COMPUTER GRAPHICS

501472-3
Lecture 6- 2D/3D transformation
Transformation
• Transformation means changing some graphics into
something else by applying rules.
• We can have various types of transformations such as
translation, scaling up or down, rotation, shearing, etc.
• Transformations play an important role in computer
graphics to reposition the graphics on the screen and
change their size or orientation.

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Why use transformations?

• Position objects in a scene (modeling)

• Change the shape of objects
• Create multiple copies of objects
• Projection for virtual cameras
• Animations

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Why Transformations?

■ In graphics, once we have an object described, transformations are used to move

that objects, scale it and rotate it.

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ObjectsTransformation
• 5-type of object Transformation:
• Translation or Shifting: moving objects
• Scaling: change size.
• Rotation: moving about a point by an angle.
• Reflection: mirror of an object.
• Shearing: distort the object (deformation).

Reflect Translate Scale Rotate Shear

TRANSLATION
Translation

■ Simply moves an object from position to another

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Translation

• The object is shifted or moved to a new coordinates.

• The shift is by (dx, dy).
• x’ = x + dx
• y’ = y + dy.
• In matrix forms: é x'ù é xù édxù
ê y'ú = ê yú + êdyú
ë û ë û ë û
Translation

• To translate a line, translate its end points.

• To translate a polygon, translate its vertices.
• To translate a circle or ellipse, its center coordinates are
transformed, then the object is drawn (boundary points)
using new coordinates.

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Translation Example 1

• Draw a square of length 2. One side of the square vertices'

coordinates are (0,0), (2,0), (2,2), (0,2), then translate it toward y-
axis by 1.
• Solution: x’=x, y’=y+1

• dx=0, dy=1. Old New

(0,3)
Coordinates Coordinates (2,3)
(0,0) (0,1)
(0,2) (2,2)
(2,0) (2,1)
(2,2) (2,3) (0,1) (2,1)
(0,2) (0,3)
(0,0)
(2,0)

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Translation Example 2:

■ A triangle with vertices coordinates: (1,0), (3,0), (2,2). What its new coordinates
after shifting it by 1 in the x-axis and y-axis.
■ Solution:
(3,3)
– x’= x + 1 and y’ = y+1.
(2,2)

Old New
Coordinate Coordinate (4,1)
(2,1)
s s
(1,0) (2,1) (1,0) (3,0)
(3,0) (4,1)
(2,2) (3,3)

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Translation In OpenGL
glTranslate[fd](Tx, Ty, Tz): It will shift the whole coordinates by Tx, Ty, Tz.
■ glTranslated() takes double-precision floating-point values.
■ glTranslatef() takes single-precision floating-point values.

glColor3ub(255, 0, 0);
glBegin(GL_LINE_LOOP);
glVertex2i(0,0); glVertex2i(2,0); glVertex2i(2,2);
glVertex2i(0,2);
glEnd();
glColor3ub(0, 0, 255);
glTranslatef(1.0,1.0,0); // Translate the coordinate system by
(1.0, 1.0, 0)
glBegin(GL_LINE_LOOP);
glVertex2i(0,0); glVertex2i(2,0); glVertex2i(2,2);
glVertex2i(0,2);
glEnd();
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SCALING
Scaling

• Scaling: increase/decrease size of object

• To scale an object, we do the following:
1. Shift object to point (0,0).
2. Scale by (sx, sy) using the equations: x’ = x*sx; y’ = y*sy;
3. Shift object back to its original location.

• In general, scaling equations are:

• x’ = xf + (x-xf)*sx; y’ = yf + (y-yf)*sy;
• Where (xf, yf) is the reference point.
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Scaling

• Scaling is called uniform if sx=sy, otherwise it is called differential.

• In matrix form:
x′ Sx 0 x
=
y′ 0 Sy y
■ To apply the scaling transformation to a point (x,y,1), we use matrix multiplication

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Scaling Example:

• A rectangle with coordinates (0,0), (4,0), (4,3), (0,3) what it is the

new coordinates if:
• Sx=Sy=2
• Sx=Sy=0.5 Old New
Coordinates Coordinates (8,6)
• Solution: (0,6)
(0,0) (0,0)
• Sx=Sy=2:
(4,0) (8,0) (4,3)
(0,3)
(4,3) (8,6)
(0,3) (0,6)

(0,0) (4,0) (8,0)

Scaling Example:

■ Sx=Sy=0.5:

Old New
Coordinates Coordinates
(0,0) (0,0)
(4,0) (2,0) (4,3)
(0,4)
(4,3) (2,1.5)
(0,3) (0,1.5) (0,1.5) (2,1.5)

(0,0) (2,0) (4,0)

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Scaling Example 2

■ A square with coordinates: (1,1), (2,1), (2,2) ,(1,2).

