0

0

T= 0

1 0

b 1

S= 0

s 0

0 1

0

p' = M1 M2 M3 p

0

0 0 1

Shearing Transformation • Add part of X to Y and vice versa! • Squares turn into parallelograms! • May be applied in one direction only.

1

shx

0

shy

1

0

0

0

1

Transformations: Uniform Form p=

cos θ -sin θ R(θ ) = sin θ cos θ 0 0

x/w x y = y/w 1 w

1

0

a

s

0

0

T= 0

1 0

b 1

S= 0

s 0

0 1

0

0

p' = M1 M2 M3 p

0 0 1

1 shx 0 Sh = shy 1 0 0 0 1

3 Equivalent Viewpoints p' = M p Point p gets transformed by M and stays in the same coordinate frame as p' Coordinate frame changes by M-1 giving a stationary point p new coordinates as p'

Coordinate frame changes by M giving a stationary point p' new coordinates as p !!

Why do we want to transform a coordinate frame ?? • Sometimes we want to think in terms of a reference frame which is natural to the problem • For example, consider a wheel which is rolling on the ground • There can be 3 coordinate frames. One fixed to the ground, the second fixed to the wheel but not rotating, the third fixed to the spokes of the wheel and rotating.

Rolling Wheel P'' = (-k, 0) Y' X''

Y

lm

X' Y''

X l

m

θ P

P' = (k cos θ , -k sin θ ) P = (l + k cos θ , m - k sin θ ) L=Rθ , θ =ω t

If P starts to move P' = R(180 - θ ) P''

Y' X''

Y

lm

X' Y''

X l

θ P

P = T(l,m)P' = T(l,m)R(180 - θ )P'' P'' = (-k - vt, 0)

m

We have to think of P in the X''-Y'' frame only.

Composite Operations P' = M1M2M3M4 P • Can think of transforming the coordinate frame by M1 followed by M2. • Point P is transformed by M4 followed by M3 and comes into the above coord frame. • Take Care: Once M1 is applied, the definition of X and Y axes are modified. Old axes are forgotten!

Example 1 • Rotation by θ about a point (a, b) • T(a, b) R(θ ) T(-a, -b) • Object brought to origin, rotated and moved back.

Y

a b

X

Example 1b Rotation by θ about a point (a, b) • T(a, b) R(θ ) T(-a, -b) • Coordinate frame moved to (a, b), rotated about new origin and moved to (-a, -b) of the new frame.

Y

a b

X

Example 1c Y

a b

X

Example 2 In place scaling and rotation : T(v) R(θ ) S(sx,sy) T(v)

Rotations: Another View Y

Y

X θ cos θ -sin θ R(θ ) = sin θ cos θ 0 0

cos θ

-sin θ

sin θ

cos θ

are unit vectors to which X and Y axes move X 0

cos θ

sin θ

-sin θ

cos θ

0 1

are unit vectors which move to X and Y axes

Making Rotation Matrices • Columns give the vectors to which the axes will move after rotation. • Rows give the vectors (before rotation) that will sit on the axes after rotation. • This is alternate way to arrive at the rotation matrix from what is known.

Example 3 • Given a triangle (0, 0), (a, b), and (c, d), how to align the side with the Y axis? • [a' b']T should sit on Y axis. Orthogonal direction is [-b' a']T. • They form the rows!

(a, b) (c, d) -b' a' 0 a' b' 0 0 0 1

Summary • Rotations, translation, scaling, shearing represented using a 3x3 matrix. • Composite transformations obtained by multiplying matrices together. • Transforming a point Vs. transforming the coordinate frame. (Doing it in reverse) • Understand well. Very important in 3D !