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# Cmpe 466 computer graphics. Graphics output primitives. (Chapter 4)

## 1. CMPE 466 COMPUTER GRAPHICS

1CMPE 466

COMPUTER GRAPHICS

Chapter 4

Graphics Output Primitives

Instructor: D. Arifler

Material based on

- Computer Graphics with OpenGL®, Fourth Edition by Donald Hearn, M. Pauline Baker, and Warren R. Carithers

- Fundamentals of Computer Graphics, Third Edition by by Peter Shirley and Steve Marschner

## 2. Output primitives and reference frames

2Output primitives and reference frames

• Output primitives are functions that we use to describe the

various picture components

• World-coordinate (WC) reference frame describes objects

in the picture by giving their geometric specs in terms of

positions in WC

• Scene info is processed by viewing routines

• These routines identify visible surfaces

• Scan conversion stores info about the scene at the

appropriate locations in frame buffer

• Finally, objects are displayed on output device

## 3. Screen coordinates

3Screen coordinates

• Screen coordinates (integers): Locations on a video

monitor

• Coordinates correspond to pixel positions in frame buffer

• Pixel positions: Scan line number (y-value), column

number (x-value along a scan line)

• Hardware processes, such as screen refreshing, typically

address pixel positions with respect to the top-left corner

of the screen

• With software commands, we can set up any convenient

reference frame for screen positions

## 4. Specifying a 2D WC reference frame

4Specifying a 2D WC reference frame

Figure 4-2 World-coordinate limits for a display window, as specified in the

glOrtho2D function.

glMatrixMode (GL_PROJECTION);

glLoadIdentity ( );

gluOrtho2D (xmin, xmax, ymin, ymax);

## 5. OpenGL point functions

5OpenGL point functions

Figure 4-3 Display of three point positions generated with glBegin

(GL_POINTS).

glBegin (GL_POINTS);

glVertex2i (50, 100);

glVertex2i (75, 150);

glVertex2i (100, 200);

glEnd ( );

## 6. OpenGL point functions: alternative code

6OpenGL point functions: alternative code

Figure 4-3 Display of three point positions generated with glBegin

(GL_POINTS).

int point1 [ ] = {50, 100};

int point2 [ ] = {75, 150};

int point3 [ ] = {100, 200};

…

glBegin (GL_POINTS);

glVertex2iv (point1);

glVertex2iv (point2);

glVertex2iv (point3);

glEnd ( );

## 7. More on point functions

7More on point functions

• Specifying positions in 3D using floating-point coordinates

glBegin (GL_POINTS);

glVertex3f (-78.05, 909.72, 14.60);

glVertex3f (261.91, -5200.67, 188.33);

glEnd ( );

• Using class or struct to specify point positions

class wcPt2D {

public:

GLfloat x, y;

};

wcPt2D pointPos;

pointPos.x = 120.75;

pointPos.y = 45.30;

glBegin (GL_POINTS);

glVertex2f (pointPos.x, pointPos.y);

glEnd ( );

## 8. OpenGL line functions

8OpenGL line functions

Figure 4-4 Line segments that can be displayed in OpenGL using a list of five endpoint coordinates. (a) An unconnected

set of lines generated with the primitive line constant GL_LINES. (b) A polyline generated with GL_LINE_STRIP. (c) A

closed polyline generated with GL_LINE_LOOP.

glBegin (GL_LINES);

glVertex2iv (p1);

glVertex2iv (p2);

glVertex2iv (p3);

glVertex2iv (p4);

glVertex2iv (p5);

glEnd ( );

glBegin (GL_LINE_STRIP); glBegin (GL_LINE_LOOP);

glVertex2iv (p1);

glVertex2iv (p1);

glVertex2iv (p2);

glVertex2iv (p2);

glVertex2iv (p3);

glVertex2iv (p3);

glVertex2iv (p4);

glVertex2iv (p4);

glVertex2iv (p5);

glVertex2iv (p5);

glEnd ( );

glEnd ( );

## 9. Polygon Fill Areas

9Polygon Fill Areas

Figure 4-8 A convex polygon (a), and a concave polygon (b).

Concave polygons may present

problems when implementing fill

algorithms

## 10. Identifying concave polygons

10Identifying concave polygons

Figure 4-9 Identifying a concave polygon by calculating cross-products of

successive pairs of edge vectors.

For concave polygons, some cross-products are positive, some are negative

## 11. Splitting concave polygons: vector method

11Splitting concave polygons: vector method

• Form the edge vectors Ek=Vk+1 - Vk

• Calculate the cross-products of successive edge vectors

in a counter-clockwise manner

• Use the cross-product test to identify concave polygons

• If any cross-product has a negative z-component, the

polygon is concave and we can split it along the line of the

first edge vector in the cross-product pair

## 12. Splitting example

12Splitting example

Figure 4-10 Splitting a concave

polygon using the vector method.

