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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_MODELVIEWGL_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