Transcript OpenGL Transformations

OpenGL
Computer Graphics
Programming with
Transformations
Topics
Transformations in OpenGL
 Saving Current Transformation
 Drawing 3D Scenes with OpenGL
 OpenGL Functions for Modeling and
Viewing

2
Transformations in OpenGL
CT: current transformation
 Simplified graphics pipeline

V
CT
V
Q
Window-to-Viewport
Transformation
S
S
Q
Viewport
Model (Master)
Coordinate System

World Window
World
Coordinate System
Screen
Coordinate System
OpenGL maintains so-called modelview
matrix
• Every vertex passed down the graphics
pipeline is multiplied by this matrix
3
Transformations in OpenGL

OpenGL is a 3D graphics package
• Transformations are 3D

How does it work in 2D?
• 2D drawing is done in the xy-plane, z
y
coordinate is 0.
• Translation: dz = 0
• Scaling: Sz = 1
• Rotation: z-roll
z
x
4
Transformations in OpenGL

Fundamental Transformations
• Translation: glTranslated(dx, dy, dz)
for 2D: glTranslated(dx, dy, 0)
• Scaling: glScaled(sx, sy, sz)
for 2D: glScaled(sx, sy, 1.0)
• Rotation: glRotated(angle, ux, uy, uz)
for 2D: glRotated(angle, 0, 0, 1)

Transformations does not set CT
directly, a matrix is postmultiplied to CT
• CT = CT  M
5
Transformations in OpenGL

Canvas functions
• void Canvas:: initCT(void)
{
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
}
• void Canvas:: scale2D(double sx, double sy)
{
glMatrixMode(GL_MODELVIEW);
glScaled(dx, dy, 1.0);
}
6
Transformations in OpenGL

Canvas functions
• void Canvas:: translate2D(double dx, double dy)
{
glMatrixMode(GL_MODELVIEW);
glTranslated(dx, dy, 0);
}
• void Canvas:: rotate2D(double angle)
{
glMatrixMode(GL_MODELVIEW);
glRotated(angle, 0.0, 0.0, 1.0);
}
7
Transformations Example

Draw a house. Draw another house by
rotating it through -30° and then
translating it through (32, 25)
• cvs.initCT();
house();
cvs.translate2D(32, 25);
cvs.rotate2D(-30.0);
house();
8
Transformations Example
9
Transformations Example

Think of it in two different ways
• Q =T(32, 25)R(-30)P
 CT = CT  T(32, 25)  R(-30)
• Translate the coordinate system through
(32, 25) and then rotate it through –30°

The code generated by these two ways
is identical.
10
Saving Current Transformation
We can save and restore CTs using
glPushMatrix() and glPopMatrix()
 Manipulation of a stack of CT

Before
After
pushCT()
After
popCT()
After
rotate2D()
= CT3  Rot
CT4
CT4
CT3
CT3
CT3
CT3
CT2
CT2
CT2
CT2
CT1
CT1
CT1
CT1
11
Saving Current Transformation

Canvas functions
• void Canvas:: pushCT(void)
{
glMatrixMode(GL_MODELVIEW);
glPushMatrix();
}
• void Canvas:: popCT(void)
{
glMatrixMode(GL_MODELVIEW);
glPopMatrix();
}
12
Saving CT Examples
• Master coordinate system: where an object
is defined
• Modeling transformation: transforms an
object from its master coordinate system
to world coordinate system to produce an
instance
• Instance: a picture of an object in the
scene
13
Drawing 3D Scenes with OpenGL
The concept of “camera” (eye) is used
for 3D viewing
 Our 2D drawing is a special case of 3D
drawing y
far plane

view volume
near plane
z
eye
x
window
Viewport
14
Drawing 3D Scenes with OpenGL

Camera to produce parallel view of a 3D
scene
15
Drawing 3D Scenes with OpenGL

Simplified OpenGL graphics pipeline
VM
modelview
matrix
P
projection
matrix
clip
Vp
viewport
matrix
16
Drawing 3D Scenes with OpenGL

