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Figures based on regular polygons (lab3) A polygon is regular if it is simple, if all its sides have equal lengths, and if adjacent sides meet at equal interior angles. name n-gon to a regular polygon having n sides. Familiar examples are the 4-gon (a square), a 5-gon (a regular pentagon), 8-gon (a regular octagon), If the number of sides of an n-gon is large the polygon approximates a circle in appearance. In fact this is used later as one way to implement the drawing of a circle. The vertices of an n-gon lie on a circle, the so-called parent circle of the n-gon, and their locations are easily calculated.

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Finding vertices of ngon

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ngon void ngon(int n, float cx, float cy, float radius, float rotAngle) { // assumes global Canvas object, cvs if(n < 3) return; // bad number of sides double angle = rotAngle * / 180; // initial angle double angleInc = 2 * /n; //angle increment …// moveTo(radius * cos(angle) + cx, radius * sin(angle) + cy); for(int k = 0; k < n; k++) // repeat n times { angle += angleInc; … //connect (radius * cos(angle) + cx, radius * sin(angle) + cy); }

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Variation of n-gons stellation formed by connecting every other vertex. rosette, formed by connecting each vertex to every other vertex.

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Rosette void Rosette(int N, float radius) { Point2 pt[big enough value for largest rosette]; generate the vertices pt[0],...,pt[N-1], as in Figure 3.43 for(int i = 0; i < N - 1; i++) for(int j = i + 1; j < N ; j++) { …//.moveTo(pt[i]); … // connect all the vertices alternatively }

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Drawing Circles void drawCircle(Point2 center, float radius) { const int numVerts = 50; // use larger for a better circle ngon(numVerts, center.getX(), center.getY(), radius, 0); }

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Drawing Arcs void drawArc(Point2 center, float radius, float startAngle, float sweep) { // startAngle and sweep are in degrees const int n = 30; // number of intermediate segments in arc float angle = startAngle * / 180; // initial angle in radians float angleInc = sweep * /(180 * n); // angle increment float cx = center.getX(), cy = center.getY(); …//moveTo(cx + radius * cos(angle), cy + radius * sin(angle)); for(int k = 1; k < n; k++, angle += angleInc) ….//connect all the points: (cx + radius * cos(angle), cy + radius * sin(angle)); }

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Order of Transformation Matters Scale, translate Translate scale Rotate, Translate Translate, Rotate

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An Example Line (A,B), A=(0,0,0) B=(1,0,0) S=(2,2,1) T=(2,3,0) First Scale, then Translate A’=(0,0,0) B’=(2,0,0); A’’=(2,3,0); B’’=(4,3,0) First Translate, then Scale A’=(2,3,0) B’=(3,3,0); A’’=(4,6,0); B’’=(6,6,0)

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10 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 OpenGL Transformations

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11 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Objectives Learn how to carry out transformations in OpenGL Rotation Translation Scaling Introduce OpenGL matrix modes Model-view Projection

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12 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 OpenGL Matrices In OpenGL matrices are part of the state Multiple types Model-View ( GL_MODELVIEW ) Projection ( GL_PROJECTION ) Texture ( GL_TEXTURE ) (ignore for now) Color( GL_COLOR ) (ignore for now) Single set of functions for manipulation Select which to manipulated by glMatrixMode(GL_MODELVIEW); glMatrixMode(GL_PROJECTION);

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13 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Current Transformation Matrix (CTM) Conceptually there is a 4 x 4 homogeneous coordinate matrix, the current transformation matrix (CTM) that is part of the state and is applied to all vertices that pass down the pipeline The CTM is defined in the user program and loaded into a transformation unit CTMvertices p p’=Cp C

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14 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 CTM operations The CTM can be altered either by loading a new CTM or by postmutiplication Load an identity matrix: C I Load an arbitrary matrix: C M Load a translation matrix: C T Load a rotation matrix: C R Load a scaling matrix: C S Postmultiply by an arbitrary matrix: C CM Postmultiply by a translation matrix: C CT Postmultiply by a rotation matrix: C C R Postmultiply by a scaling matrix: C C S

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15 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Rotation about a Fixed Point Start with identity matrix: C I Move fixed point to origin: C CT 1 Rotate: C CR Move fixed point back: C CT 2 Note that T 2 = T 1 -1 Result of Postmultiplication : C = T 1 R T 2 which is not we want. (remember that the first matrix applied is on the far right)

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16 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Reversing the Order We want C = T 2 R T 1 so we must do the operations in the following order C I C CT 2 C CR C CT 1 Each operation corresponds to one function call in the program. Note that the last operation specified is the first matrix applied to vertex (far right matrix)

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17 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 CTM in OpenGL OpenGL has a model-view and a projection matrix in the pipeline which are concatenated together to form the CTM Can manipulate each by first setting the correct matrix mode

