Who Am I?

1
Who Am I?

• Prof Stephen Chenney
• These notes will be online after the lecture in
  fact theyre online already
• http//www.cs.wisc.edu/schenney/courses/guest/cs5
  59-sept1603.ppt

2
Sampling and Drawing

• Today well talk about rasterization
• A regular, rectangular grid is referred to as a
  raster
• A piece of history there are also vector
  displays
• The raster name comes from the ways the pixels
  are drawn to the screen left to right, top to
  bottom
• Rasterization is the problem of determining which
  pixels to color which samples to use to draw
  an object

3
Drawing Points

• You can draw points at any continuous position,
  not just at a pixel center (a sample point)
• How do we know which pixel to turn on?

4
Drawing Points

• You can draw points at any continuous position,
  not just at a pixel center (a sample point)
• How do we know which pixel to turn on?
• Solution is the simple one
• Find the closest sample and turn it on fill the
  pixel
• Can also specify a radius fill a square of that
  size, or fill a circle
• Square is faster
• The best solution is anti-aliased points, where
  you partially color multiple pixels

5
Drawing Lines

• Lines have a similar problem to points they are
  not likely to hit any pixel centers exactly
• What might we do instead?
• What are the aesthetic/perceptual issues in
  drawing lines?

6
Drawing Lines

• Task Decide which pixels to fill (samples to
  use) to represent a line
• Issues
• If slope between -1 and 1, one pixel per column.
  Otherwise, one pixel per row
• Constant brightness? Lines of the same length
  should light the same number of pixels (we
  normally ignore this)
• Anti-aliasing? (Getting rid of the jaggies)

7
Line Drawing Algorithms

• Consider lines of the form ym x c, where
  m?y/?x, 0ltmlt1, integer coordinates
• All others follow by symmetry
• Variety of slow algorithms (Why slow?)
• step x, compute new y at each step by equation,
  rounding
• step x, compute new y at each step by adding m to
  old y, rounding

8
Bresenhams Algorithm Overview

• Aim For each x, plot the pixel whose y-value is
  closest to the line
• Given (xi,yi), must choose from either
  (xi1,yi1) or (xi1,yi)
• Idea compute a decision variable
• Value that will determine which pixel to draw
• Easy to update from one pixel to the next
• Bresenhams algorithm is the midpoint algorithm
  for lines
• Other midpoint algorithms for conic sections
  (circles, ellipses)

9
Midpoint Methods

• Consider the midpoint between (xi1,yi1) and
  (xi1,yi)
• If its above the line, we choose (xi1,yi),
  otherwise we choose (xi1,yi1)
• Amazing thing You can do this with only integer
  add/subtract and if/then

yi1
yi1
yi
yi
xi1
xi
xi1
xi
Choose (xi1,yi)
Choose (xi1,yi1)
10
Midpoint Decision Variable

• Write the line in implicit form
• The value of F(x,y) tells us where points are
  with respect to the line
• F(x,y)0 the point is on the line
• F(x,y)lt0 The point is above the line
• F(x,y)gt0 The point is below the line
• The decision variable is the value of di
  2F(xi1,yi0.5)
• The factor of two makes the math easier

11
What Can We Decide?

• di negative gt next point at (xi1,yi)
• di positive gt next point at (xi1,yi1)
• At each point, we compute di and decide which
  pixel to draw
• How do we update it? What is di1?

12
Updating The Decision Variable

• dk1 is the old value, dk, plus an increment
• If we chose yi1yi1
• If we chose yi1yi
• What is d1 (assuming integer endpoints)?
• Notice that we dont need c any more

13
Bresenhams Algorithm

• For integers, slope between 0 and 1
• xx1, yy1, d2dy - dx, draw (x, y)
• until xx2
• xx1
• If dgt0 then yy1, draw (x, y), dd2?y - 2?x
• If dlt0 then yy, draw (x, y), dd2?y
• Compute the constants (2?y-2?x and 2?y ) once at
  the start
• Inner loop does only adds and comparisons
• Floating point has slightly more difficult
  initialization, but is otherwise the same
• Care must be taken to ensure that it doesnt
  matter which order the endpoints are specified in
  (make a uniform decision if d0)

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Example (2,2) to (7,6)
?x5, ?y4 x y d
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Example (2,2) to (7,6)
?x5, ?y4 x y d 2 2 3 3 3 1 4 4 -1 5 4 7 6 5 5 7
6 3
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5
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2
1
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16
Filling Triangles

• Sampling triangles
• When is a pixel inside a triangle?
• Given a pixel, which triangle does it lie in?
  Point location
• Triangle representation
• Triangle defined by three vertices, in known
  order (clockwise, ccw)
• We care most about triangles for 3D graphics
• We care most about rectangles for 2D graphics
• Filling more general shapes is difficult

17
What is inside?

• Assume sampling with an array of spikes
• If spike is inside, pixel is inside

18
What is inside?

• Assume sampling with an array of spikes
• If spike is inside, pixel is inside

19
Ambiguous Case

• Ambiguous cases What if a pixel lies on an edge?
• Problem because if two polygons share a common
  edge, we dont want pixels on the edge to belong
  to both
• Ambiguity would lead to different results if the
  drawing order were different
• Rule if (x?, y?) is in, (x,y) is in (? is a
  small number)
• What if a pixel is on a vertex? Does our rule
  still work?

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Ambiguous Case 1

• Rule
• On edge? If (x?, y?) is in, pixel is in
• Which pixels are colored?

21
Ambiguous Case 1

• Rule
• Keep left and bottom edges
• Assuming y increases in the up direction
• If rectangles meet at an edge, how often is the
  edge pixel drawn?

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22
Ambiguous Case 2
23
Ambiguous Case 2
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or
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Really Ambiguous

• We will accept ambiguity in such cases
• The center pixel may end up colored by one of two
  polygons in this case
• Which two?

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2
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25
Exploiting Coherence

• When filling a triangle (or polygon)
• Several contiguous pixels along a row tend to be
  in the triangle - a span of pixels
• Scanline coherence
• Consider whole spans, not individual pixels
• The pixels required dont vary much from one span
  to the next
• Edge coherence
• Incrementally update the span endpoints

26
Sweep Fill Algorithms

• Algorithmic issues
• Reduce to filling many spans
• Which edges define the span of pixels to fill?
• How do you update these edges when moving from
  span to span?
• What happens when you cross a vertex?

27
Spans

• Process - fill the bottom horizontal span of
  pixels move up and keep filling
• Have xmin, xmax for each span
• Define
• floor(x) largest integer lt x (not standard
  floor)
• ceiling(x) smallest integer gtx
• Fill from ceiling(xmin) up to floor(xmax)
• Consistent with convention

28
Updating Edges

• Each edge is a line of the form
• Next row is
• So, each current edge can have its x position
  updated by adding a constant stored with the edge
• Other values may also be updated, such as depth
  or color information
• For max efficiency, use a version of the midpoint
  algorithm

29
When are Edges Relevant?

• Edge is relevant when ygtymin and yltymax of edge
• What about horizontal edges?
• m is infinite

30
Avoiding Floating Point

• For edge, m?x/?y, which is a rational number
• View x as xixn/?y, with xnlt?y. Store xi and xn
• Then x-gtxm is given by
• xnxn?x
• if (xngt?y) xixi1 xnxn- ?y
• Advantages
• no floating point
• can tell if x is an integer or not, and get
  floor(x) and ceiling(x) easily, for the span
  endpoints