CAP4730: Computational Structures in Computer Graphics

1
CAP4730 Computational Structures in Computer
Graphics
2D Basics, Line Drawing, and Clipping

• Chapter 3 Hearn Baker
• Portions obtained from Leonard McMillans COMP136
  Notes
• www.cs.unc.edu/mcmillan/comp136/Lecture 6

2
Definitions

• CG API Computer Graphics Application
  Programming Interface (OpenGL, DirectX)
• Graphics primitives functions in the API that
  describe picture components
• How could we describe an object?
• Typically focus on object shape
• Define an objects shape with geometric
  primitives
• Span of primitives are defined by the API
• What are some types?
• Lines, Triangles, Quadrics, Conic sections,
  Curved surfaces

3
Two Dimensional Images
Y

• Use Cartesian coordinates
• We label the two axes as
• X (horizontal)
• Y (vertical)
• Origin is in the lower left
• How big is the space?
• So what is the image we see on a screen?
• We call this space the world coordinate system

Y Axis
(0,0)
X Axis
X
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Partition the space into pixels
1. Define a set of points (vertices) in 2D space.
2. Given a set of vertices, draw lines between
consecutive vertices. 3. If you were writing
OpenGL yourself, lets talk about low level
calls 4. What about 2D vs 3D?
Y
(2,7)
(9,7)
(2,1)
(9,1)
X
Screen Coordinates references to frame buffer
locations
Q True or Flase Screen Coordinates World
Coordinates
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Pixels

• ?glSetPixel(?)
• ?glGetPixel(?)
• Scan line number y
• Column number x

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Absolute and Relative Coordinate Specifications
Y

• Absolute coordinates location specified as a
  relationship to the origin
• Relative coordinates location specified as a
  relationship to other points
• Good for pen/plotters
• Publishing/layout
• Allows for a very object oriented approach
• For this class we will always use absolute
  coordinates

(0,6)
(7,0)
(2,1)
(0,-6)
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Specifying a World Coordinate System in OpenGL
Y
X
gluOrtho2D (xmin, xmax, ymin, ymax) What should
our xmin, xmax, ymin, ymax values be? Equivalent
to the size of the framebuffer
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What is a pixel
From a geometry point of view, a pixel is a
point. Q Where is (2,1)?
Q What is a pixel? A square or a point?
3
2
1
2
1
3
4
5
0
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But when we think about images, a pixel is a
rectangle. Q Where is (2,1)? A. The center of
a pixel
2
1
0
2
1
0
3
4
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Basic OpenGL Point Structure

• In OpenGL, to specify a point
• glVertex()
• In OpenGL, some functions require both a
  dimensionality and a data type
• glVertex2i(80,100), glVertex2f(58.9, 90.3)
• glVertex3i(20,20,-5), glVertex3f(-2.2,20.9,20)
• Must put within a glBegin/glEnd pair
• glBegin(GL_POINTS)
• glVertex2i(50,50)
• glVertex2i(60,60)
• glVertex2i(60,50)
• glEnd()
• Lets draw points in our assignment 1
• Next up? Lines

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Draw a line from 0,0 to 4,2
How do we choose between 1,0 and 1,1? What would
be a good heuristic?
(4,2)
2
1
(0,0)
0
2
1
0
3
4
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What are lines composed of?Write
glBegin(GL_LINES)
(4,2)
2
1
(0,0)
0
2
1
0
3
4
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What we are working with
V1 (6,8)
V2 (13,8)
V0 (6,2)
V3 (13,2)

• We are still dealing with vertices
• Draws a line between every pair of vertices
• glBegin(GL_LINES)
• glVertex2i(6,2)
• glVertex2i(6,8)
• glEnd()

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Lets draw a triangle
(0,2)
(4,2)
2
1
(2,0)
0
2
1
0
3
4
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Consider a translation
(-0.2,2)
(3.8,2)
2
1
(1.8,0)
0
2
1
0
3
4
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The Ideal Line
(17,8)
What do we want?

• Continuous appearance
• Uniform thickness and brightness
• Pixels near the ideal line are on
• Speed

(2,2)
Discretization - converting a continuous signal
into discrete elements. Scan Conversion -
converting vertex/edges information into pixel
data for display
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Slope-Intercept Method

• From algebra y mx b
• m slope b y intercept Lets write some
  code

class Point public int x, y int
r,g,b unsigned byte framebufferIMAGE_WIDTHIM
AGE_HEIGHT3
DrawLine (Point point1, Point point2)
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Slope-Intercept Method

• From algebra y mx b
• m slope b y intercept Lets write some
  code

DrawLine (Point point1, Point point2) m(point2.
y-point1.y) / (point2.x-point2.x) bpoint1.y
(-point1.x) m for ipoint1.x to
point2.x SetPixel(i , round(mib)), pixel1.r,
pixel1.g, pixel1.b
SetPixel(int x, int y, int r, int g, int
b) framebuffer(y IMAGE_WIDTHx) 3
0r framebuffer(y IMAGE_WIDTHx) 3
1g framebuffer(y IMAGE_WIDTHx) 3
2b
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Example 1 Point1 V(2,2) C(255,102,0) Point2
V(17,8) C(255,102,0) What if colors were
different?
(0,9)
(17,8)
(2,2)
(0,0)
(18,0)
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How do we change the framebuffer?
Whats the index into GLubyte framebuffer?
Point is 9,5
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Example
(0,9)
(0,0)
(18,0)
What are the problems with this method? Slopegt1
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Revised Slope Intercept
DrawLine (Point point1, Point point2) m(point2.
y-point1.y) / (point2.x-point2.x) bpoint1.y
(-point1.x) m if (mgt1) for ipoint1.x to
point2.x SetPixel(i , round(imb)) else
for ipoint1.y to point2.y SetPixel(i ,
round(i-b/m)) Which one should we use if
m1? What is the cost per pixel?
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Optimization (DDA Algorithm)

• Since we increment y by the same amount, we can
  also inline rounding
• New cost one floating point addition, one
  integer addition, one cast.

DrawLine (Point point1, Point point2) m(point2.
y-point1.y) / (point2.x-point2.x) jpoint1.y
(-point1.x) m 0.5 for ipoint1.x to
point2.x SetPixel(i , (int)jm))
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Bresenhams Line Drawing

• In general
• Addition and Subtraction are faster than
  Multiplication which is faster than Division
• Integer calculations are faster than Floating
  point
• Made all math just integers (Section 3.5)
• How?

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• What you need to know about Bresenham LDA
• Why we use it
• Major idea of integer-izing a decision point
• How this reduces things to just integer form.

(0,9)
(17,8)
(2,2)
(0,0)
(18,0)
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Recap

• DDA/Line Intercept Algorithm
• Slope Intercept y mx b
• Easy to implement
• Slow
• Bresenham
• No floating point math
• Fast
• Why do we spend so much time optimizing this?

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Other Primitive Drawing Solutions

• What other shapes might we want to draw quickly?
• Circles (and thus) Ovals
• Curves
• Fill?