1. Geometric Transformations

2. Class 3차원 Vector와 4x4 Matrix Class를 만들어 본다 class Vector3 { public:
float p[3]; };
class Mat4x4 {
public:
float p[4][4]; };

3. OpenGL Geometric Transformations
glMatrixMode(GL_MODELVIEW);

4. OpenGL Geometric Transformations
Basic Transpormation:
glLoadIdentity();
glTranslatef(tx, ty, tz);
glRotatef(theta, vx, vy, vz); angle-axis
(vx, vy, vz) is automatically normalized
glScalef(sx, sy, sz);
glLoadMatrixf(Glfloat elems[16]);
Multiplication
glMultMatrixf(Glfloat elems[16]);
The current matrix is postmultiplied by the matrix
Column major

5. OpenGL Geometric Transformations
Getting the current matrix value:
glGetFloatv (GL_MODELVIEW_MATRIX, GLfloat elems[16]);
Column major 4 8 12 1 5 9 13 2 6 10 14 3 7 11 15
GLfloat mat [16];
glGetFloatv (GL_MODELVIEW_MATRIX, mat);

6. OpenGL Geometric Transformations
Matrix Direct Manipulation:
glLoadMatrixf(GLfloat elems[16]);
Column major
glMultMatrixf(GLfloat elems[16]);
The current matrix is postmultiplied by the matrix 4 8 12 1 5 9 13 2 6 10 14 3 7 11 15
glLoadIdentity();
glMultMatrixf (M1);
glMultMatrixf (M2);
M = M1∙M2

7. OpenGL GLUT Animation Function
GLUT Idle Callback fuction:
Idling: when there is nothing to do.
Redraw the scene:
void MyIdle() {
… // things to do
… // when Idling }
glutIdleFunc ( MyIdle );
glutPostRedisplay ( );

8. 바람개비(풍차)만들기

9. Hierarchical Modeling
A hierarchical model is created by nesting the descriptions of subparts into one another to form a tree organization

10. OpenGL Matrix Stacks Stack processing
The top of the stack is the “current” matrix
glPushMatrix(); // Duplicate the current matrix at the top
glPopMatrix(); // Remove the matrix at the top

11. Homework #2 다각형을 이용하여 움직이는 2차원 아름다운 애니메이션 만들기
Push/Pop Matrix를 사용하여 2단계 이상의 계층적인 움직임 디자인을 할 것
Ex): 태양계 시스템 (태양지구달) 사람의 움직임 (몸어깨팔꿈치손목손)
숙제제출: 이메일 제출 (Screenshot, code, report)
숙제마감: 4월 28일 수요일 23시59분까지

12. Viewing Part I: Two-Dimensional Viewing
Sang Il Park
Sejong University
Lots of slides are stolen from Jehee Lee’s

13. Viewing Pipeline

14. Two-Dimensional Viewing
Two dimensional viewing transformation
From world coordinate scene description to device (screen) coordinates

15. Normalization and Viewport Transformation
World coordinate clipping window
Normalization square: usually [-1,1]x[-1,1]
Device coordinate viewport

16. OpenGL 2D Viewing Projection Mode GLU clipping-window function
glMatrixMode(GL_PROJECTION);
GLU clipping-window function
gluOrtho2D(xwmin,xwmax,ywmin,ywmax);
Normalized to [-1,1]x[-1,1]
OpenGL viewport function
glViewport(xvmin,xvmax,yvmin,yvmax);

17. GLUT window Functions:
glutInitWindowPosition
glutInitWindowSize
glutCreateWindow
glutDestroyWindow
glutSetWindow/glutGetWindow
….. And more!
See the text book pp 346~354

18. Clipping Remove portion of output primitives outside clipping window
Two approaches
Clip during scan conversion: Per-pixel bounds check
Clip analytically, then scan-convert the modified primitives

19. Two-Dimensional Clipping
Point clipping – trivial
Line clipping
Cohen-Sutherland
Cyrus-beck
Liang-Barsky
Fill-area clipping
Sutherland-Hodgeman
Weiler-Atherton
Curve clipping
Text clipping

20. Line Clipping Basic calculations:
Is an endpoint inside or outside the clipping window?
Find the point of intersection, if any, between a line segment and an edge of the clipping window.
Both endpoints inside:
trivial accept
One inside: find
intersection and clip
Both outside: either
clip or reject

21. Cohen-Sutherland Line Clipping
One of the earliest algorithms for fast line clipping
Identify trivial accepts and rejects by bit operations
0000
1000
1001
0001
0101
0100
0110
0010
1010
Clipping window
< Region code for each endpoint >
above
below
right
left
Bit

22. Cohen-Sutherland Line Clipping
Compute region codes for two endpoints
If (both codes = 0000 ) trivially accepted
If (bitwise AND of both codes  0000) trivially rejected
Otherwise, divide line into two segments
test intersection edges in a fixed order.
(e.g., top-to-bottom, right-to-left)
0000
1000
1001
0001
0101
0100
0110
0010
1010
Clipping window

23. Cohen-Sutherland Line Clipping
Fixed order testing and clipping cause needless clipping (external intersection)

24. Cohen-Sutherland Line Clipping
This algorithm can be very efficient if it can accept and reject primitives trivially
Clip window is much larger than scene data
Most primitives are accepted trivially
Clip window is much smaller than scene data
Most primitives are rejected trivially
Good for hardware implementation

25. Cyrus-Beck Line Clipping
Use a parametric line equation
Reduce the number of calculating intersections by exploiting the parametric form
Notations
Ei : edge of the clipping window
Ni : outward normal of Ei
An arbitrary point PEi on edge Ei

26. Cyrus-Beck Line Clipping

27. Cyrus-Beck Line Clipping
Solve for the value of t at the intersection of P0P1 with the edge
Ni · [P(t) - PEi] = 0 and P(t) = P0 + t(P1 - P0)
letting D = (P1 - P0),
Where
Ni  0
D  0 (that is, P0  P1)
Ni · D  0 (if not, no intersection)

28. Cyrus-Beck Line Clipping
Given a line segment P0P1, find intersection points against four edges
Discard an intersection point if t  [0,1]
Label each intersection point either PE (potentially entering) or PL (potentially leaving)
Choose the smallest (PE, PL) pair that defines the clipped line

29. Cyrus-Beck Line Clipping
Cyrus-Beck is efficient when many line segments need to be clipped
Can be extended easily to convex polygon (rather than upright rectangle) clip windows

30. Liang-Barsky Line Clipping
Liang-Barsky optimized Cyrus-Beck for upright rectangular clip windows
Q(x2.y2) tR tT
P(x1,y1) tL tB

31. Liang-Barsky Line ClippingtT T B tB

32. Nicholl-Lee-Nicholl Line Clipping
Divide more cases and reduce the computation.
(3)
(2)
(1)

33. General Clipping Window
Line clipping using nonrectangular polygon clip windows
Convex polygon
Cyrus-Beck algorithm can be readily extended
Concave polygon
Split the concave polygon into convex polygons
Vector Method for Concave Splitting
Calculate edge-vector cross products in a counterclockwise order
If any z component turns out to be negative, the polygon is concave