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Chapter 4 2 - D Transformations Cartesian coordinates PowerPoint PPT Presentation


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Chapter 4 2 - D Transformations Cartesian coordinates Linear transformations, affine transformations Homogeneous coordinates Translations Scaling Rotation Reflection Combinations of transformations Animations. ( x, y ). y. x. ( x, y ). y. x. Cartesian coordinates

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Chapter 4 2 - D Transformations Cartesian coordinates

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Chapter 4 2 d transformations cartesian coordinates

Chapter 4

2-D Transformations

  • Cartesian coordinates

  • Linear transformations, affine transformations

  • Homogeneous coordinates

  • Translations

  • Scaling

  • Rotation

  • Reflection

  • Combinations of transformations

  • Animations


Chapter 4 2 d transformations cartesian coordinates

(x, y)

y

x

(x, y)

y

x

Cartesian coordinates

  • A 2D point is represented by a 2-tuple . For convenience, the 2-tuple often written as (x, y).

  • A 2-tuple can also be interpreted as a vector. The 2-tuple defines the direction and length of the vector, but not the position.


Chapter 4 2 d transformations cartesian coordinates

(3,13)

(7,7)

Translation

  • A translation moves a point to a new position by adding a displacement vector to it.

    Eg, The following translates the point to


Chapter 4 2 d transformations cartesian coordinates

(3,13)

(2,3)

Linear Transformations

  • A linear transformation moves a point to a new position by multiplying it with a non-singularmatrix.

    Eg, The following transforms the point (2, 3) to (3, 13)

A non-singular matrix, M, has the inverse M-1 such that MM-1 =M-1M= I. It denotes an one-to-one mapping.


Affine transformations

Affine Transformations

  • An affine transformation is a linear transformation followed by a translation. Its 2D general form is

  • Affine transformations preserve lines.

  • Many geometric movements of objects, eg, translations, rotations, and scalings are affine transformations.


Chapter 4 2 d transformations cartesian coordinates

y

(x/w, y/w, 1)

x

w = 1

Homogeneous coordinates

  • A point in homogeneous coordinates (x, y, w), w ≠ 0, corresponds to the 2D point (x/w, y/w) in Cartesian coordinates.

  • Conceive that the Cartesian coordinates axes lies on the plane of w = 1. The intersection of the plane and the line connecting the origin and (x, y, w) gives the corresponding Cartesian coordinates.

w

(x, y, w)

y

x

w = 0


Chapter 4 2 d transformations cartesian coordinates

  • For example, both the points (6, 9, 3) and (4, 6, 2) in the homogeneous coordinates corresponds to (2, 3) in the Cartesian coordinates.

    Conversely, the point (2, 1) of the Cartesian corresponds to (2, 1, 1), (4, 2, 2) or (6, 3, 3) of the homogeneous

  • Homogeneous coordinates are used in many graphics systems because all practical transformations can be expressed as matrix multiplications. Such uniformity enhances efficiency.


Chapter 4 2 d transformations cartesian coordinates

(x’, y’)

(3,4)

(1,3)

(x,y)

(7,2)

(5,1)

(3,2)

(1,1)

  • Translation

    move an object to a new position

  • The coord. of the new position: x’ = x + 2, y’ = y + 1

    In general, x’ = x + Tx, y’ = y + Ty

  • In homogeneous coordinates


Chapter 4 2 d transformations cartesian coordinates

(x’, y’)

(1,3)

(x,y)

(2,3/2)

(10,1/2)

(2,1/2)

(5,1)

(1,1)

  • Scaling relative to the origin (changing size)

  • The new position: x’ = 2x, y’ = y/2

    In general, x’ = Sx  x, y’ = Sy  y

  • In homogeneous coordinates


Chapter 4 2 d transformations cartesian coordinates

(x’,y’)

r

r

(x,y)

x=r cos

The angle, , is measured from the original vector to the new one in counterclockwise.

The parameters of sin() and cos() are provided in radians.

 radians is 180o.

  • Rotation about the origin

  • In Cartesian, x’ = r cos(+)

    = r cos cos  r sin sin

    = x cos - y sin

    y’ = r sin(+)

    = r sin cos + r cos sin

    = x sin + y cos


Chapter 4 2 d transformations cartesian coordinates

(2.8, 4.2)

(-1.4, 2.8)

(1,3)

(x’, y’)

(x,y)

(0,1.4)

(5,1)

(1,1)

  • Eg, to rotate 45o (/4 radian)

  • sin /4 = cos /4 = 0.7071

    x’ = x cos - y sin = 0.7071 (x - y)

    y’ = x sin + y cos = 0.7071 (x + y)

  • In homogeneous coordinates


Chapter 4 2 d transformations cartesian coordinates

(1,3)

(2.8,2.8)

(0,2)

(-1.4,1.4)

(5,1)

(4,0)

(0,0)

(1,1)

  • Conduct a sequence of transformations

    • Translate the right-angle vertex

      to the origin (Tx = -1, Ty = -1)

    • Rotate 45o (/4 radian)

      sin /4 = cos /4 = 0.7071


Chapter 4 2 d transformations cartesian coordinates

  • The computation of [ ] from [ ] [ ] is called matrix multiplication. The general form is:

  • A sequence of transformations can be lumped in a single matrix via matrix multiplications


Chapter 4 2 d transformations cartesian coordinates

Vertices

Vertices

CTM

Stack

  • In OpenGL, all the model transformations are accumulated in the current transformation matrix (CTM). All vertices of an object will be transformed via this matrix before the object is drawn.

