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2 d transformations
1. Computer Graphics
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)
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2. Computer Graphics
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).
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3. Computer Graphics
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
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