Computer Graphics

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Computer Graphics

• Chapter 5
• Geometric Transformations

Andreas Savva
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2D Translation

• Repositioning an object along a straight line
  path from one co-ordinate location to another
• (x,y) (x,y)
• To translate a 2D position, we add
  translation distances tx and ty to the original
  coordinates (x,y) to obtain the new coordinate
  position (x,y)
• x x tx , y y ty

Matrix form
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2D Translation

• Moving a polygon from position (a) to position
  (b) with the translation vector (-5, 10), i.e.

(a)
(b)
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Translating a Polygon

• class Point2D
• public
• GLfloat x, y
• void translatePoly(Point2D P, GLint n,
• GLfloat tx, GLfloat ty)
• GLint i
• for (i0 iltn i)
• Pi.x Pi.x tx
• Pi.y Pi.y ty
• glBegin (GL_POLYGON)
• for (i0 iltn i)
• glVertex2f(Pi.x, Pi.y)
• glEnd()

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2D Rotation

• Repositioning an object along a circular path in
  the xy-plane

The original coordinates are
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2D Rotation

• Substituting

Matrix form
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2D Rotation about a Pivot position

• Rotating about pivot position (xr, yr)

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Translating a Polygon

• class Point2D
• public
• GLfloat x, y
• void rotatePoly(Point2D P, GLint n,
• Point2D pivot, GLdouble theta)
• Point2D V
• V new Point2Dn
• GLint i
• for (i0 iltn i)
• Vi.x pivot.x (Pi.x pivot.x)
  cos(theta)
• - (Pi.y pivot.y)
  sin(theta)
• Vi.y pivot.y (Pi.x pivot.x)
  sin(theta)
• - (Pi.y pivot.y)
  cos(theta)
• glBegin (GL_POLYGON)
• for (i0 iltn i
• glVertex2f(Vi.x, Vi.y)

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2D Scaling

• Altering the size of an object. Sx and Sy are the
  scaling factors. If Sx Sy then uniform scaling.

Matrix form
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2D Scaling relative to Fixed point

• Scaling relative to fixed point (xf, yf)

OR
where the additive terms xf(1-Sx) and yf(1-Sy)
are constants for all points in the object.
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Translating a Polygon

• class Point2D
• public
• GLfloat x, y
• void scalePoly(Point2D P, GLint n, Point2D
  fixedPt,
• GLfloat Sx, GLfloat Sy)
• Point2D V
• V new Point2Dn
• GLfloat addx fixedPt.x (1 Sx)
• GLfloat addy fixedPt.y (1 Sy)
• GLint i
• for (i0 iltn i)
• Vi.x Pi.x Sx addx
• Vi.y Pi.y Sy addy
• glBegin (GL_POLYGON)
• for (i0 iltn i
• glVertex2f(Vi.x, Vi.y)

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Matrix Representation

• Use 33 matrices to combine transformations
• Translation
• Rotation
• Scaling

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Inverse Transformations

• Translation
• Rotation
• Scaling

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Example

• Consider the line with endpoints (10, 10) and
  (30, 25). Translate it by tx -20, ty -10 and
  then rotate it by ? 90º.

Right-to-left
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Solution
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Solution (continue)
Point (10, 10)
Point (30, 25)
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Result
Step-by-step
T(-20, -10)
R(90º)
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Exercises

• Consider the following object
• Apply a rotation by 145º then scale it by Sx2
  and Sy1.5 and then translate it by tx20 and
  ty-30.
• Scale it by Sx½ and Sy2 and then rotate it by
  30º.
• Apply a rotation by 90º and then another rotation
  by 45º.
• Apply a rotation by 135º.

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Exercises

• Composite 2D Transformations
• Translation Show that
• Rotation Show that
• Scaling Show that

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General 2D Pivot-Point Rotation
Original position and Pivot Point
Translate Object so that Pivot Point is at origin
Rotation about origin
Translate object so that Pivot Point is return to
position (xr , yr)
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General Pivot-point Rotation
Using Matrices
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Exercises

• Consider the following object
• Apply a rotation by 60 on the Pivot Point (-10,
  10) and display it.
• Apply a rotation by 30 on the Pivot Point (45,
  10) and display it.
• Apply a rotation by 270 on the Pivot Point (10,
  0) and then translate it by tx -20 and ty 5.
  Display the final result.

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General 2D Fixed-Point Scaling
Original position and Fixed Point
Translate Object so that Fixed Point is at origin
Scale Object with respect to origin
Translate Object so that Fixed Point is return to
position (xf , yf)
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General 2D Fixed-Point Scaling
Using Matrices
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Exercises

• Consider the following object
• Scale it by sx 2 and sy ½ relative to the
  fixed point (140, 125) and display it.
• Apply a rotation by 90 on the Pivot Point (50,
  60) and then scale it by sx sy 2 relative to
  the Fixed Point (0, 200). Display the result.
• Scale it sx sy ½ relative to the Fixed Point
  (50, 60) and then rotate it by 180 on the Pivot
  Point (50, 60). Display the final result.

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Order of Transformations
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Reflection
About the x axis
About the y axis
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Reflection
Relative to the coordinate origin
With respect to the line y x
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2D Shear

• x-direction shear

Matrix form
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2D Shear

• x-direction relative to other reference line

Matrix form
y
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2D Shear

• y-direction shear

Matrix form
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2D Shear

• y-direction relative to other reference line

Matrix form
shy ½, xref -1
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Transformations between 2D Coordinate Systems

• To translate object descriptions from xy
  coordinates to xy coordinates, we set up a
  transformation that superimposes the xy axes
  onto the xy axes. This is done in two steps
• Translate so that the origin (x0, y0) of the xy
  system is moved to the origin (0, 0) of the xy
  system.
• Rotate the x axis onto the x axis.

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Transformations between 2DCoordinate Systems

• i.e.
• 1)
• 2)
• Concatenating

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Example

• Find the xy-coordinates of the xy points (10,
  20) and (35, 20), as shown in the figure below

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(No Transcript)
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Exercise

• Find the xy-coordinates of the rectangle shown
  in the figure below

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3D Translation

• Repositioning an object along a straight line
  path from one co-ordinate location to another
• (x,y,z) (x,y,z)
• To translate a 3D position, we add
  translation distances tx ty and tz to the
  original coordinates (x,y,z) to obtain the new
  coordinate position (x,y)
• x x tx , y y ty , z z tz

Matrix form (4 4)
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3D Rotation

• z-axis
• The 2D z-axis rotation equations are extended to
  3D.

Matrix form
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3D Rotation

• x-axis

Matrix form
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3D Rotation

• y-axis

Matrix form
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3D Scaling
Matrix form
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Other 3D Transformations

• Reflection z-axis

• Shears z-axis