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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

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

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

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

Pixels

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

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)

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

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

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

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

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

What are lines composed of? Write

glBegin(GL_LINES)

(4,2)

2

1

(0,0)

0

2

1

0

3

4

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()

Lets draw a triangle

(0,2)

(4,2)

2

1

(2,0)

0

2

1

0

3

4

Consider a translation

(-0.2,2)

(3.8,2)

2

1

(1.8,0)

0

2

1

0

3

4

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

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)

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

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)

How do we change the framebuffer?

Whats the index into GLubyte framebuffer?

Point is 9,5

Example

(0,9)

(0,0)

(18,0)

What are the problems with this method? Slopegt1

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?

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))

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?

- 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)

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?

Other Primitive Drawing Solutions

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