Graphical primitives

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Lecture 4

• Graphical primitives
• Rasterization algorithmic approach
• Rasterization geometric approach
• 2D discrete lines, triangles
• Discrete planes
• 3D discrete lines
• Meshes of triangles
• Graphical primitives in OpenGL

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Main elements of vector model

• Lines
• Polylines, polygons
• Curves
• Text

3
Main elements of vector model

• Lines
• Polylines, polygons (simple, regular)
• Curves
• Text

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Main elements of vector model

• Lines
• Polylines, polygons (simple, regular)
• Curves
• Text

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Main elements of vector model

• Lines
• Polylines, polygons (simple, regular)
• Curves
• Text

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Main elements of vector model

• Lines
• Polylines, polygons (simple, regular)
• Curves
• are represented by their coordinates
• (x1,y1), (x2,y2), , (xn,yn)
• Coordinate system origin, orientation of the
  axes, units

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Main elements of vector model

• Attributes
• Color
• Thickness
• Line Style
• Fill Style
• Text Font/Style

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Main elements of vector model

• Attributes
• Color
• Thickness
• Line Style
• Fill Style
• Text Font/Style

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Main elements of raster model

• Pixels/voxels
• pixel picture element
• voxel volume element

Represented as a bitmap
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Main elements of mixed model

• Lines, curves, text are rasterized

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Main elements of mixed model

• Lines, curves, text are rasterized

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Approaches to rasterization Algorithmic Approach

• Developing algorithms for the main geometrical
  primitives
• Line segment
• Circle
• Curve
• Filled area

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Approaches Algorithmic Approach. Deficiencies

• It is difficult to determine whether a pixel
  belongs to an object

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Approaches Algorithmic Approach. Deficiencies

• It is difficult to determine the intersection
  of two objects

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Approaches Algorithmic Approach. Deficiencies

• In some cases the quality of the picture is
  not satisfactory

Algorithm
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Approaches Algorithmic Approach. Deficiencies

• In some cases the quality of the picture is
  not satisfactory

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Approaches Algorithmic Approach. Deficiencies

• In some cases the quality of the picture is
  not satisfactory

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Approaches Geometric Approach.

• Objectives of the Geometric Approach
• Developing a relevant discrete counterpart of the
  analytical geometry (Discrete Analytical
  Geometry).
• Developing efficient algorithms for generation of
  the basic primitives of the discrete analytical
  geometry.
• Advantages
• Makes up the deficiencies of the algorithmic
  approach.
• Exact arithmetic.
• Raising the computational efficiency of some
  algorithms.
• Data compression.

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Approaches Geometric Approach. Difficulties
The intersection of two discrete lines may be 0,
1, or infinitely many points.
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Approaches Geometric Approach. Difficulties
The objects have thickness.
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Approaches Geometric Approach. Difficulties
In 3D the things become much more complex. The
intersection of two planes is not a line. It may
be even a disconnected set.
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Basic Definitions. Coordinate Systems, Pixels,
Voxels

• Def. 2D Coordinate system Only the integer points
  are included.
• Pixels are unit squares centered about the
  integer points.
• Voxels are unit cubes centered about the integer
  points.

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Basic Definitions. Discrete Adjacency

• Def. 2D adjacency Let P1 and P2 be two pixels.
  They are
• 8-adjacent, if they share a vertex or an edge
• 4-adjacent, if they share an edge.
• Remark 4-adjacent pixels are also 8-adjacent.

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Basic Definitions. Discrete Adjacency

• Def. 3D adjacency Let V1 and V2 be two voxels.
  They are
• 26-adjacent, if they share a vertex, an edge or a
  side.
• 18-adjacent, if they share an edge or a vertex.
• 6-adjacent, if they share a side.
• Remark The 6-adjacent voxels are also 18- and
  26-adjacent and the 18-adjacent voxels are also
  26-adjacent.

