2D Transformations in Computer Graphics

8/12/2019 2D Transformations in Computer Graphics

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Machiraju/Zhang/Mller

2D Transformations

CMPT 361

Introduction to Computer GraphicsTorsten Mller


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2

Graphics Pipeline

Hardware

Modelling

Transform Visibility

Illumination +

Shading

ColorPerception,

Interaction

Texture/

Realism


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3

Reading

Chapter 3 of Angel

Chapter 5 of Foley, van Dam,


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4

Schedule

Geometry basics

Affine transformations

Use of homogeneous coordinates Concatenation of transformations

3D transformations

Transformation of coordinate systems Transform the transforms

Transformations in OpenGL


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5

Basic transformations

The most basic ones

Translation

Scaling

Rotation

Shear

And others, e.g., perspective transform, projection, etc.

Basic types of transformationsRigid-body: preserves length and angle

Affine: preserves parallel lines, not angles or lengths

Free-form: a

nything goes


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Translation in 2D

P = P+ T

x = x+ dx

y = y + dy x

y=

xy+ dx

dy


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Scaling in 2D

x

y

=

sx 0

0 sy

x

y

P= S P

x

= sx x

y= sy y

Uni orm sx = sy

Non-uniform sx = sy


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Positive angles:

counterclockwise

For negative angles

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Rotation about origin

P= R P

x = x cos(

)

ysin(

)y

= x sin() + ycos()

cos() = cos()

sin() = sin()

x

y

=

cos() sin()

sin() cos()

x

y


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Make use of polar coordinates:

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Derivation of rotation matrix

x= r cos

y= r sin()

x= r cos +

y= r sin(+ )x

= r cos(+ )

= r cos()cos() r sin() sin()

= x cos() ysin()

y = r sin( + )

= r sin() cos() + r cos() sin()

= ycos() + x sin()

(r,) (x, y)

(r,) (x, y)


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Shearing in 2D

P= SHy P

P= SHx P

x

y

=

1 a

0 1

x

y

xy

=

1 0b 1

xy

x= x+ ay

y= y


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11

Schedule

Geometrybasics



3D transformations




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12

Homogeneous coordinates

Only translation cannot be expressed as

matrix-vector multiplications

But we can add a third coordinate

Homogeneous coordinatesfor 2D points

(x, y) turns into (x, y, W)

if (x, y, W) and (x', y', W ) are multiples of one

another, they represent the same point typically, W !0.

points with W = 0 are points at infinity


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2D Homogeneous Coordinates

Cartesian coordinatesof the homogenous point

(x, y, W): x/W, y/W (divide through by W)

Our typical homogenized points: (x, y, 1) Connection to 3D?

(x, y, 1) represents a 3D

point on the plane W = 1

A homogeneous point is a

linein 3D, through the origin


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Transformation in

homogeneous coordinates General form of affine (linear) transformation

E.g., 2D translation inhomogeneous coordinates


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2D line equation

Duality of 3D points to a line

3D plane equation:

Duality of 4D points to a plane

Homog. Coords in 1D?

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Homogeneous Coords (3)

ax+ y +cx + = 0

a b c d

xy

z

1

= 0

ax+by +c = 0

a b c

x

y1

= 0


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Basic 2D transformations


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Inverse of transformations

Inverse of T(dx, dy) = T("dx, "dy)

R(#)!

1= R("#)

S(sx, sy)!1= S(1/sx, 1/sy)

Sh(ax)!1= Sh("ax)

T("dx, "dy) = T(dx, dy)!1


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Schedule

Geometrybasics



3D transformations




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Compound transformations

Often we need many transforms to build

objects (or to direct them)

e.g., rotation about an arbitrary point/line?

Concatenate basic transforms sequentially

This corresponds to multiplication of the

transform matrices, thanks to homogeneous

coordinates


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Compound translation

What happens when a point goes through

T(dx1, dy1) and then T(dx2, dy2)?

Combined translation: T(d

x1

+dx2

, dy1

+dy2

)

Concatenation of transformations: matrix multiplication

1 0 dx2

0 1 dy2

0 0 1

1 0 dx1

0 1 dy1

0 0 1

=

1 0 dx1 +dx2

0 1 dy1 +dy2

0 0 1


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Compound rotation

What happens when a point goes through

R(#) and then R($)?

Combined rotation: R(#+ $)

Scaling or shearing (in one direction) is similar

These concatenations are commutative


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Commutativity

Transformations that do commute:

Translate translate

Rotate rotate (around the same axis)

Scale scale

Uniform scale rotate

Shear in x (y) shear in x (y), etc.

In general, the order in which transformationsare composed is important

After all, matrix multiplications are not

commutative


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Commutativity

the order in which we compose matrices is

important: matrix multiplication does not

commute

x

y

z

x

y

z x

y

z

x

y

z x

y

z

X then Z

Z then X


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Rotation about an

arbitrary point Break it down into easier problems:

Translate to the origin: T("x1, "y1)

Rotate: R(q)

And translate back: T(x1, y1)


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Rotation about an

arbitrary point Compound transformation (non-commutative):

T(x1, y1) R(q) T("x1, "y1)

P = T(x1, y1)R(q)T("x1, "y1)P = AP

Why combine these matrices into a single A?


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Another example

Compound transformation:

T(x2, y2) R(90o) S(sx, sy) T("x1, "y1)


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Rigid-body transformation

A transformation matrix

where the upper 2 %2 submatrix is

orthonormal, preserves angles and lengths

called rigid-bodytransformation

any sequence of translation and rotation give a

rigid-body transformation


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Affine transformation

Preserves parallelism, not lengths or angles

Product of a sequence of translation, scaling,

rotation, and shear is affine

Unit cube Rot("45o) Scale in x


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Types of Transforms

Continuous/Discrete, one to one ; invertible

Classification

Euclidean (preserves angles)

translation, rotation, uniform scale

Affine (preserves parallel lines)

Euclidean plus non-uniform scale, shearing

Projective (preserves lines) Isometry (preserves distance)

Reflection, rigid body motion

Nonlinear (lines become curves)

Twists, Bends, warps, morphs


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Window-To-Viewport

Transformation Important example of compound transform

World coords: space users work in

need to map world to the screen window in world-coordinate space and

viewport in screen coordinate space


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one can specify non-uniform scaling

results in distorted images

What property of window and viewportindicates whether the mapping will distort the

images?

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Window-To-Viewport

Transformation (2)


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Window-To-Viewport

Transformation (3) Specify transform in matrix form

Mwv = T

uminvmin

S

umaxumin

xmaxxminvmaxvmin

ymaxymin

T

xminymin

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2D Transformations in Computer Graphics