1. 3. Transformation

2. Transformations Understanding the shapes of objects and the figures
Properties that could measure
Ex) angles, distance, area, and volume
Logical relationships among shapes
Ex) equivalence, similarity, constructability
Geometric transformations
Translations, rotations, reflection, scaling, and combination
Invariant(=unchanged) properties

3. The Geometries of Transformations
The Geometries of Transformations (Fig. 3.1)
Congruent geometry
Identical size and shape
Translation and rotation
Conformal(=similar) geometry
Size is different, but corresponding angles are equal
Affine geometry
Parallel lines remain parallel (parallelism)
Angles are not preserved
Projective geometry
Parallel lines are not preserved
Straight lines remain straight
Topological geometry (=topology)
Connectivity and order are preserved
Euclidean Geometry

4. The Geometries of Transformations
Invariant Geometric properties
Fixed point
If a transformation maps it onto itself
Collineation (fig. 3.2)
A transformation preserves lines ( T(L) = L’ )
Distance-preserving transformations
Preserve angles, area, and volume
Ex) Translation and rotation
Conformal transformations (fig. 3.3)
Preserve the angle measures of a figure but not distance (size)
Orientation (fig. 3.4)
Orientation-preserving transformations (=direct transformations)
Orientation-reversing transformations (=opposite transformations)
Ex) Reflection
Clockwise or Counterclockwise

5. The Geometries of Transformations
Isometries
Isometry
from the Greek words isos and metron meaning equal measure
A rigid-body motion a figure
Algebraic equations in the plane (not including translation)
If determinant is equal to -1  orientation reversing
Ex) Reflection, Rotation, Translation, Glide reflection

6. The Geometries of Transformations
Similarities (fig. 3.5)
Similarity transformation (=conformal transformation)
Change the size of a figure but does not change its shape
A combination of a scaling transformation and an isometry
k>0 for a direct similarity
k<0 for an opposite similarity
The term k is the ration of expansion or contraction

7. The Geometries of Transformations
Affinities (fig. 3.6)
Affine transformation
Collinearity and parallelism are important invariant properties

8. The Geometries of Transformations
Transformation in Three Dimensions
Algebraic equations in 3D (not including translation)
Algebraic equations in 3D (including translation)

9. Affine Transformations
Affine Transformation of a Line
The equation of a line in the plane
Straight lines transform into straight line under affine transformation

10. Matrix Form
Matrix form P’ A’ P
P’=AP  P=A-1P’

11. Translation Rigid body transformation Translation of a point
Translation and rotation
Translation of a point
including translation
Not homogeneous equations
(not square matrix, no multiplicative inverse)

12. Translation Two points Vectors and Translations
Translation that takes P1(x1, y1) into P2 (x2, y2)
Vectors and Translations
A Vector Translation by another vector
Translate the straight Line
Succession of translation t1, t2, …, tn

13. Rotations in the plane A rotation in the plane about origin
Use a right-hand coordinate system
Positive rotations  counterclockwise
Fig. 3.10, eq
Successive Rotation in the plane (eq )

14. Rotations in the plane Rotation about an Arbitrary point (fig. 3.11)
Translate pc to the origin 
Rotate the results about the origin 
Reverse step 1 
Rotation of the Coordinate System
Oppositely directed rotation of points in the original system (fig. 3.12, eq. 3.51)

15. Rotations in space The simplest rotations in space
Rotations about the principal axes
Right-hand rule (fig. 3.13)
Look toward the origin from a point on the axis
Positive rotation : counterclockwise z y x

16. Rotations in space Successive Rotations
Different orders produce very different results
Example
Perform the rotation about the z axis,
Perform the rotation about the y axis,
Perform the rotation about the x axis,

17. Rotations in space Rotation of the Coordinate System (eq. 3.60-62)
Rotate the x, y, z system of axes through about the z axis
This moves the x axis to x1 and y axis to y1
The z1 axis is identical to the z axis
Rotate the x1, y1, z1 system of axes through about the y1 axis
This moves the x1 axis to x2 and z1 axis to z2
The y2 axis is identical to the y1 axis
Rotate the x2, y2, z2 system of axes through about the x2 axis
This moves the y2 axis to y3 and z2 axis to z3
The x3 axis is identical to the x2 axis
z, z1
z, z1
z, z1 z2 z2 y3 z3 y1
y1, y2
y1, y2 y y y x1 x1 x1 x x x
Step 1
Step 2 x2
Step 3
x2, x3