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18 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Rotation, Translation, Scaling glRotatef(theta, vx, vy, vz) glTranslatef(dx, dy, dz) glScalef( sx, sy, sz) glLoadIdentity() Load an identity matrix: Multiply on right: theta in degrees, ( vx, vy, vz ) define axis of rotation Each has a float (f) and double (d) format ( glScaled )

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19 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Example Rotation about z axis by 30 degrees with a fixed point of (1.0, 2.0, 3.0) Remember that last matrix specified in the program is the first applied glMatrixMode(GL_MODELVIEW); glLoadIdentity(); glTranslatef(1.0, 2.0, 3.0); glRotatef(30.0, 0.0, 0.0, 1.0); glTranslatef(-1.0, -2.0, -3.0);

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20 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Arbitrary Matrices Can load and multiply by matrices defined in the application program The matrix m is a one dimension array of 16 elements which are the components of the desired 4 x 4 matrix stored by columns In glMultMatrixf, m multiplies the existing matrix on the right glLoadMatrixf(m) glMultMatrixf(m)

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21 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Matrix Stacks In many situations we want to save transformation matrices for use later Traversing hierarchical data structures (Chapter 10) Avoiding state changes when executing display lists OpenGL maintains stacks for each type of matrix Access present type (as set by glMatrixMode) by glPushMatrix() glPopMatrix()

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22 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Reading Back Matrices Can also access matrices (and other parts of the state) by query functions For matrices, we use as glGetIntegerv glGetFloatv glGetBooleanv glGetDoublev glIsEnabled double m[16]; glGetFloatv(GL_MODELVIEW, m);

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23 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Using Transformations Example: use idle function to rotate a cube and mouse function to change direction of rotation Start with a program that draws a cube ( colorcube.c ) in a standard way Centered at origin Sides aligned with axes Will discuss modeling later

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24 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 main.c void main(int argc, char **argv) { glutInit(&argc, argv); glutInitDisplayMode(GLUT_DOUBLE | GLUT_RGB | GLUT_DEPTH); glutInitWindowSize(500, 500); glutCreateWindow("colorcube"); glutReshapeFunc(myReshape); glutDisplayFunc(display); glutIdleFunc(spinCube); glutMouseFunc(mouse); glEnable(GL_DEPTH_TEST); glutMainLoop(); }

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25 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Idle and Mouse callbacks void spinCube() { theta[axis] += 2.0; if( theta[axis] > ) theta[axis] -= 360.0; glutPostRedisplay(); } void mouse(int btn, int state, int x, int y) { if(btn==GLUT_LEFT_BUTTON && state == GLUT_DOWN) axis = 0; if(btn==GLUT_MIDDLE_BUTTON && state == GLUT_DOWN) axis = 1; if(btn==GLUT_RIGHT_BUTTON && state == GLUT_DOWN) axis = 2; }

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26 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Display callback void display() { glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT); glLoadIdentity(); glRotatef(theta[0], 1.0, 0.0, 0.0); glRotatef(theta[1], 0.0, 1.0, 0.0); glRotatef(theta[2], 0.0, 0.0, 1.0); colorcube(); glutSwapBuffers(); } Note that because of fixed from of callbacks, variables such as theta and axis must be defined as globals Camera information is in standard reshape callback

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27 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Using the Model-view Matrix In OpenGL the model-view matrix is used to Position the camera Can be done by rotations and translations but is often easier to use gluLookAt Build models of objects The projection matrix is used to define the view volume and to select a camera lens

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28 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Model-view and Projection Matrices Although both are manipulated by the same functions, we have to be careful because incremental changes are always made by postmultiplication For example, rotating model-view and projection matrices by the same matrix are not equivalent operations. Postmultiplication of the model-view matrix is equivalent to premultiplication of the projection matrix

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29 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Smooth Rotation From a practical standpoint, we are often want to use transformations to move and reorient an object smoothly Problem: find a sequence of model-view matrices M 0,M 1,…..,M n so that when they are applied successively to one or more objects we see a smooth transition For orientating an object, we can use the fact that every rotation corresponds to part of a great circle on a sphere Find the axis of rotation and angle Virtual trackball (see text)

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30 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Incremental Rotation Consider the two approaches For a sequence of rotation matrices R 0,R 1,…..,R n, find the Euler angles for each and use R i = R iz R iy R ix Not very efficient Use the final positions to determine the axis and angle of rotation, then increment only the angle Quaternions can be more efficient than either

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31 Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009 Interfaces One of the major problems in interactive computer graphics is how to use two- dimensional devices such as a mouse to interface with three dimensional obejcts Example: how to form an instance matrix? Some alternatives Virtual trackball 3D input devices such as the spaceball Use areas of the screen Distance from center controls angle, position, scale depending on mouse button depressed

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