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Basic Definitions. 2D Lines

• Def. 2D arithmetic line L(a,b, µ,w) is a set of
  pixels (x,y) satisfying the conditions
• 0 ? ax by µ lt w
• w - arithmetical thickness
• µ - internal translation constant

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Basic Definitions. 2D Lines

• 0 ? ax by µ lt w
• 0 ? 3x - 5y lt 5
• x y x y
• ------- --------
• 0 0 8 4
• 1 0 9 5
• 2 1 10 6
• 3 1 11 6
• 4 2 12 7
• 5 3 13 7
• 6 3 14 8
• 7 4

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Basic Definitions. 2D Lines

• 0 ? 3x - 5y lt 5
• Let P 3x - 5y
• x y P x y P
• ---------- ----------
• 0 0 0 8 4 4
• 1 0 3 9 5 2
• 2 1 1 10 6 0
• 3 1 4 11 6 3
• 4 2 2 12 7 1
• 5 3 0 13 7 4
• 6 3 3 14 8 2
• 7 4 1

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Basic Definitions. Types of 2D Lines

• If wltmax(a, b) disconnected
• If wmax(a, b) naïve
• If wab standard
• If wgtab thick

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Basic Definitions. Types of 2D Lines

• If wltmax(a, b) disconnected
• If wmax(a, b) naïve
• If wab standard
• If wgtab thick

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Basic Definitions. Types of 2D Lines

• If wltmax(a, b) disconnected
• If wmax(a, b) naïve
• If wab standard
• If wgtab thick

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Basic Definitions. Types of 2D Lines

• If wltmax(a, b) disconnected
• If wmax(a, b) naïve
• If wab standard
• If wgtab thick

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Basic Definitions. Tunnels in 2D Lines

• A line L (a,b,µ,w) has an k-tunnel, if there
  exist two k-adjacent pixels A and B, such that
• axAbyAµlt0 and
• axBbyBµ?w

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Basic Definitions. Bresenham Line

• Bresenham line corresponding to the Euclidean
  line
• ax by c 0 is
• 0? ax by cw/2 lt w,
• where wmax(a, b)

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Discrete Triangles

• Problems
• Def. Let A, B, C be given.
• Border union of the Bresenham lines AB, BC, CA
• Interior All the integer points, belonging to
  the Euclidean triangle ABC, which do not belong
  to the border.

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Discrete Planes

• Def. Discrete plane P(a,b,c,µ,w) is a set of
  voxels (x,y,z) satisfying the conditions
• 0 ? ax by cz c lt w
• w - arithmetical thickness
• µ - internal translation constant
• Naïve plane
• wmax(a, b, c
• Standard plane
• wabc

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Discrete Planes

• Def. Discrete plane P(a,b,c,µ,w) is a set of
  voxels (x,y,z) satisfying the conditions
• 0 ? ax by cz c lt w
• w - arithmetical thickness
• µ - internal translation constant
• Naïve plane
• wmax(a, b, c
• Standard plane
• wabc

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Discrete Planes

• Def. A plane P has a k-tunnel (n6,18,26) if
  there exist two k-adjacent voxels A and B such
  that
• axA byA czA µ lt0
• and
• axB byB czB µ ? w
• Tunnel-free is equivalent to 26-tunnel-free.

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Functionality

• Def. A plane P is functional on a coordinate
  plane, e.g. Oxy, if for any voxel (x,y) from Oxy
  there is exactly one voxel belonging to P. The
  plane Oxy is called functional coordinate plane
  for P.

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3D Lines

• Def. A 3D discrete line is defined as an
  intersection of two discrete planes.
• 0 ? - cx az µ lt w
• 0 ? - cy bz µ lt w

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Graphical Primitives in OpenGL

• OpenGL uses primitives to draw all the vector
  primitives that we studied points, lines,
  polygons, rectangles, etc.
• 2D and 3D
• Remark For 3D primitives we have to call
  projection function, e.g. glOrtho as in the
  example of drawing triangle.
• All functions for drawing primitives start with
• glBegin( PRIMITIVE_NAME ) and end with
• glEnd().