18. Rotations in space Rotation about an Arbitrary Axis
Rotate about some arbitrary a through angle
Rotate about the z axis through an angle
The axis a lies in the yz plane
Rotate about the x axis through an angle
The axis a coincident with the z axis
Rotate about the a axis through
Same as a rotation about the now coincident z axis
Reverse the second rotation about the x axis
Reverse the first rotation about the z axis
Returning the axis a to its original position z a y x

19. Rotations in space Equivalent Rotations (fig. 3.16)
Euler (in 1972) proved
No matter how many times we twist and turn an object with respect to a reference position, we can always reach every possible new position with a single equivalent rotation

20. Reflection Reflection
A reflection in the plane reverses the orientation of a figure (fig. 3.18)
If a line is parallel to the axis of refection, then so is its image
If a line is perpendicular to the axis of reflection, then so is its image
A reflection is an isometry, preserving distance, angle, parallelness, perpendicularity, betweenness, and midpoints

21. Reflection Inversion in a point in the plane (fig. 3.19)
Preserve orientation
Inversion(=half turn)
Identical to the image produced if we rotate it 180 about P
Two successive half-turns about the same point
Identity transformation
Two successive half-turns about two different points
Produces a translation (Fig. 3.20)
Two Reflections in the plane
Equivalent to a rotation about the point of intersection of the lines and through an angle equal to twice the angle between lines (Fig. 3.21)

22. Reflection
Transformation Equations for inversion and Reflections in the plane
Inversion in the origin (fig.3.22)
x’=-x y’=-y
Inversion in an arbitrary point (a, b)
x’=-x+2a y’=-y+2b
Reflection in x or y axis (fig 3.23)
x’=x y’=-y (x-axis)
x’=-x y’=y (y-axis)
Reflection in an arbitrary line through the origin (fig. 3.24)
X’=xcos2a + y sin2a
Y’=xsin2a – ycos2a

23. Reflection
Transformation Equations for inversion and Reflections in space
Inversion through the origin (fig.3.25)
x’=-x y’=-y z’=-z
Inversion in an arbitrary point (a, b, c)
x’=-x+2a y’=-y+2b z’=-z+2c
Reflection across the xy plane (z=0 plane) (fig. 3.26)
x’=x y’=y z’=-z

24. Homogeneous Coordinates
A translation destroys the homogeneous character of the transformation equations
Homogeneous Coordinates
The coordinates of points in projective space
Allow us to combine several transformation into one matrix (include translation)
Reduce to a series of matrix multiplication
No matrix addition

25. Homogeneous Coordinates
2D Translation represented by a 3 x 3 matrix
Point represented with homogeneous coordinate

26. Homogeneous Coordinates
Add a 3rd coordinate to every 2D point
(x, y, w) represents a point at location (x/w, y/w)
(x, y, 0) represents a point at infinity
(0, 0, 0) is not allowed

27. Homogeneous Coordinates
Translation
Rotation (eq )
Scaling (eq. 3.84)
Projection (chap. 18)
translation
scaling,
rotation,
reflection and
shearing
scaling factor
perspective
transform
TR=RT ??

28. Linear Transformations
Linear transformations are combinations of …
Scale
Rotation
Shear
Mirror (=Reflection)
Properties of linear transformations
Satisfies:
Origin maps to origin
Lines map to lines
Parallel line remain parallel
Ratios are preserved
Closed under composition

29. Affine Transformations
Affine Transformations are combinations of …
Linear transformation
Translation
Properties of affine transformations
Origin does not necessarily map to origin
Lines map to lines
Parallel line remain parallel
Ratios are preserved
Closed under composition

30. Projective Transformations
Affine transformation
Projective warps
Properties of projective transformations
Origin does not necessarily map to origin
Lines map to lines
Parallel line do not necessarily remain parallel
Ratios are not preserved (but “cross-ratios” are)
Closed under composition

31. Cross-ratio
Given any four points A, B, C, and D, taken in order, define their cross ratio as
Cross Ratio (ABCD) =
cross ratio (ABCD) = cross ratio (A'B'C'D')