■ Scale it by Sx=Sy=2, with (xf,yf)=(1,1).

■ Solution:
Old New
◦ Sx=Sy=2:
Coordinates Coordinates
◦ Shift point (1,1) to (0,0)
(1,1) (1,1)
■ (Xf,yf) = (1,1).
(2,1) (3,1)
◦ x’ = xf + (x-xf)*sx;
(2,2) (3,3)
◦ y’ = yf + (y-yf)*sy
(1,2) (1,3)

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Scaling Example 2

(1,3) (3,3)
(1,2) (2,2) (1,2)
(2,2)

(1,1) (2,1) (3,1)

(1,1) (2,1)

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OpenGL Scaling

glScale[fd](Sx, Sy, Sz)

glBegin(GL_LINE_LOOP);
glVertex2i(0,0); glVertex2i(4,0); glVertex2i(4,4); glVertex2i(0,4);
glEnd();

glScaled(2,2,1); // Apply scaling with scaling factors (2, 2, 1)

glBegin(GL_LINE_LOOP);
glVertex2i(0,0); glVertex2i(4,0); glVertex2i(4,4); glVertex2i(0,4);
glEnd();

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OpenGL Scaling:
ROTATION
Rotation
• In linear algebra, a rotation matrix is used to perform
a rotation in Euclidean space.

• For example, rotating points in the xy plane

counterclockwise through an angle θ with respect to
the positive x axis about the origin of a 2D Cartesian
coordinate system.
• By means of the following matrix multiplication,

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Rotation

• Thus, the new coordinates (xʹ, yʹ) of a point (x, y) after a 2D rotation
around the origin (0,0) by an angle θ in the counterclockwise
direction is:
• x' = x cos θ - y sin θ ; y' = x sin θ + y cos θ

• The equation for rotation about a reference point (xf,yf) is:

(x’,y’)
• x' = x f + (x - x f ) cosq - ( y - y f ) sinq (x,y)
q
• y' = y f + (x - x f ) sinq + ( y - y f ) cosq
(xf,yf)
Rotation Example

• Consider a rectangle with coordinates (1,1), (3,1), (3,5), (1,5).

Rotate it 45o about the point (1,1).

Old Coordinates New Coordinates

• Solution:
(1,1) (1,1)
• (xf,yf)=(1,1), q = 45o.
(3,1) (2.4142, 2.4142)
• cos 45 = 0.71, sin 45 = 0.71,
(3,5) (-0.4142 5.2426)
• So, x’ = xf + (x-xf) * cos q - (y-yf)sin q (1,5) (-1.828, 3.828)
• And y’ = yf + (x-xf) * sin q + (y-yf)*cosq

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Rotation In Opengl

• glRotatef(angle, x, y, z).

• Rotation 45o about the z-axis:

• glRotatef(45, 0.0, 0.0, 1.0).

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Rotation In Opengl

glBegin(GL_LINE_LOOP);
glVertex2i(0,0); glVertex2i(4,0); glVertex2i(4,4); glVertex2i(0,4);
glEnd();
glRotatef(45, 0.0, 0.0, 1.0);
glBegin(GL_LINE_LOOP);
glVertex2i(0,0); glVertex2i(4,0); glVertex2i(4,4); glVertex2i(0,4);
glEnd();

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Rotation In Opengl

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REFLECTION
Reflection

• The equation for reflection with respect to x-axis

• For each point (x, y)
• New point (x, -y).

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Reflection

• The equation for reflection with respect to y-axis

• For each point (x, y)
• New point (-x, y).

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Reflection

• The equation for reflection with respect to origin

• For each point (x, y)
• New point (-x, -y).

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Reflection Example

Consider a rectangle with coordinates (1,1), (3,1), (3,5), (1,5). Reflect it with
respect to x-axis

Solution:
Old Coordinates New Coordinates
(1,1) (1, -1)
(3,1) (3, -1)
(3,5) (3, -5)
(1,5) (1, -5)

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Reflection Example

Consider a rectangle with coordinates (1,1), (3,1), (3,5), (1,5). Reflect it with
respect to y-axis

Solution:
Old Coordinates New Coordinates
(1,1) (-1, 1)
(3,1) (-3, 1)
(3,5) (-3, 5)
(1,5) (-1, 5)

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Reflection Example

Consider a rectangle with coordinates (1,1), (3,1), (3,5), (1,5). Reflect it with
respect to origin

Solution: Old Coordinates New Coordinates

(1,1) (-1, -1)
(3,1) (-3, -1)
(3,5) (-3, -5)
(1,5) (-1, -5)

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