6.2Dtform

Translation

x' = x + t_x
y' = y + t_y

Translation

x' = x + t_x
y' = y + t_y

In matrix form:
P' = P + T

where T = [ t_x
t_y ]

Repeat transformation for each point in figure (rigid body transformation)

Scaling

x' = x * s_x
y' = y * s_y

Scaling

x' = x * s_x
y' = y * s_y

In matrix form:
P' = S^.P

where S = [ s_x 0
0 s_y ]

Uniform and Nonuniform Scaling

S₁ =

[

2 0
0 2

]

S₁ =

[

3 0
0 0.5

]

Rotation

x' = r cos[theta]
y' = r sin[theta]

Rotation

x' = x cos[theta] - y sin [theta]
y' = x sin[theta] + y cos [theta]

In matrix form:
P' = R^.P
where

R = [ cos[theta] -sin[theta] ]

sin[theta] cos[theta]

Homogeneous Coordinates

Want to do all standard transformations as matrix multiplications

P=[

x
y

] => [

x/h
y/h
1

] or [

x
y
h

]

Composite Transformations

Translation

P' = T₂^.T₁^.P = T(tx₂, ty₂)^.T(tx₁, ty₁)^.P
= T(tx₂ + tx₁, ty₂ + ty₁)^.P

Composite Transformations

Rotation

P' = R([theta]₂)^.R([theta]₁)^.P = R([theta]₂ + [theta]₁)^.P

Composite Transformations

Scaling

P' = S(sx₂, sy₂)^.S(sx₁, sy₁)^.P
= S(sx₂ + sx₁, sy₂ + sy₁)^.P

General Pivot Point Rotation

A transformation sequence for rotating an object about a specified pivot point using the rotation matrix

	Original Position of Object and Pivot Point
	Translation of Object so that Pivot Point is at Origin
	Rotation about Origin
	Translation of Object so that Pivot Point is Returned to original Position

P' = T(x_r, y_r) ^. R([theta]) ^. T(-x_r, -y_r) ^. P

General Pivot Point Scaling

A transformation sequence for scaling an object with respect to a specified fixed position using the scaling matrix

	Original Position of Object and Fixed Point
	Translation of Object so that Fixed Point is at Origin
	Scale Object with Respect to Origin
	Translation of Object so that Fixed Point is Returned to original Position