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Graphical Primitives in OpenGL

• Example The following fragment draws one 3D
  point.
• glBegin( GL_POINTS )
• glVertex3f( 0.2, 0.4, 0.3 )
• glEnd()
• Example The following fragment draws four 3D
  point.
• glBegin( GL_POINTS )
• glVertex3f( 0.1, 0.1, 0 )
• glVertex3f( 0.1, 0.5, 0 )
• glVertex3f( 0.5, 0.5, 0 )
• glVertex3f( 0.5, 0.1, 0 )
• glEnd()

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Graphical Primitives in OpenGL

• Example The following fragment draws a square.
• glColor3f( 1.0, 0.0, 0.0 )
• glBegin( GL_POLYGON )
• glVertex3f( 0.1, 0.1, 0 )
• glVertex3f( 0.1, 0.5, 0 )
• glVertex3f( 0.5, 0.5, 0 )
• glVertex3f( 0.5, 0.1, 0 )
• glEnd()

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Graphical Primitives in OpenGL

• Z-coordinate is zero. We can skip it.
• glColor3f( 1.0, 0.0, 0.0 )
• glBegin( GL_POLYGON )
• glVertex2f( 0.1, 0.1 )
• glVertex2f( 0.1, 0.5 )
• glVertex2f( 0.5, 0.5 )
• glVertex2f( 0.5, 0.1 )
• glEnd()

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Graphical Primitives in OpenGL

• Remark the order of the vertices is important.
• The program that we considered will draw the
  following window

My first OpenGL program
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Graphical Primitives in OpenGL

• Remark the order of the vertices is important.
• The program
• glColor3f( 1.0, 0.0, 0.0 )
• glBegin( GL_POLYGON )
• glVertex2f( 0.1, 0.5 )
• glVertex2f( 0.1, 0.1 )
• glVertex2f( 0.5, 0.5 )
• glVertex2f( 0.5, 0.1 )
• glEnd()

My first OpenGL program
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List of the Primitives
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GL_POINTS

• glColor3f( 0.0, 0.0, 1.0 )
• glBegin(GL_POINTS)
• glVertex3f( 0.1, 0.9, 0 )
• glVertex3f( 0.1, 0.2, 0 )
• glVertex3f( 0.9, 0.9, 0 )
• glVertex3f( 0.9, 0.2, 0 )
• glEnd()

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GL_POINTS

• glColor3f( 0.0, 0.0, 1.0 )
• glBegin(GL_POINTS)
• glVertex3f( 0.1, 0.9, 0 )
• glVertex3f( 0.1, 0.2, 0 )
• glVertex3f( 0.9, 0.9, 0 )
• glVertex3f( 0.9, 0.2, 0 )
• glEnd()

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GL_LINES connects two points to form a line. Odd
number of vertices last ignored
glColor3f( 0.0, 0.0, 1.0 ) glBegin(GL_LINES) gl
Vertex3f( 0.1, 0.2, 0 ) glVertex3f( 0.1,
0.9, 0 ) glVertex3f( 0.9, 0.9, 0 )
glVertex3f( 0.9, 0.2, 0 ) glEnd()

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GL_LINE_STRIP draws lines without lifting the
pen

glColor3f( 0.0, 0.0, 1.0 ) glBegin(GL_LINE_STRIP)
glVertex3f( 0.1, 0.2, 0 ) glVertex3f(
0.1, 0.9, 0 ) glVertex3f( 0.9, 0.9, 0 )
glVertex3f( 0.9, 0.2, 0 ) glEnd()
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GL_LINE_LOOP do the same as GL_LINE_STRIP and
connect the first and last points

glColor3f( 0.0, 0.0, 1.0 ) glBegin(GL_LINE_LOOP)
glVertex3f( 0.1, 0.2, 0 ) glVertex3f(
0.1, 0.9, 0 ) glVertex3f( 0.9, 0.9, 0 )
glVertex3f( 0.9, 0.2, 0 ) glEnd()
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GL_TRIANGLES take vertices in groups of three and
draw triangles. If the number of Vertices is not
divisible by 3, those vertices at the end are
ignored.

