Two-Dimensional Geometric Transformations ch5. 참조 Subjects : Basic Transformations Homogeneous Coordinates Composite Transformations Other Transformations Properties of Transformation matrix

Basic Transformations : Translation (1/2) Translation ( 이동 )  Definition : repositioning objects along a straight line path from one position to another  Let : original position, : new position  We need translation distance (translation vector) for x direction for y direction  Then,

Translation (2/2)  In matrix form  Translation is rigid-body Transformation that moves objects without deformation every point on the object is translated by the same amount  For straight line applying translation distance to each line end points  For polygon, curves

Basic Transformations : Rotation (1/3) Rotation  Definition : repositioning objects along a circular path in the xy plane  We need to specify rotation angle rotation point (pivot point) direction (clockwise (-), counter clockwise(+))  a rotation about a rotation axis Z

Rotation (2/3) Rotation angle : rotation point : origin (0,0) direction : c.c.w  Then where original points are so,

Rotation (3/3)  In matrix form called Rotation matrix  Rotation about an arbitrary pivot position  As with translations, rotations are rigid-body transformations

Basic Transformations : scaling (1/2) Scaling  Definition : alters the size of an objects  we need scaling factors : for x value : for y value  In matrix form  Then  if : Uniform scaling : differential scaling

scaling (2/2)  if : uniform compression move objects to the coordinate origin  if : uniform Enlargement move objects farther from the origin  Fixed point scaling : scaling based on a fixed point An object is scaled relative to the fixed point by scaling distance from each vertex to fixed point when

Homogeneous Coordinates (1/4)  Let’s consider combination of two transformation, translation after rotation  We can combine two matrices into a single matrix by expanding the 2x2 matrix to 3x3 matrix i.e.

Homogeneous Coordinates (2/4)  To utilize above 3x3 matrix, represent Cartesian coordinate point (x,y) with the Homogeneous Coordinate (x h, y h, h) where  Homogeneous Coordinate  Homogeneous Coordinate : (x h, y h, h) first developed in geometry (1946) applied in graphics by Roberts (1965) usually h=1 (called weight value) then (2,3) = (2,3,1) = (1,1.5,0.5)

Homogeneous Coordinates (3/4) there is infinite number of equivalent representation for point (x,y) origin case (0,0,1) for (x h, y h, h), at least one of triple must be nonzero, i.e. (0,0,0) is not allowed if (x,y,0) called point at infinity  For Translation, Rotation, Scaling Translation:Scaling: Rotation:

Homogeneous Coordinates (4/4)  with the H.G.C we can perform uniform scaling with scaling factor S when S>1 : Uniform compression 0<S<1 : Uniform enlargement  with the H.G.C the transformation matrix be 3x3, i.e. m, n : translation factor s : scaling factor

Composite Transformation (1/2) Translation  If two successive translation factor (t x1, t y1 ) and (t x2, t y2 ) are applied to a coordinate point P then ex) i.e,  Two successive Translation are additive

Composite Transformation (2/2) Rotation  two successive rotation  two successive rotations are also additive Scaling  successive scalings are multiplicative

General pivot-point Rotation (1/2) Rotation about arbitrary point p (x r, y r )  step1) Translate P to origin  step2) Rotation about origin  step3) Retranslation to position P.

General pivot-point Rotation (2/2)  Composite transformation matrix

General Fixed-Point Scaling Scaling about arbitrary point p(x r, y r )  step 1) translate p to origin  step 2) scaling about origin  step 3) Retranslate to position P

General Scaling Directions  Normal scaling performed along x and y directions  To accomplish the scaling for arbitrary direction without changing object orientation Rotate direction by Perform scaling Rerotate by

Concatenation properties.  Matrix multiplication : associative ex) For three matrices A, B and C Translation or Rotation : additive property commutative scaling : multiplicative property commutative  However, Translation and Rotation : non commutative order of transformation matrix multiplication is important

Other Transformations : Reflection (1/4) Reflection : produce a mirror images of an object  We need axis of reflection  rotating the objects 180  about reflection axis  Reflection about line y=0, the x axis. x coordinate values are unchanged the transformation matrix

Reflection (2/4)  Reflection about line x=0, the y axis y coordinate values are unchanged  Reflection about origin both x and y values are changed

Reflection (3/4)  Reflection through arbitrary point P (x r, y r ) translate point p to origin : Tr perform reflection about origin : Ro retranslate to original position : Tr  Reflection through y=x line

Reflection (4/4)  Reflection through an arbitrary line y=Lx+b

Other Transformations: Shear (1/3) Shear : distorts the shape of an object  Shearing (slide over) can be done either x or y direction  x direction shearing by shearing factor sh x point is shifted by horizontally by an amount proportional to its distance from x-axis y values are unchanged

Shear (2/3) So, Transformed positions  y direction shearing  x-direction shearing relative to line

Shear (3/3)  y-direction shearing relative to line

Properties of Transformation matrix (1/5) Properties of Rotation matrix  case1) rotate point p p’ c.c.w direction by  case2) rotate point p’ p c.w. direction by same result to rotate ( ) to c.c.w direction is a Inverse matrix of i.e,

Properties of Transformation matrix (2/5)  Examine (A) & (B) then Inverse matrix = Transpose matrix of R I.e,  So, the inverse of the general rotation matrix [R] is its transpose  Def) An nxn matrix A is an orthogonal matrix if  Def) Every 2x2 orthogonal matrix R with det[R]=1 is pure rotation matrix

Properties of Transformation matrix (3/5) Properties of Reflection matrix  Determinant of Reflection matrix = -1  Def) Every 2x2 orthogonal matrix R with det(R) = -1 is a pure reflection matrix  If two pure reflections about line passing through the origin are applied successively, the result is a pure rotation about the origin ex) reflection through x axis (R X ) and reflection through y axis (R Y )

Properties of Transformation matrix (4/5) Rigid-body Transformation  concept : moves object without deformation i.e, perpendicular lines transformed as perpendicular lines or unit square remains a unit square  Translation, rotation, or combination of both

Properties of Transformation matrix (5/5) Affine Transformation  Concepts : parallel lines transformed as parallel lines finite points maps to finite points but not length and angle Rot, Trans, Ref preserves angle and length Tra, Rot, Sca, Ref, and Shr or combination of those