Modelview matrix = CT
• Object transformation + camera transformation
• Applying model matrix M then viewing matrix V
17
Drawing 3D Scenes with OpenGL

Projection matrix
• Shifts and scales view volume into a
standard cube (extension from –1 to 1)
• Distortion can be compensated by viewport
transformation later
18
Drawing 3D Scenes with OpenGL

Viewport matrix
• Maps surviving portion of objects into a 3D
viewport after clipping is performed
• Standard cube 
block w/ x and y extending across viewport
and z from 0 to 1
19
OpenGL Modeling and Viewing Functions

Modeling transformation
• Translation: glTranslated(dx, dy, dz)
• Scaling: glScaled(sx, sy, sz)
• Rotation: glRotated(angle, ux, uy, uz)

Camera for parallel projection
• glMatrixMode(GL_PROJECTION);
glLoadIdentity();
glOrtho(left, right, bottom, top, near, far)
• Example
• near=2: near plane is 2 units in front of eye
far=20: far plane is 20 units in front of eye
20
OpenGL Modeling and Viewing Functions

Positioning and aiming camera
• glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
glutLookAt(eye.x, eye.y, eye.z, // eye position
look.x, look.y, look.z, // look at point
up.x, up.y, up.z)
// up vector
• Up vector is often set to (0, 1, 0)

glutLookAt() builds a matrix that converts
world coordinates into eye coordinates.
21
Set up a Typical Camera - Example

glMatrixMode(GL_PROJECTION);
glLoadIdentity();
glOrtho(-3.2, 3.2, -2.4, 2.4, 1, 50)
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
(4, 4, 4)
glutLookAt(4, 4, 4,
0, 1, 0,
0, 1, 0)
(0, 1, 0)
22
Transformation Matrix for LookAt

Camera coordinate system
• Axes: u, v, n
n = eye – look
u = up  n
v=nu
• Origin: eye (looking in the direction –n)

Transformation matrix
u x

v x
V 
nx

0

uy uz
vy vz
ny nz
0
0
 eye  u 

 eye  v 
 eye  n 


1

23
Transformation Matrix for LookAt
24
Elementary 3D Shapes Provided by
OpenGL

Cube
• glutWireCube(GLdouble size)
• size = length of a side

Sphere
• glutWireSphere(GLdouble radius, GLint
nSlices, GLint nStacks)
• Approximated by polygonal faces
• nSlices = #polygons around z-axis
• nStacks = #bands along z-axis
25
Elementary 3D Shapes Provided by
OpenGL

Torus
• glutWireTorus(GLdouble inRad, GLdouble
outRad, GLint nSlices, GLint nStacks)
• Approximated by polygonal faces

Teapots
• glutWireTeapot(GLdouble size)

There are solid counterparts of the wire
objects
26
Plantonic Solids Provided by OpenGL

Tetrahedron
• glutWireTetrahedron()

Octahedron
• glutWireOctahedron()

Dodecahedron
• glutWireDodecahedron()

Icosahedron
• glutWireIcosahedron()

All of them are centered at the origin
27
Plantonic Solids Provided by OpenGL
28
Cone Provided by OpenGL

Cone
• glutWireCone(GLdouble baseRad, GLdouble
height, GLint nSlices, GLint nStacks)
Axis coincides with the z-axis
 Base rests on xy-plane and extends to z
= height
 baseRad: radius at z = 0

29
Tapered Cylinder Provided by OpenGL

Tapered cylinder
• gluCylinder(GLUquadricObj *qobj,
GLdouble baseRad, GLdouble topRad,
GLdouble height, GLint nSlices, GLint
nStacks)
Axis coincides with the z-axis
 Base rests on xy-plane and extends to z
= height
 baseRad: radius at z = 0
 topRad: radius at z = height

30
Tapered Cylinder Provided by OpenGL
A family of shapes distinguished by the
value of topRad
 To draw, we have to

• Deifne a new quadric object
• Set drawing style
• GLU_LINE: wire frame
• GLU_FILL: solid rendering
• Draw the object
31
Tapered Cylinder Provided by OpenGL

