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Transformations. Jehee Lee Seoul National University. Transformations. Local moving frame. Linear transformations Rigid transformations Affine transformations Projective transformations. T. Global reference frame. Linear Transformations.

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Transformations l.jpg

Transformations

Jehee Lee

Seoul National University


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Transformations

Local moving frame

  • Linear transformations

  • Rigid transformations

  • Affine transformations

  • Projective transformations

T

Global reference frame


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Linear Transformations

  • A linear transformationT is a mapping between vector spaces

    • T maps vectors to vectors

    • linear combination is invariant under T

  • In 3-space, T can be represented by a 3x3 matrix


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Examples of Linear Transformations

  • 2D rotation

  • 2D scaling


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Examples of Linear Transformations

  • 2D shear

    • Along X-axis

    • Along Y-axis


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Examples of Linear Transformations

  • 2D reflection

    • Along X-axis

    • Along Y-axis


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Properties of Linear Transformations

  • Any linear transformation between 3D spaces can be represented by a 3x3 matrix

  • Any linear transformation between 3D spaces can be represented as a combination of rotation, shear, and scaling

  • Rotation can be represented as a combination of scaling and shear

    • Is this combination unique ?


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Properties of Linear Transformations

  • Rotation can be computed as two-pass shear transformations


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Changing Bases

  • Linear transformations as a change of bases


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Changing Bases

  • Linear transformations as a change of bases


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Affine Transformations

  • An affine transformationT is a mapping between affine spaces

    • T maps vectors to vectors, and points to points

    • T is a linear transformation on vectors

    • affine combination is invariant under T

  • In 3-space,T can be represented by a 3x3 matrix together with a 3x1 translation vector


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Properties of Affine Transformations

  • Any affine transformation between 3D spaces can be represented by a 4x4 matrix

  • Any affine transformation between 3D spaces can be represented as a combination of a linear transformation followed by translation


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Properties of Affine Transformations

  • An affine transf. maps lines to lines

  • An affine transf. maps parallel lines to parallel lines

  • An affine transf. preserves ratios of distance along a line

  • An affine transf. does not preserve absolute distances and angles


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Examples of Affine Transformations

  • 2D rotation

  • 2D scaling


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Examples of Affine Transformations

  • 2D shear

  • 2D translation


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Examples of Affine Transformations

  • 2D transformation for vectors

    • Translation is simply ignored


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Examples of Affine Transformations

  • 3D rotation

How can we rotate about an arbitrary line ?


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Pivot-Point Rotation

  • Rotation with respect to a pivot point (x,y)


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Fixed-Point Scaling

  • Scaling with respect to a fixed point (x,y)


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Scaling Direction

  • Scaling along an arbitrary axis


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Composite Transformations

  • Composite 2D Translation


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Composite Transformations

  • Composite 2D Scaling


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Composite Transformations

  • Composite 2D Rotation


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Composing Transformations

  • Suppose we want,

  • We have to compose two transformations


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Composing Transformations

  • Matrix multiplication is not commutative

Translation

followed by

rotation

Translation

followed by

rotation


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Composing Transformations

  • R-to-L : interpret operations w.r.t. fixed coordinates

  • L-to-R : interpret operations w.r.t local coordinates

(Column major convention)


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Review of Affine Frames

  • A frame is defined as a set of vectors {vi | i=1, …, N} and a point o

    • Set of vectors {vi} are bases of the associate vector space

    • o is an origin of the frame

    • N is the dimension of the affine space

    • Any point p can be written as

    • Any vector v can be written as


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Changing Frames

  • Affine transformations as a change of frame


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Changing Frames

  • Affine transformations as a change of frame


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Rigid Transformations

  • A rigid transformationT is a mapping between affine spaces

    • T maps vectors to vectors, and points to points

    • T preserves distances between all points

    • T preserves cross product for all vectors (to avoid reflection)

  • In 3-spaces, T can be represented as


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Rigid Body Transformations

  • Rigid body transformations allow only rotation and translation

  • Rotation matrices form SO(3)

    • Special orthogonal group

(Distance preserving)

(No reflection)


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Summary

  • Linear transformations

    • 3x3 matrix

    • Rotation + scaling + shear

  • Rigid transformations

    • SO(3) for rotation

    • 3D vector for translation

  • Affine transformation

    • 3x3 matrix + 3D vector or 4x4 homogenous matrix

    • Linear transformation + translation

  • Projective transformation

    • 4x4 matrix

    • Affine transformation + perspective projection


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Questions

  • What is the composition of linear/affine/rigid transformations ?

  • What is the linear (or affine) combination of linear (or affine) transformations ?

  • What is the linear (or affine) combination of rigid transformations ?


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