3. Transformation 2005. 4

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

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

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

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

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

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

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

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

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

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)

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

Rotations in the plane A rotation in the plane about origin Use a right-hand coordinate system Positive rotations  counterclockwise Fig. 3.10, eq. 3.33-35 Successive Rotation in the plane (eq. 3.47-49)

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)

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

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,

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

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

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

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

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)

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

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

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

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

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

Homogeneous Coordinates Translation Rotation (eq. 3.81-83) Scaling (eq. 3.84) Projection (chap. 18) translation scaling, rotation, reflection and shearing scaling factor perspective transform TR=RT ??

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

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

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

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')