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FUNDAMENTALS OF OBJECT REPRESENTATION

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These, too, are called half spaces. ... (half space) ... If the coordinates of a point produce h(x,y) 0 then the point is said to outside the half space ... – PowerPoint PPT presentation

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Title: FUNDAMENTALS OF OBJECT REPRESENTATION


1
  • FUNDAMENTALS OF OBJECT REPRESENTATION
  •  
  • Half Spaces
  •  
  • An unbounded straight line, plane curve (1
    dimensional in nature) divides the two
    dimensional space into two semi-infinite regions
    called half spaces
  •  
  • Infinite Line
    Infinite Curve
  •  
  •  
  •  
  •  
  • In 3D, similarly an unbounded plane or surface
    divides the 3D into two semi-infinite regions.
    These are also called half spaces.
  •  

2
  • In the computer, we want to simulate a real world
    3D object. But in the computer it is just a set
    of point. We need to be able to give it some
    properties. Need to define boundaries so that
    you can classify any point as either outside,
    inside or on the object.
  • Outside

  •  




3
  • A square is bounded by 4 edges infinite lines
  • A square is formed by the intersection of 4 - ½
    spaces
  •  
  •  
  •  
  •  
  • Closed curves and surfaces can divide spaces into
    ½ spaces as well. As for examples, circles,
    spheres, cones and torus can divide their spaces
    into regions, one finite and other infinite.
    These, too, are called half spaces.
  •  
  •  

4
  • The same can be said for 3 D. A block is
    bounded by 6 ½ spaces. To get other shapes, you
    can take smaller blocks and subtract or add them
    to the original blocks.
  •  
  • The first 3D system used a defined building
    blocks. blocks, cones, torus (primitives) to
    build objects.
  •  

5
  • Two dimensional half spaces
  •  
  • The general equation of a straight line in the
    Cartesian coordinate system is
  • Ax ByC0
  •  
  •  
  • For a half space bounded by a straight line we
    define the half space as follows
  •  
  • h(x,y) Ax By C
  •  
  • (half space)
  •  
  • h(x,y) says that points in the ½ space are a
    function of (or are dependant on) the values of x
    and y. Any combination of the values for x and y
    that satisfies the equations so that h(x,y)0 are
    on the line, the boundary of the half space.
    Other values provide an inequality, either
    h(x,y)gt0 or h(x,y)lt0

6
  • Classification of points in regards to ½ Spaces
  •  
  • Use the following conventions to classify a point
    with respect to agiven half space.
  •  
  • 1.     If the coordinates of a point produce
    h(x,y)0 then the point is on the boundary
  •  
  • 2.     If the coordinates of a point produce
    h(x,y)gt0, then the point is said to be inside
    the half space.
  •  
  • 3.     If the coordinates of a point produce
    h(x,y)lt0 then the point is said to outside the
    half space

7
  • Examples
  •  
  • Problem1
  •  
  • h X-2Y-6
  •  
  • 1) Draw the line X-2Y-60
  • Pick any point and test it against x-2y-6. Lets
    pick 0,0 H(0,0) -6
  • Therefore, above the curve must be out side and
    below must be inside.
  • Note
  • Changing the sign of h(x,y), inside/outside
    classification gets reversed. The complement of
    h(x,y) is h(x,y)

8
  • Problem2
  •  
  • Now try h(x,y) X2-Y1
  •  
  •  
  • Yx21
  •  
  • Pick 0,0
  •  
  •  
  •  
  •  
  •  

9
  • Problem3
  •  
  • Now let h(x,y) (x-2)2 (y-2)2 1
  • This is the equation of a circle with center of
    (2,2) and a radius of 1
  •  
  •  
  • Lets try 0,0
  • Y h 4 4 1

10
  • REMEMBER
  •  
  • Complement of ½ space
  •  
  • If you want to reverse what is inside to what is
    outside, multiple h(x,y) by 1
  •  

11
  • What is the need of a half Space?
  •  
  • Half spaces define regions (2D) and volumes
    (3D).
  •  
  • If you would do it with points it could be quite
    cumbersome if not impossible
  •  
  • Look at this simple cube
  •  
  •  
  •  
  •  
  •  
  • It is bounded by 6 faces each face is bounded
    by 4 edges and each edge is bounded by two
    points.

12
  • With half space
  • An object is bounded by half-spaces.
  •  
  •  
  •  
  • 2 Ways to express an object

  • 1.     Geometrically express exactly (x,y)
  • 2.     Define the object as a combination of a
    lot of ½ spaces
  •  
  • Extend that thought to 3 dimensions
  • Use primitives to define objects.
  •  
  • Primitives
  • Torus 4th order
  • Sphere - 2nd order
  • Plane - 1st order
  • Cone - 2nd order
  • Cylinder 2nd order

13
  • Block a unique object bounded by planer spaces.
  •  
  • Review
  •  
  • Boolean Operators
  •  
  • Union REGLARIZED BOOLEAN
  • Intersection OPERATOR
  • Difference


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