Transformations

168 471 Computer Graphics, KKU. Lecture 5

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Transformations

Given two frames in an affine space of dimensionn , we can find a (n+1 ) x (n +1 ) matrix that conver

- ts the coor dinates of a point in the first frame to the coordinates of the same point in the second f

- rame. The opera tion are called transformation . The matrix is called transformation matrix . I

n 3 D space, for example, the general form of th e matrix is


2

Transformations: The Relationship

• The matrix can be applied to - the coordinate,leaving the frame fixed.- the frame, leaving the coordinate fixed.

• It is useful to allow us to shift a point between the local coordinates of the point, or between the frames

• In 3D space


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Cramer’s Rule

Given a frame and any point having coordinates (u,v,w), we can define a vector

Cramer’s Rule states that1

2

3

Du DDv DDw D

and


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Finding a transformation matrix

Consider two frames denoted as and By Cramer’s rule, let

1. If we give , then we can get 1,1 1,2 1,3, , ande u e v e w

2. If we give , then we can get 2,1 2,2 2,3, ,ande u e v e w

3. If we give , then we can get 3,1 3,2 3,3, , ande u e v e w

4 . If we give , then we can get4,1 4,2 4,3, , ande u e v e w


5

168 471 Computer Graphics (First Semester, 2001)Assignment#2 (Due on June 27, 2001)

1. For three points in 2D cartesian frame, 1 1( , )x y , 2 2( , )x y and 3 3( , )x y , show that thedeterminant

1 2 3

1 2 3

1 1 1

x x xy y y

is proportional to the area of the triangle whose corner are the three points.

2. By using Cramer’s rule, determine the transformation matrix of the example given at the endof the lecture 4.


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Simple Transformations

• Translation• Scaling• Rotation• Shearing


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Translation

• Moves all points of an objects a fixed distance in a specified direction.

• Translation of frames

• Translation of points within the frame

the origin is moved but the vectors stay the same.


8

Translation (Cont.)

• A translation matrix is

• The matrix is most frequently applied to all points of an object in the local system to move the object within the system.


9

Scaling

• Scales the coordinates of an objects• Scaling a frame - expands or contracts the lengths of the vectors

• Scaling a point


10

Scaling (Cont.)

• A Scaling matrix

• By scaling, you multiply each point of an object by a factor which effectively scaling the object about the origin.


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Scaling (Cont.)

• If the center of the object is not at the origin, the operation will move the object away from the origin of the frame.


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Scaling (Cont.)

• For the case of scaling about other points, we need to combine the scaling transformation with two translation transformations in order to correctly get a transformation matrix.


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• performed about an axis which is usually specified by a point P and a vector direction .

Rotation

• Rotating about a point in two dimensions

Rotation matrix


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Rotation about the X-Axis

;xR


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Rotation about the Y-Axis

;yR


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Rotation about the Z-Axis

;zR


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Rotation about an arbitrary axis

• Assuming the axis of rotation is represented by and

then a rotation of degrees about this axis can be defined by concatenating the following transformations 1. Translate so that the point P moves to the origin


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Rotation about an arbitrary axis (Cont.)

2. Rotate the vector until it is in the yz ppppp pp ppppp p pppppppp

where


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3. Rotate the vector until it coincides with the x axis

where


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4. Rotate about the z axis.


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5. Use rotations and translations to reverse the first three steps


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The matrix representation of the general rotation is given by the product of the above transformations.


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Shearing

• Shearing transformations in three-dimensions alter two of the three coordinate values proportionally to the value of the third coordinate.


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X-Shearing

• X-shearing a frame: We ``x-shear'' a frame by modifying the first vector of the frame by adding to it a linear combination of the other two vectors.


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- X Shearing (Cont.)

• X-shearing a point

• The X-shear transformation matrix can be defined by


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Y-Shearing

• Y-shearing a frame


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- Y Shearing (Cont.)

• Y-shearing a point

• The Y-shear transformation matrix can be defined by


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Z-Shearing

• Z-shearing a frame


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- Z Shearing (Cont.)

• Z-shearing a point

• The Z-shear transformation matrix can be defined by

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Transformations

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