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H331: Computer Graphics

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Today

- Graphics programming
- Book: Chapters 3, 4, 10

Announcements

- Practicum 1 available
- http://www.cs.kuleuven.ac.be/~graphics/H331/

- Practicum 1 & 2: 7 points / 20
- Best: 4
- 2nd best: 3

Announcements

- SIGGRAPH Los Angeles August 8-12
- http://www.siggraph.org/s2004/
- Student volunteers! (Deadline: February 25)

3D 2D

- How to transform the 3D world to a 2D image?
- 3 aspects:
- Objects: exist in space, independent of viewer
- Viewer: camera, human, ….
- Lights: shading, shadows, …

3D 2D

Objects (points, lines, polygons)

Described by vertices

Lights (e-m spectrum)

(350-780 nm)

Viewer = camera

Synthetic Camera

Projection plane in front of center-of-projection:

Synthetic Camera

Clipping: looking through a window

3D APIs

- Synthetic camera is basis of 3D API
- OpenGL, PHIGS, Direct 3D, VRML,JAVA-3D, GKS, …

- We need to functions to specify:
- Objects: vertices that describe points, lines, polygons
- Camera: position, orientation, width, height
- Light sources: position, color
- Materials: reflection characteristics

3D APIs

glBegin(GL_POLYGON)

glVertex3f(0.0, 0.0, 0.0);

glVertex3f(0.0, 1.0, 0.0);

glVertex3f(0.0, 0.0, 0.1);

glEnd();

…

gluLookAt(posx, posy, posz, atx, aty, atz, …);

glPerspective(view_angle, …);

3D APIs

- 3D API performs modeling + rendering
- But … modeling can also be done ‘off-line’
- Write model to file
- Read file in 3D API and transform to 3D API modeling commands

- RenderMan (Pixar)
- Prepares off-line model for rendering
- Rendering takes ‘converted’ model

Graphics hardware

- 3D ‘world’ coordinates 2D ‘screen’ coordinates

Graphics hardware

- 3D vertex 2D pixel

Transform to camera coordinate system

Clip away things we don’t see in the camera window

3D coordinates 2D coordinates

Transform to pixels in the frame buffer

How to draw things?

- Given: window on the screen
- Graphics API (e.g. OpenGL) has something of the form:
plotPixel(int x, int y)

How to draw things?

- plotPixel(289,190)
- plotPixel(320,128)
- plotPixel(239,67)
- plotPixel(194,101)
- plotPixel(129,83)
- plotPixel(75,73)
- plotPixel(74,74)
- plotPixel(20,10)

Why is this impractical?

- Coordinates are expressed in screen space, but objects live in (3D) world space
- Resizing window implies we have to change coordinates of objects to be drawn
- We want to make a separation between:
- values to describe geometrical objects
- values needed to draw these objects on the screen

How to draw things?

- Specify points to OpenGL

glVertex*( … )

glVertex2i( … ) glVertex3i( … )

glVertex2f( … ) glVErtex3f( … )

glBegin(GL_LINES);

glVertex2f(x1, y1);

glVertex2f(x2, y2);

glEnd();

glBegin(GL_POINTS);

glVertex2f(x1, y1);

glVertex2f(x2, y2);

glEnd();

How to draw things?

For (k=0; k<500; k++) {

…

// compute point k

x = …;

y = …;

glBegin(GL_POINTS);

glVertex2f(x, y);

glEnd();

}

glFlush();

More about OpenGL…

- OpenGl = set of libraries

More about OpenGL…

- OpenGl supports geometric primitives and raster primitives

Geometric Primitives in OpenGL

- Geometric primitives are defined by vertices
- GL_POINTS
- GL_LINES
- GL_LINE_STRIP, GL_LINE_LOOP

Geometric Primitives in OpenGL

- Closed loops = polygons
- Polygons: describe surfaces

Geometric Primitives in OpenGL

- GL_POLYGON, GL_QUADS, …

Viewing in OpenGL

- Scene is independent of camera
- gluOrtho2D(left, tight, bottom, top)