Example – wire frame cylinder
• GLUquadricObj *qobj;
qobj = gluNewQuadric();
gluQuadricDrawStyle(qobj, GLU_LINE);
gluCylinder(qobj, baseRad, topRad,
height, nSlices, nStacks);
32
33
34
#include <gl/glut.h>
//<<<<<<<<<<<<<<<<<<< axis >>>>>>>>>>>>>>
void axis(double length)
{ // draw a z-axis, with cone at end
glPushMatrix();
glBegin(GL_LINES);
glVertex3d(0, 0, 0); glVertex3d(0,0,length); // along the
z-axis
glEnd();
glTranslated(0, 0,length -0.2);
glutWireCone(0.04, 0.2, 12, 9);
glPopMatrix();
}
35
//<<<<<<<<<<<<<< displayWire >>>>>>>>>>>>>>
void displayWire(void)
{
glMatrixMode(GL_PROJECTION); // set the view volume
shape
glLoadIdentity();
glOrtho(-2.0*64/48.0, 2.0*64/48.0, -2.0, 2.0, 0.1, 100);
glMatrixMode(GL_MODELVIEW); // position and aim the
camera
glLoadIdentity();
gluLookAt(2.0, 2.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0);
// to obtain the picture shown in Figure 5.59 we have to
// change the eye location as follows
// gluLookAt(1.0, 1.0, 2.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0);
36
glClear(GL_COLOR_BUFFER_BIT); // clear the screen
glColor3d(0,0,0); // draw black lines
axis(0.5);
// z-axis
glPushMatrix();
glRotated(90, 0, 1, 0);
axis(0.5);
// x-axis
glRotated(-90, 1, 0, 0);
axis(0.5);
// y-axis
glPopMatrix();
glPushMatrix();
glTranslated(0.5, 0.5, 0.5); // big cube at (0.5, 0.5, 0.5)
glutWireCube(1.0);
glPopMatrix();
37
glPushMatrix();
glTranslated(1.0,1.0,0); // sphere at (1,1,0)
glutWireSphere(0.25, 10, 8);
glPopMatrix();
glPushMatrix();
glTranslated(1.0,0,1.0); // cone at (1,0,1)
glutWireCone(0.2, 0.5, 10, 8);
glPopMatrix();
glPushMatrix();
glTranslated(1,1,1);
glutWireTeapot(0.2); // teapot at (1,1,1)
glPopMatrix();
38
glPushMatrix();
glTranslated(0, 1.0 ,0); // torus at (0,1,0)
glRotated(90.0, 1,0,0);
glutWireTorus(0.1, 0.3, 10,10);
glPopMatrix();
glPushMatrix();
glTranslated(1.0, 0 ,0); // dodecahedron at (1,0,0)
glScaled(0.15, 0.15, 0.15);
glutWireDodecahedron();
glPopMatrix();
39
glPushMatrix();
glTranslated(0, 1.0 ,1.0); // small cube at (0,1,1)
glutWireCube(0.25);
glPopMatrix();
glPushMatrix();
glTranslated(0, 0 ,1.0); // cylinder at (0,0,1)
GLUquadricObj * qobj;
qobj = gluNewQuadric();
gluQuadricDrawStyle(qobj,GLU_LINE);
gluCylinder(qobj, 0.2, 0.2, 0.4, 8,8);
glPopMatrix();
glFlush();
}
40
//<<<<<<<<<<<<<<<< main >>>>>>>>>>>>>>>>
void main(int argc, char **argv)
{
glutInit(&argc, argv);
glutInitDisplayMode(GLUT_SINGLE | GLUT_RGB );
glutInitWindowSize(640,480);
glutInitWindowPosition(100, 100);
glutCreateWindow("Transformation testbed - wireframes");
glutDisplayFunc(displayWire);
glClearColor(1.0f, 1.0f, 1.0f,0.0f); // background is white
glViewport(0, 0, 640, 480);
glutMainLoop();
}
41