OpenGL Computer Graphics Programming with Transformations.

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OpenGL Computer Graphics

Programming with Transformations

Topics

Transformations in OpenGL Saving Current Transformation Drawing 3D Scenes with OpenGL OpenGL Functions for Modeling

and Viewing

Transformations in OpenGL CT: current transformation Simplified graphics pipeline

OpenGL maintains so-called modelview matrix• Every vertex passed down the graphics

pipeline is multiplied by this matrix

CTWindow-to-Viewport Transformation

World Window

V Q SS

Viewport

Model (Master)Coordinate System

WorldCoordinate System

ScreenCoordinate System

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

coordinate is 0.

• Translation: dz = 0

• Scaling: Sz = 1

• Rotation: z-roll

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

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);

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);

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();

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.

Saving Current Transformation

We can save and restore CTs using glPushMatrix() and glPopMatrix()

Manipulation of a stack of CT

Before

After pushCT()

After rotate2D()

CT4 = CT3 Rot

After popCT()

Saving Current Transformation Canvas functions

• void Canvas:: pushCT(void){glMatrixMode(GL_MODELVIEW);glPushMatrix();}

• void Canvas:: popCT(void){glMatrixMode(GL_MODELVIEW);glPopMatrix();}

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

Drawing 3D Scenes with OpenGL

The concept of “camera” (eye) is used for 3D viewing

Our 2D drawing is a special case of 3D drawingy

eyewindow

near plane

far plane

Viewport

view volume

Camera to produce parallel view of a 3D scene

Simplified OpenGL graphics pipeline

VM P clip Vp

modelviewmatrix

projectionmatrix

viewportmatrix

Drawing 3D Scenes with OpenGL Modelview matrix = CT

• Object transformation + camera transformation

• Applying model matrix M then viewing matrix V

Projection matrix• Shifts and scales view volume into a

standard cube (extension from –1 to 1)

• Distortion can be compensated by viewport transformation later

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

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

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.

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();glutLookAt(4, 4, 4,

0, 1, 0, 0, 1, 0)

(4, 4, 4)

(0, 1, 0)

Transformation Matrix for LookAt Camera coordinate system

• Axes: u, v, n n = eye – looku = up nv = n u

• Origin: eye (looking in the direction –n)

Transformation matrix

eyennn

eyevvv

eyeuuu

Transformation Matrix for LookAt

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

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

Plantonic Solids Provided by OpenGL

Tetrahedron• glutWireTetrahedron()

Octahedron• glutWireOctahedron()

Dodecahedron• glutWireDodecahedron()

Icosahedron• glutWireIcosahedron()

All of them are centered at the origin

Plantonic Solids Provided by OpenGL

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

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

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

Example – wire frame cylinder• GLUquadricObj *qobj;

qobj = gluNewQuadric();gluQuadricDrawStyle(qobj, GLU_LINE);gluCylinder(qobj, baseRad, topRad,

height, nSlices, nStacks);

#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-axisglEnd();glTranslated(0, 0,length -0.2); glutWireCone(0.04, 0.2, 12, 9);glPopMatrix();

//<<<<<<<<<<<<<< displayWire >>>>>>>>>>>>>>void displayWire(void){

glMatrixMode(GL_PROJECTION); // set the view volume shapeglLoadIdentity();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 cameraglLoadIdentity();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);

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-axisglRotated(-90, 1, 0, 0);axis(0.5); // y-axisglPopMatrix();

glPushMatrix();glTranslated(0.5, 0.5, 0.5); // big cube at (0.5, 0.5, 0.5)glutWireCube(1.0);glPopMatrix();

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();

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();

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();

//<<<<<<<<<<<<<<<< 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 whiteglViewport(0, 0, 640, 480);glutMainLoop();

OpenGL Computer Graphics Programming with Transformations.

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