Cmpe 466 computer graphics. 2d geometric transformations. (Chapter 7)

Basic geometric transformations

Translation
Rotation
Scaling

2D translation

Figure 7-1 Translating a point from position P to position

2D translation equations

Translation is a rigid-body transformation: Objects are moved without
deformation.

2D translation example

Figure 7-2 Moving a polygon from position (a) to

2D translation example program

2D rotation

All points of the object are transformed to new positions

2D rotation

Figure 7-4 Rotation of a point from position (x, y

2D rotation in matrix form

Rotation is a rigid-body transformation: Objects are

Rotation about a general pivot point

Figure 7-5 Rotating a point from

2D rotation example

// Make necessary allocations!!

2D scaling

sx and sy are scaling factors
sx scales an object in

2D scaling

Figure 7-6 Turning a square (a) into a rectangle (b)

2D scaling

Figure 7-7 A line scaled with Equation 7-12 using sx

Scaling relative to a fixed point

Figure 7-8 Scaling relative to a

2D scaling relative to a fixed point

2D scaling example

// Make necessary allocations!!

Matrix representations and homogeneous coordinates

Multiplicative and translational terms for a 2D

2D translation matrix

2D rotation matrix

2D scaling matrix

Inverse transformations

Inverse translation

Inverse rotation

Inverse scaling

Composite transformations

Composite transformation matrix is formed by calculating the product of

Composite transformations

Successive rotations (additive)
Successive scaling (multiplicative)

2D pivot-point rotation

Figure 7-9 A transformation sequence for rotating an object

2D pivot-point rotation

Note the order of operations:

2D fixed-point scaling

Figure 7-10 A transformation sequence for scaling an object

2D fixed-point scaling

Matrix concatenation properties

Multiplication is associative
Multiplication is NOT commutative
Unless the sequence of

Computational efficiency

Formulation of a concatenated matrix may be more efficient
Requires fewer

Other transformations: reflection

Figure 7-16 Reflection of an object about the x

Reflection

Figure 7-17 Reflection of an object about the y axis.

Reflection

Figure 7-18 Reflection of an object relative to the coordinate origin.

Reflection

Figure 7-19 Reflection of an object relative to an axis perpendicular

Reflection

Figure 7-20 Reflection of an object with respect to the line

Other transformations: shear

Distorts the shape of an object such that the

Shear

Figure 7-24 A unit square (a) is transformed to a shifted

Shear

Figure 7-25 A unit square (a) is turned into a shifted

Transformations between 2D coordinate systems

Figure 7-30 A Cartesian x' y' system

Transformations

x’y’ system can be obtained by rotation of xy
by Θ counter-clockwise

xy

Transformations between coordinate systems

Alternative method

Figure 7-32 Cartesian system x' y' with origin at P0

Transformations

Figure 7-33 A Cartesian x ' y' system defined with two

Example: Rotating points vs. rotating coordinate systems

Consider the following transformation:
Rotation of

Example continued

You may think of this as mapping the origin and

Example continued

Now consider the point P=(x(2),y(2),1)T in System 2
What are the

OpenGL matrix operations

glMatrixMode ( GL_MODELVIEW )
Designates the matrix that is to

OpenGL transformation example

OpenGL transformation example

Figure 7-34 Translating a rectangle using the OpenGL function

OpenGL transformation example

Figure 7-35 Rotating a rectangle about the z axis