1. drawing curved surfaces
Class 3:Chaps 2 cont.
trig review
drawing curves
depth buffer
perspective
drawing curved surfaces

2. Trig Review:
Definition of sine and cosine
(Unit circle, radius=1)
(cos sin )
(cos sin ) 
From Wikipedia, modified

3. cos(0) =
sin(0) =
1.0
0.0
(cos(0), sin(0))

4. cos(PI/2) =
sin(PI/2) =
0.0
1.0
(cos(PI/2), sin(PI/2))

5. cos(PI) =
sin(PI) =
-1.0
0.0
(cos(PI), sin(PI))

6. Change the radius from 1 to R
(Rcos(), Rsin()) R 

7. Move the center from (0,0) to (X,Y)
(X+Rcos(), Y+Rsin()) R 
(X,Y) Y X
(0,0)

8. Curved Objects Run the program circle.cpp. Observe
instructions in comment and console screen
global variables
new version of color: glColor3ub
use of rand()
glutPostRedisplay in keyInput

9. parametric equation of circle

10. The book also has parabola.cpp,
another example of a curve described
parametrically. Run it on your own.

11. 3-D Depth Buffer Perspective

12. circularAnnuluses.cpp (modified)

13. Overwritten void drawDisc(float R, float X, float Y, float Z);
glColor3f(0.0, 0.0, 1.0);
drawDisc(20.0, 25.0, 75.0, 0.0);
glColor3f(1.0, 1.0, 1.0);
drawDisc(10.0, 25.0, 75.0, 0.0);

14. The real deal if (isWire) glPolygonMode(GL_FRONT, GL_LINE);
else glPolygonMode(GL_FRONT, GL_FILL);
glColor3f(1.0, 0.0, 0.0);
glBegin(GL_TRIANGLE_STRIP);
for(i = 0; i <= N; ++i) {
angle = 2 * PI * i / N;
glVertex3f(50 + cos(angle) * 10.0, 30 + sin(angle) * 10.0, 0.0);
glVertex3f(50 + cos(angle) * 20.0, 30 + sin(angle) * 20.0, 0.0); }
glEnd();

15. Floating glEnable(GL_DEPTH_TEST); // Enable depth testing.
glColor3f(0.0, 1.0, 0.0);
drawDisc(20.0, 75.0, 75.0, 0.0);
glColor3f(1.0, 1.0, 1.0);
drawDisc(10.0, 75.0, 75.0, 0.5);
// Compare this z-value with that of the green disc.
glDisable(GL_DEPTH_TEST); // Disable depth testing.

16. To use the depth buffer, you must initialize it in main.
glutInitDisplayMode(GLUT_DOUBLE |
GLUT_RGB |
GLUT_DEPTH);
And clear it
glClear(GL_COLOR_BUFFER_BIT |
GL_DEPTH_BUFFER_BIT);

17. Hidden surface removal.
glOrtho(0.0, 100.0, 0.0, 100.0, -1.0, 1.0);

18. What is it? x = Rcos(t) -10*PI <= t <= 10*PI y = Rsin(t)
z = t
-10*PI <= t <= 10*PI

19. Helix.cpp x = Rcos(t) -10*PI <= t <= 10*PI y = Rsin(t)
z = t
-10*PI <= t <= 10*PI
_____ _____ z:

20. Helixmod.cpp cont
after a modification
of view

21. How do we indicate depth in pictures?

22. glFrustum(left, right, bottom, top, near, far)

23. glFrustum(-5.0,5.0,-5.0,5.0,4.0,100.0);