Computer Graphics

K. Adisesha

1

2D Transformations

A transformation is any operation on a point in space (x, y) that maps the point's coordinates into a new set

of coordinates (x1,y1).

Translation

In translation an object is displaced a given distance and direction from its original position. If the

displacement is given by the vector v = txI + tyJ, the new object point P'(x

', y

') can be found by applying

the transformation Tv to P(x, y). See the figure below

P' = Tv(P)

where x' = x + tx and y

' = y + ty.

As an example, consider a triangle defined by three vertices (20,0), (60, 0), and (40, 100) being translated

100 units to the right along the x-axis ( tx = 100) and 10 units up along the y-axis (ty = 10). The new

vertices are (120, 10), (160, 10), and (140, 110), see figure below:

Rotation

In rotation, the object is rotated ø° about the origin. The convention is that the direction of the rotation is

CCW if ø is a positive angle and CW if the ø is a negative angle. The transformation for rotation Rø is

P' = Rø(P)

where x' = x cos(ø) - y sin(ø) and y

' = x sin(ø) + y cos(ø)

For example a triangle (20,0), (60,0), (40,100) rotated 45° clockwise about the origin is (14.14, -14.14),

(42.43, -42.43), (98.99, -42.43)

Computer Graphics

K. Adisesha

2

Scaling

Scaling is the process of expanding or compressing the dimensions of an object. Positive scaling

constants Sx and Sy are used to describe changes in length with respect to the x direction and y direction. A

scaling constant > 1 creates an expansion (magnification) of length, and < 1 a compression (reduction) of

length. Scaling occurs along the x-axis and y-axis to create a new point from the original. This is achieved

using the following transformation:

P' = TSx,Sy (P),

where x' = Sx * x ,and y

' = Sy * y

If Sx and Sy are not equal, they have the effect of distorting pictures by elongating or shrinking them along

the directions parallel to the coordinate axes. The mirror image of an object can be generated by using the

negative values for Sxand Sy.

Homogeneous Coordinates

Translation, scaling and rotation were expressed non-homogenously:

P´ = P + T

P´ = S × P

P´ = R × P

Composition is difficult to express using the standard notation above. Homogeneous coordinates allow all

three to be expressed homogeneously, using multiplication by 3 × 3 matrices.

Add a third coordinate to a point P(x,y). So instead of representing the point using an (x,y) coordinate

pair, each point is represented by three values, (x, y, W).

P2d (x, y) -> Ph (wx, wy, w), w xb9 0

Given Ph (x, y, w), w xb9 0

Then P2d (x, y) = P2d (x/w, y/w)

W is 1 for affine transformations in graphics. Affine transformations have the property of preserving

parallism of lines, but not the lengths and angles. See example in figure 5.6 on page 207 in your Computer

Graphics text.

Shear an affine transformation

A shear is a transformation that distorts the shape of an object along either or both of the axies. Like scale

and translate, a shear can be done along just one or along both of the coordinate axes. A shear along one

axis (say, the x-axis) is performed in terms of the point's coordinate in the other axis (the y-axis). Thus a

shear of 1 in the x-axis will cause the x-coodinate of the point ot distort by 1*(y-coordinate).

Computer Graphics

K. Adisesha

3

To shear in the x direction the equation is:

x1 = x + ay

y1 = y

Where b = 0

Where x1 and y1 are the new values, x and y are the original values, and a is the scaling factor in the x

direction. The matrix is as follows.

Shearing in the y direction is similar except the roles are reversed.

x1 = x

y1 = y + bx

Where a = 0.

Where x1 and y1 are the new values, x and y are the original values, and b is the scaling factor in

the y direction. The matrix is as follows.

Example

Original Y-Shear X-Shear

Computer Graphics

K. Adisesha

4

Composition

R(ø) rotates about the origin; to rotate about P1

� Translate P1 to origin

� Rotate

Translate origin back to P1