## 13. Splitting polygons

13Splitting polygons

## 14. Splitting a convex polygon into a set of triangles

14Splitting a convex polygon into a set of triangles

• Triangles make several important processing routines

simple

Define any sequence of three consecutive vertices to be a

new polygon (triangle)

Delete the middle triangle vertex from the original vertex

list

Apply the same procedure to the modified list to strip off

another triangle

Continue until the original polygon is reduced to just three

vertices (the last triangle)

## 15. Inside-outside tests

15Inside-outside tests

• Odd-even rule

• Draw a line from any position P to a distant point outside

the coordinate extents of the closed polyline

• Count the number of line segment crossings along this

line

• If odd, P is an interior point

• Otherwise, P is an exterior point

• Note that the line must not intersect any line segment

end-points

## 16. Nonzero winding-number rule

16Nonzero winding-number rule

• Object edges and the line must be vectors

• Count the number of times that the boundary of an object

“winds” around a particular point in the counter-clockwise

direction

• Interior points are those that have nonzero value for the winding

number

• Draw a line from P to a distant point

• As we move along the line from P, count the number of object

line segments that cross the reference line in each direction

• Add 1 to the winding number every time we intersect a segment that

crosses the line from right to left

• Subtract 1 if crossed from left to right

• If the final value of winding number is nonzero, P is an interior point;

otherwise, it is an exterior point

• For simple objects, both rules give the same results

## 17. Inside-outside test examples

17Inside-outside test examples

Figure 4-12 Identifying interior and exterior regions of a closed polyline that contains self-intersecting

segments.

## 18. Variations of nonzero winding-number rule

18Variations of nonzero winding-number rule

Figure 4-13 A fill area defined as a region that has a positive value for the winding number. This fill

area is the union of two regions, each with a counterclockwise border direction.

## 19. Variations of nonzero winding-number rule

19Variations of nonzero winding-number rule

Figure 4-14 A fill area defined as a region with a winding number greater than 1. This fill area is

the intersection of two regions, each with a counterclockwise border direction.

## 20. Variations of nonzero winding-number rule

20Variations of nonzero winding-number rule

Figure 4-15 A fill area defined as a region with a positive value for the winding number. This fill area is the

difference, A − B, of two regions, where region A has a positive border direction (counterclockwise) and

region B has a negative border direction (clockwise).

## 21. Polygon tables

21Polygon tables

Figure 4-16 Geometric data-table representation for two adjacent polygon

surface facets, formed with six edges and five vertices.

A surface shape can

be defined as a

mesh of polygon

patches. Geometric

data for objects can

be arranged in 3 lists

## 22. Plane equations

22Plane equations

• Often, information about spatial orientation of surface

components is needed

• This info is obtained from the equations that describe

polygon surfaces

• Each polygon in a scene is contained within a plane of

infinite extent

• The general equation of a plane is

where (x, y, z) is any point on the plane

## 23. Solutions to plane equations

23Solutions to plane equations

• We obtain A, B, C, and D by solving a set of three plane

equations

• Select 3 successive convex polygon vertices in a counter-clockwise

manner and solve the following set of simultaneous linear plane

equations for A/D, B/D, and C/D

• Solution to this set of equations can be obtained using Cramer’s

rule

## 24. Solutions

24Solutions

• Then

## 25. Front and back polygon faces

25Front and back polygon faces

• Faces can be determined by the sign of Ax+By+Cz+D

Figure 4-19 The normal vector N for a plane described with the equation Ax + By +Cz + D

= 0 is perpendicular to the plane and has Cartesian components (A, B, C) .

## 26. OpenGL polygon fill-area functions

26OpenGL polygon fill-area functions

Figure 4-22 Displaying polygon fill areas using a list of six vertex positions. (a) A single convex

polygon fill area generated with the primitive constant GL_POLYGON. (b) Two unconnected

triangles generated with GL_ TRIANGLES. (c) Four connected triangles generated with

GL_TRIANGLE_STRIP. (d) Four connected triangles generated with GL_TRIANGLE_FAN.

## 27. Fill-area examples

27Fill-area examples

## 28. Quadrilateral fill-areas

28Quadrilateral fill-areas

Figure 4-23 Displaying quadrilateral fill areas

using a list of eight vertex positions. (a) Two

unconnected quadrilaterals generated with

GL_QUADS. (b) Three connected

quadrilaterals generated with

GL_QUAD_STRIP.

## 29. A complex scene might require hundreds or thousands of OpenGL calls!

29A complex scene might require hundreds or

thousands of OpenGL calls!

Figure 4-24 A cube with an

edge length of 1.

## 30. Using a vertex array

30Using a vertex array

Figure 4-25 Subscript values for array pt

corresponding to the vertex coordinates

for the cube shown in Figure 4-24.

{

}

## 31. Display lists

31Display lists

const double TWO_PI = 6.2831853;

GLuint regHex;

GLdouble theta;

GLint x, y, k;

/* Set up a display list for a regular

hexagon.

* Vertices for the hexagon are six

equally spaced

* points around the circumference of

a circle.

*/

regHex = glGenLists (1); // Get an

identifier for the display list.

glNewList (regHex, GL_COMPILE);

glBegin (GL_POLYGON);

for (k = 0; k < 6; k++) {

theta = TWO_PI * k / 6.0;

x = 200 + 150 * cos (theta);

y = 200 + 150 * sin (theta);

glVertex2i (x, y);

}

glEnd ( );

glEndList ( );

glCallList (regHex);

## 32. OpenGL display-window reshape function

32OpenGL display-window reshape function

• glutReshapeFunc (winReshapeFcn);

• Activated whenever display-window size is changed