3D primitives

void triangle(point3 a, point3 b, point3 c) {

glBegin(GL_POLYGON)

glVertex3fv(a);

glVertex3fv(b);

glVertex3fv(c);

glEnd();

}

void tetrahedron () {

glColor3f(1.0,0.0,0.0);

triangle(v[0], v[1], v[2]);

…

}

3D primitives

- Hidden surfaces?
- Z-buffer
- Keep depth for each pixel

- Initialize!
- glClear(GL_COLOR_BUFFER_BIT);
- glClear(GL_DEPTH_BUFFER_BIT);
- …
- glFlush();

World window & viewport

- World window:specifies what part of the world should be drawn
- Viewport:rectangular area in the screen window in which we will draw

Mapping: world window to viewport

- If x = Wl, then sx = Vl
- If x = Wr, then sx = Vr
- If x = f*(Wr-Wl), then sx = f*(Vr-Vl)
- If x < Wl, then sx < Vl
- If x > Wr, then sx > Vr
- … also for y and sy

Clipping

- Lines outside of world window are not to be drawn.
- Graphics API clips them automatically.
- But clipping is a general tool in graphics!

Clipping

A

B

clipSegment(…):

- Return 1 if line within window
- Return 0 if line outside window
- If line partially inside, partially outside: clip and return 1

C

E

D

Cohen-Sutherland region outcodes

- 4 bits:
TTFF

Left of window?

Above window?

Right of window?

Below window?

Cohen-Sutherland region outcodes

- Trivial accept: both endpoints are FFFF
- Trivial reject: both endpoints have T in the same position

TTFF

FTFF

FTTF

TFFF

FFFF

FFTF

TFFT

FFFT

FFTT

Cohen-Sutherland: chopping

- If segment is neither trivial accept or reject:
- Clip against edges of window in turn

Cohen-Sutherland: chopping

Trivial accept

Cohen-Sutherland line clipper

- int clipSegment (point p1, point p2)
Do {

If (trivial accept) return (1)

If (trivial reject) return (0)

If (p1 is outside)

if (p1 is left) chop left

else if (p1 is right) chop right

…

If (p2 is outside)

…

} while (1)

Raster Graphics

- What is an image?
- Array of pixels

- How to convert lines and polygons to pixels?
- Continuous to discrete
- Scan conversion

Displays

- Early displays were vector displays
- Electron beam traces lines
- Image is sequence of endpoints
- Wireframes, no solid fills

Displays

- Raster displays
- Electron beam traces regular pattern
- Image is 2D array of pixels
- Fast, but discretisation errors

- Every pixel has b bits for color
- B&W: 1 bit
- Basic colors: 8, 15, 16, 24 bits
- High-end: 96 bits

Displays and Framebuffers

- Raster image is stored in memory as a 2D array of pixels = framebuffer
- The color of each pixel determines the intensity of the beam
- Video hardware scans framebuffer at 60Hz
- Changes in framebuffer show on screen => double buffering
- Switch buffers when one buffer is finished

Displays and Framebuffers

Framebuffer

(double buffer)

display

Video controller

Graphics software (rasterizer)

Rasterizer: Example

- How to rasterize a line, once its 2D screen coordinates are known?
- Given: endpoints of a line
- What pixels to draw?

Scan converting lines

- find the pixels closest to the ideal line
- assume slope m 1: illuminate one pixel per column, work incrementally
- if m1 : x y.

Scan converting lines

- Inefficient: compute round(y) for each integer x and floating point addition
- Bresenham’s algorithm: only integer arithmetic
- Standard for most HW+SW rasterizers

Scan converting lines

- What’s the next pixel?
- Decision variable d = a – b if (d>0) … else …
- Or d = Dx(a-b)

Scan converting lines

- dk+1 = dk – 2Dy ordk+1 = dk – 2(Dy-Dx)

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