What will this program fragment draw? glColor3f(
0.0, 0.0, 1.0 ) glBegin(GL_TRIANGLES) glVertex3
f( 0.1, 0.9, 0 ) glVertex3f( 0.1, 0.2, 0 )
glVertex3f( 0.9, 0.9, 0 ) glVertex3f( 0.9,
0.2, 0 ) glEnd()
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GL_TRIANGLE_STRIP also does this with the
exception that every vertex after the first three
forms a connected triangles( i.e., v0, v1, v2,
then v2, v1 and v3, then v2, v3 and v4).
glColor3f( 0.0, 0.0, 1.0 ) glBegin(GL_TRIANGLE_ST
RIP) glVertex3f( 0.1, 0.2, 0.0 ) glVertex3f(
0.1, 0.5, 0.0 ) glColor3f ( 1.0, 0.0, 0.0
) glVertex3f( 0.5, 0.5, 0.0 ) glVertex3f(
0.5, 0.2, 0.0 ) glEnd()

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GL_TRIANGLE_FAN forms triangles that share the
common first vertex.
glColor3f( 0.0, 0.0, 1.0 ) glBegin(GL_TRIANGLE_
FAN) glVertex3f( 0.1, 0.2, 0 ) glVertex3f(
0.1, 0.9, 0 ) glVertex3f( 0.9, 0.5, 0
) glVertex3f( 0.5, 0.2, 0 ) glEnd()

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GL_QUADS takes vertices in-groups of four to draw
four-side polygons. If the number of points is
not divisible by 4, the last a few points are
ignored.
glColor3f( 0.0, 0.0, 1.0 ) glBegin(GL_QUADS) gl
Vertex3f( 0.1, 0.2, 0 ) glVertex3f( 0.1, 0.5, 0
) glVertex3f( 0.5 ,0.5,0) glVertex3f(
0.5,0.2, 0 ) glColor3f( 0.0, 1.0, 0.0 )
glVertex3f( 0.6,0.2, 0 ) glVertex3f(
0.6,0.6, 0 ) glVertex3f( 0.9,0.6, 0 )
glVertex3f( 0.9,0.2, 0 ) glEnd()

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GL_QUADS takes vertices in-groups of four to draw
four-side polygons. If the number of points is
not divisible by 4, the last a few points are
ignored.

GL_QUAD_STRIP draws quads(four sided polygons)
using vertices v0, v1, v2, v3 then v4, v5, v6,
v7.
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GL_POLYGON draws a polygon with the specified
vertices

• glColor3f( 0.0, 0.0, 1.0 )
• glBegin(GL_POLYGON)
• glVertex3f( 0.1, 0.2, 0 )
• glVertex3f( 0.1, 0.9, 0 )
• glVertex3f( 0.9, 0.9, 0 )
• glVertex3f( 0.9, 0.2, 0 )
• glEnd()

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Approaches Algorithmic Approach. Line Segment
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Approaches Algorithmic Approach. Line Segment
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Approaches Algorithmic Approach. Line Segment
Back
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Discrete Circles

• Approaches
• ysqrt(r2 x2) bad quality of the image
• xr.cos(f), yr.sin(f) slow

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Discrete Circles

• Symmetry
• The formula ysqrt(r2 x2) is used for the
  second octant.
• The symmetry of the circle is explored

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Discrete Circles
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Discrete Circles
Back
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Approaches Algorithmic Approach. Deficiencies

• for i 1 to 200 do
• Circle (x, y, i)

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