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**OpenGL Computer Graphics**

Programming with Transformations

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**Topics Transformations in OpenGL Saving Current Transformation**

Drawing 3D Scenes with OpenGL OpenGL Functions for Modeling and Viewing

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**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 V Q Window-to-Viewport Transformation S CT S V Q Viewport World Window Screen Coordinate System Model (Master) Coordinate System World Coordinate System

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**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 y z x

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**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

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**Transformations in OpenGL**

Canvas functions void Canvas:: initCT(void) { glMatrixMode(GL_MODELVIEW); glLoadIdentity(); } void Canvas:: scale2D(double sx, double sy) glScaled(dx, dy, 1.0);

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**Transformations in OpenGL**

Canvas functions void Canvas:: translate2D(double dx, double dy) { glMatrixMode(GL_MODELVIEW); glTranslated(dx, dy, 0); } void Canvas:: rotate2D(double angle) glRotated(angle, 0.0, 0.0, 1.0);

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**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);

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**Transformations Example**

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**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.

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**Saving Current Transformation**

We can save and restore CTs using glPushMatrix() and glPopMatrix() Manipulation of a stack of CT After rotate2D() After popCT() Before After pushCT() CT4 CT = CT3 Rot CT3 CT3 CT3 CT3 CT2 CT2 CT2 CT2 CT1 CT1 CT1 CT1

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**Saving Current Transformation**

Canvas functions void Canvas:: pushCT(void) { glMatrixMode(GL_MODELVIEW); glPushMatrix(); } void Canvas:: popCT(void) glPopMatrix();

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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

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**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 far plane y view volume near plane z x eye Viewport window

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**Drawing 3D Scenes with OpenGL**

Camera to produce parallel view of a 3D scene

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**Drawing 3D Scenes with OpenGL**

Simplified OpenGL graphics pipeline VM P clip Vp modelview matrix projection matrix viewport matrix

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**Drawing 3D Scenes with OpenGL**

Modelview matrix = CT Object transformation + camera transformation Applying model matrix M then viewing matrix V

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**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

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**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

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**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

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**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.

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**Set up a Typical Camera - Example**

glMatrixMode(GL_PROJECTION); glLoadIdentity(); glOrtho(-3.2, 3.2, -2.4, 2.4, 1, 50) glMatrixMode(GL_MODELVIEW); glutLookAt(4, 4, 4, 0, 1, 0, 0, 1, 0) (4, 4, 4) (0, 1, 0)

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**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

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**Transformation Matrix for LookAt**

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**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

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**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

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**Plantonic Solids Provided by OpenGL**

Tetrahedron glutWireTetrahedron() Octahedron glutWireOctahedron() Dodecahedron glutWireDodecahedron() Icosahedron glutWireIcosahedron() All of them are centered at the origin

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**Plantonic Solids Provided by OpenGL**

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**Cone Provided by OpenGL**

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

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**Tapered Cylinder Provided by OpenGL**

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

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**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

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**Tapered Cylinder Provided by OpenGL**

Example – wire frame cylinder GLUquadricObj *qobj; qobj = gluNewQuadric(); gluQuadricDrawStyle(qobj, GLU_LINE); gluCylinder(qobj, baseRad, topRad, height, nSlices, nStacks);

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**#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(); }

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//<<<<<<<<<<<<<< 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 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);

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

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glPushMatrix(); glTranslated(1.0,1.0,0); // sphere at (1,1,0) glutWireSphere(0.25, 10, 8); glPopMatrix(); glTranslated(1.0,0,1.0); // cone at (1,0,1) glutWireCone(0.2, 0.5, 10, 8); glTranslated(1,1,1); glutWireTeapot(0.2); // teapot at (1,1,1)

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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(); glTranslated(1.0, 0 ,0); // dodecahedron at (1,0,0) glScaled(0.15, 0.15, 0.15); glutWireDodecahedron();

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glPushMatrix(); glTranslated(0, 1.0 ,1.0); // small cube at (0,1,1) glutWireCube(0.25); glPopMatrix(); 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); glFlush(); }

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//<<<<<<<<<<<<<<<< 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(); }

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