Computer Graphics Rendering 2D Geometry

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Computer GraphicsRendering 2D Geometry

• CO2409 Computer Graphics
• Week 2

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Todays Lecture

• 2D Geometry
• Coordinate Systems Viewports
• Rendering Points and Lines
• Polygons and Circles
• Anti-Aliasing

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2D Geometry Basics

• Will look at 2D geometry represented by points
  and lines in two dimensional Cartesian space.
• Represented on a graph with two axes, usually X
  (horizontal) and Y (vertical)
• Origin of the graph and of the 2D space is where
  the axes cross

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2D Geometry Definitions

• I will often use informal English to describe 2D
  geometry. However, here are a couple of specific
  definitions that I will use

A vertex a single point (plural vertices)
defined by coordinates on the axes An edge a
straight line joining two vertices A polygon a
single closed loop of edges
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Specifying 2D Geometry

• Example vertices
• A(10, 20), B(30, 30), C(35, 15)
• Edges
• AB, BC, AC
• Edges with direction (a form of vector, see later
  lectures)
• AB, BC, CA
• Polygons
• ABCA, or ABC (implies the final edge to A)

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Coordinate Systems

• Define a coordinate system as a particular set of
  choices for
• The location of the origin
• The orientation and scale of the axes
• A vertex may have different coordinates in two
  different coordinate systems

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Viewports

• A viewport (or window) is a rectangle of pixels
  representing a region of 2D geometry space.
• A viewport has its own coordinate system, which
  may not match that of the geometry.
• The axes will usually be X horizontal Y
  vertical
• But dont have to be rotated viewports
• The scale of the axes may be different
• The direction of the Y axis may differ.
• E.g. the geometry may be stored with ve Y up,
  but the viewport has ve Y down.
• The origin (usually in the corners or centre of
  the viewport) may not match the geometry origin.

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Viewport Example

• Example of changing coordinate system from
  geometry space to viewport space

P (20,15) in geometry space. Where is P in
viewport space?
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Rendering Points

• Rendering is the process of converting geometry
  into pixels
• To render a vertex (a point)
• Convert vertex coordinates into viewport space
• Set the colour of the pixel at those coordinates
• The colour might be stored with the geometry, or
  we can use a fixed colour (e.g. black)
• In the previous diagram the vertex P would be
  rendered by setting the colour of the pixel at
  coordinates (266, 450) in the viewport

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Rendering Lines 1st Attempt

• A possible method to render an edge (a line)
• Convert the start and end vertex into viewport
  space
• Trace along this converted edge
• Colour each pixel traced over
• Unfortunately, this method creates thick lines.
• We need to colour fewer pixels to get a cleaner
  result.

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Rendering Lines 2nd Attempt

• Consider drawing a horizontal line with previous
  method
• Trace along horizontally adjacent pixels, setting
  the colour of each. Result is fine.
• Adapt this for a diagonal line
• Start at the left end of the line
• Move rightwards, colouring each pixel
• Periodically move up/down one pixel to create
  steps of the correct gradient
• Use variations of above for lines in different
  directions

To go from start to end need to move right 9
pixels and down 4 pixels. So move down one pixel
about every 2 pixels right
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Rendering Lines Algorithms

• We basic version of this algorithm to render
  lines
• Using floating point calculations
• Most widely used method is Bresenhams Line
  Algorithm
• Highly efficient - only uses integer calculations
• We wont cover this, or other efficient 2D
  algorithms in this module instead concentrate
  on higher level graphics work
• Many 2D algorithms are implemented in hardware
  nowadays

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Rendering Polygons

• A polygon is a sequence of edges, so we simply
  need to render each edge.
• We only need the set of vertices defining the
  polygon
• Squares and rectangles are special cases of a
  polygon and can be rendered more efficiently.
• Horizontal / vertical lines are simple cases
• Will not consider filled polygons until the 3D
  section of the module.

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Rendering Circles?

• The coordinates of a circle can be written as
• X A R sin(a), Y B R cos(a), for 0 a lt
  360º
• where (A, B) is the circle centre and R the
  radius
• We could use this to render a circle
• Step a from 0 to 360 and calculate X Y
• Convert to viewport space and colour the pixels
• Not ideal though try it in the lab
• Again there is a better integer based method from
  Bresenham

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Regular Polygons

• Can use the circle equations to create regular
  polygons.
• A regular polygon has all its points on a circle
• Can calculate the vertices using the circle
  equations
• Conversely - can we use regular polygons to draw
  a circle?

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Anti-Aliasing

• Returning to the naive line algorithm
• Again we colour every pixel passed through
• This time consider how close the centre of the
  pixel is to the line we are drawing
• The colour used is more intense for pixels that
  lie nearer the line

• This is an anti-aliased line
• Aliasing is the jagged look of pixel edges
• This method smoothes these effects hence the
  name anti-aliasing