  • A system stack is provided for storing the backup copies of the CTM during execution.

  • We usually save the CTM in the stack before the drawing of a transformed object. And restore the original CTM afterwards.


Chapter 4 2 d transformations cartesian coordinates

Vertices

Vertices

CTM

.

.

.

Stack

Vertices

Vertices

CTM

.

.

.

Stack

(before)

  • To save a copy of the CTM in the stack

    glPushMatrix();

(after)


Chapter 4 2 d transformations cartesian coordinates

Vertices

Vertices

CTM

. . .

Stack

(before)

  • To overwrite the CTM with the top matrix in the stack

    glPopMatrix();

Vertices

Vertices

CTM

. . .

Stack

(after)


Chapter 4 2 d transformations cartesian coordinates

Vertices

Vertices

CTM

Vertices

Vertices

CTM

  • To specify a translation

    glTranslatef( double Tx, double Ty, 0.0)

  • The system first generates the matrix representing the translation. Then post multiplies this matrix with the CTM. Finally overwrites the CTM with the result.

Before

After


Chapter 4 2 d transformations cartesian coordinates

  • Example

    //draw a white head at (0,0)

    glColor3f( 1.0, 1.0, 1.0);

    draw_head();

    //draw a green head at (-2,-1)

    glColor3f( 0.0, 1.0, 0.0);

    glPushMatrix();

    glTranslatef( -2.0, -1.0, 0.0);

    draw_head();

    glPopMatrix();

    glColor3f( 1.0, 1.0, 1.0); //draw a title

    hkgluBitMapString( -1.9, 1.8, "Translation of (-2, -1)” );

Progressive Translation


Chapter 4 2 d transformations cartesian coordinates

  • To specify a scaling

    glScalef( double Sx, double Sy, 1.0)

    Generate a matrix for the

    scaling. Post multiply it with

    the CTM

    //Scale by Sx = 1.5, Sy = 2

    glPushMatrix();

    glScalef( 1.5, 2.0, 1.0);

    draw_head();

    glPopMatrix();

In general, an object moves away from the origin when scaled up, moves towards when scaled down

Progressive Scaling


Chapter 4 2 d transformations cartesian coordinates

  • To specify a rotation

    glRotatef( double degree, 0.0, 0.0, 1.0)

    glPushMatrix();

    glRotatef( 45.0, 0.0, 0.0, 1.0);

    draw_head();

    glPopMatrix();

    (Note that the angles provided

    to gl functions are expressed in

    degrees instead of radians.)

Progressive Rotation


Chapter 4 2 d transformations cartesian coordinates

Vertices

Vertices

CTM

  • First to scale( Sx =1.5, Sy =2) and then translate the picture to (-2, -1).

  • Note that the transformations are specified in reverse order: first call glTranslatef(...), and then glScalef(…)


Chapter 4 2 d transformations cartesian coordinates

  • The program

    glPushMatrix();

    glTranslatef( -2.0, -1.0, 0.0);

    glScalef( 1.5, 2.0, 1.0);

    draw_head();

    glPopMatrix();

Progressive (Scaling + Translation)


Chapter 4 2 d transformations cartesian coordinates

(xf, yf)

  • Scaling relative to a fix point

    //Scaling relative to the

    //apex of the nose at (xf, yf)

    glPushMatrix();

    glTranslatef( xf, yf, 0.0);

    glScalef( sx, sy, 1);

    glTranslatef( -xf, -yf, 0);

    draw_head();

    glPopMatrix();

Scaling relative to a fix point


Chapter 4 2 d transformations cartesian coordinates

(xf, yf)

  • Rotation about a pivot point

    //Rotate 90 degree about

    //the apex of the nose

    glPushMatrix();

    glTranslatef( xf, yf, 0.0);

    glRotatef( 90., 0.0, 0.0, 1.0);

    glTranslatef( -xf, -yf, 0);

    draw_head();

    glPopMatrix();

Rotation about a pivot point


Chapter 4 2 d transformations cartesian coordinates

(x,y)

(x,-y)

  • Reflection about the x axis

  • The transformation matrix is the same as scaling matrix with Sx = 1 and Sy = 1. Thus, the reflection about the x axis can be achieved by calling

    glScalef( 1.0, -1.0, 1.0)

  • Similarly, the reflection about the y axis is achieved by calling

    glScalef( -1.0, 1.0, 1.0)


Chapter 4 2 d transformations cartesian coordinates

  • Example

    glPushMatrix();

    glScalef( 1, -1, 1);

    draw_head();

    glPopMatrix();

Reflection about x-axis


Chapter 4 2 d transformations cartesian coordinates

y =mx+b

  • Reflection along a line

    • Translate (0, -b) so that the line

      passes through the origin

    • Rotate the line onto the x axis by -o

    • Reflect about the x axis

    • Backward rotate

    • backward translate

      (Be reminded that these operations must be specified in reverse order.)

b


Chapter 4 2 d transformations cartesian coordinates

// To draw the reflection of the head about y = 2x + 0.5

// (we need to convert theta from radians to degrees)

double m = 2.0, b = .5, theta = atan(m)*180.0/3.1416;

glPushMatrix();

glTranslatef( 0, b, 0.);

glRotatef( theta, 0.0, 0.0, 1.0);

glScalef( 1, -1, 1); //Reflect

glRotatef( -theta, 0.0, 0.0, 1.0);

glTranslatef( 0, -b, 0.);

draw_head();

glPopMatrix();

Reflection against an arbitrary line


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