2D Geometry

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2-D Geometry

• TRANSFORMATIONS

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• Homogeneous Coordinates
• The purpose of using homogeneous coordinates is
  to capture the concept of infinity as we cant
  represent the infinity with the conventional
  Cartesian system.
• We use two numbers a and w to represent a value
  v, va/w. If w is not zero, the value is exactly
  a/w. Otherwise, we identify the infinite value
  with (a,0). Therefore, the concept of infinity
  can be represented with a number pair like (a,
  w).
• Given a polynomial of degree n, after
  introducing w, all terms are of degree n.
  Consequently, these polynomials are called
  homogeneous polynomials and the coordinates
  (x,y,w) the homogeneous coordinates.

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For example Suppose we have a line Ax By C
0. Replacing x and y with x/w and y/w yields
A(x/w) B(y/w) C 0. Multiplying by w
changes it to Ax By Cw 0. Similarly, let
the given equation be a second degree polynomial
Ax2 2Bxy Cy2 2Dx 2Ey F 0. After
replacing x and y with x/w and y/w and
multiplying the result with w2, we have Ax2
2Bxy Cy2 2Dxw 2Eyw Fw2 0
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• Converting Homogeneous to Cartesian
• Given a point (x,y,w) in homogeneous
  coordinates, its corresponding point in the
  xy-plane is (x/w,y/w).
• e.g. a point (3,4,5) in homogeneous
  coordinates converts to point (3/5,4/5)(0.6,0.8)
  in the xy-plane.
• Similarly, a point (x,y,z,w) in homogeneous
  coordinates converts to a point (x/w,y/w,z/w) in
  space.
• Conversely, the homogeneous coordinates of a
  point (x,y) in the xy-plane is simply (x,y,1). It
  is not unique. The homogeneous coordinates of a
  point (x,y) in the xy-plane is (xw, yw, w) for
  any non-zero w.
• Converting from a homogeneous coordinates to a
  conventional one is unique but, converting a
  conventional coordinates to a homogeneous one is
  not.

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• Transformation
• It is the simulation of the manipulation of
  objects in space by a graphics system is known as
  
• Or
• It is altering the coordinate description of an
  object by changing orientation, size and shape of
  it.
• Why are they important to graphics?
• moving objects on screen / in space
• mapping from model space to world space to
  camera space to screen space
• specifying parent/child relationships

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There are 2 types of transformation Geometric
Object is transformed relative to stationary
coordinates. Coordinate Object is stationary
and coordinate system is transformed.
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Geometric transformation The basic geometric
transformations are Translation, Rotation and
Scaling
Translation A translation is applied to an
object by repositioning it along a straight line
path from one coordinate location to another. We
can translate a 2-D point by adding translation
distances tx and ty to the original coordinate
position (x,y) to move the point to new position
(x,y) i.e. x x tx y y ty
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Rotation Rotation is applied to an object by
repositioning it along a circular path in the xy
plane. We specify a rotation angle and the
position of the rotation point (called pivot
point) about which the object is to be rotated.
ve values of angle define counterclockwise
rotations -ve values of angle define clockwise
rotation.
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Using trigonometric identities, we can represent
tranformed acoordinate as follow X rcos(T F)
r cosTcos F r sinTsinF Yrsin(T F) r
cosFsinT r cosTsinF Also X rcosF and
YrsinF Therefore X xcosT ysinT Y xsinT
ycosT
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• Rotation of a point about an arbitrary pivot
  position is given as
• Xxr (x-xr)cosT ( y-yr)sinT
• Yyr (x-xr)sinT (y-yr)cosT
• Rotation is a rigid body transformation

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Scaling A scaling transformation alter the size
of an object
The operation can be carried out by multiplying
coordinate values by scaling factor.
X x.sx and Y y.sy
X sx 0 x Y
0 sy y
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Uniform scaling when both sx and sy have same
value Differential scaling when both sx and sy
have different values.