Geometrical Transformations:

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Lecture 7

• Geometrical Transformations
• 2D transformations
• 3D transformations
• Matrix representation
• OpenGL functions

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Rendering Pipeline
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Rendering Pipeline
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Rendering Pipeline
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Rendering Pipeline
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Basic 2D Transformations
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Basic 2D Transformations
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Basic 2D Transformations
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Matrix representation
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Combination of 2D transformations
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2x2 matrices
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2x2 matrices
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2x2 matrices
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2x2 matrices
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2x2 matrices

• Only rotation, scaling, and shear can be
  represented in 2x2 matrix form.
• These are linear transformations.

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Linear transformations
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2D Translation 3x3 matrix
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Homogeneous coordinates
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Basic 2D transformations in homogeneous
coordinates
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Affine transformations
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Projections
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Matrix compositions
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Matrix compositions
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Matrix compositions
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3D transformations
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Basic 3D transformations
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Basic 3D transformations
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OpenGL Matrix Stacks
OpenGL works with stacks of 4x4
matrices glMatrixMode(enum mode) mode GL_MODELV
IEWGL_PROJECTIONGL_TEXTURE Specifies whether
the modelview, projection or texture matrix will
be modified.
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OpenGL Matrix Stacks
glLoadIdentity(void) Clears the currently
modifiable matrix for future transformation
instructions. Typically we call this instruction
before specifying modeling, viewing or projection
transformations.
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OpenGL Matrix Stacks
Manipulating stacks the OpenGL calls to push,
pop, or multiply top of stack glLoadMatrix(void)
glMultMatrix(void) glPushMatrix(void) the
topmost matrix is copied glPopMatrix(void) the
topmost matrix is destroyed All vertices of the
object are multiplied by the matrix.
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OpenGL Matrix Stacks
Manipulating stacks the OpenGL calls to push,
pop, or multiply top of stack glLoadMatrix(const
type m) glMultMatrix(const type m) m1 m5
m9 m13 m2 m6 m10 m14 m3 m7 m11 m15 m4 m8
m12 m16
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OpenGL Matrix Stacks
Manipulating stacks the OpenGL calls to push,
pop, or multiply top of stack glPushMatrix(void)
the topmost matrix is copied glPopMatrix(void)
the topmost matrix is destroyed All vertices of
the object are multiplied by the matrix.
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OpenGL Transformations
glTranslate(TYPE x, TYPE y, TYPE z) Multiplies
the current matrix by a translation
matrix. glRotate(TYPE angle, TYPE x, TYPE y, TYPE
z) Multiplies the current matrix by a rotation
matrix. glScale(TYPE x, TYPE y, TYPE
z) Multiplies the current matrix by a scaling
matrix.
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Summary

• Types of transformations
• linear
• affine
• projective
• Representation of transformations
• 3x3 and 4x4 matrices
• homogeneous coordinates
• Compositions of transformations
• OpenGL Matrix Instructions