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1 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Building Models modified by Ray Wisman rwisman@ius.edu Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico

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2 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Objectives Introduce simple data structures for building polygonal models Vertex lists Edge lists OpenGL vertex arrays

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3 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Representing a Mesh Consider a mesh There are 8 nodes and 12 edges 5 interior polygons 6 interior (shared) edges Each vertex has a location v i = (x i y i z i ) v1v1 v2v2 v7v7 v6v6 v8v8 v5v5 v4v4 v3v3 e1e1 e8e8 e3e3 e2e2 e 11 e6e6 e7e7 e 10 e5e5 e4e4 e9e9 e 12

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4 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Simple Representation Define each polygon by the geometric locations of its vertices Leads to OpenGL code such as Inefficient and unstructured Consider moving a vertex to a new location Must search for all occurrences glBegin(GL_POLYGON); glVertex3f(x1, x1, x1); glVertex3f(x6, x6, x6); glVertex3f(x7, x7, x7); glEnd();

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5 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Inward and Outward Facing Polygons The order {v 1, v 6, v 7 } and {v 6, v 7, v 1 } are equivalent in that the same polygon will be rendered by OpenGL but the order {v 1, v 7, v 6 } is different The first two describe outwardly facing polygons Use the right-hand rule = counter-clockwise encirclement of outward-pointing normal OpenGL can treat inward and outward facing polygons differently

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6 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Geometry vs Topology Generally it is a good idea to look for data structures that separate the geometry from the topology Geometry: locations of the vertices Topology: organization of the vertices and edges Example: a polygon is an ordered list of vertices with an edge connecting successive pairs of vertices and the last to the first Topology holds even if geometry changes

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7 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Vertex Lists Put the geometry in an array Use pointers from the vertices into this array Introduce a polygon list x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 x 4 y 4 z 4 x 5 y 5 z 5. x 6 y 6 z 6 x 7 y 7 z 7 x 8 y 8 z 8 P1 P2 P3 P4 P5 v1v7v6v1v7v6 v8v5v6v8v5v6 topology geometry

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8 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Shared Edges Vertex lists will draw filled polygons correctly but if we draw the polygon by its edges, shared edges are drawn twice Can store mesh by edge list

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9 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Edge List v1v1 v2v2 v7v7 v6v6 v8v8 v5v5 v3v3 e1e1 e8e8 e3e3 e2e2 e 11 e6e6 e7e7 e 10 e5e5 e4e4 e9e9 e 12 e1 e2 e3 e4 e5 e6 e7 e8 e9 x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 x 4 y 4 z 4 x 5 y 5 z 5. x 6 y 6 z 6 x 7 y 7 z 7 x 8 y 8 z 8 v1 v6 Note polygons are not represented

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10 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Modeling a Cube GLfloat vertices[][3] = {{-1.0,-1.0,-1.0}, {1.0,-1.0,-1.0}, {1.0,1.0,-1.0}, {-1.0,1.0,-1.0}, {-1.0,-1.0,1.0}, {1.0,-1.0,1.0}, {1.0,1.0,1.0}, {-1.0,1.0,1.0}}; GLfloat colors[][3] = {{0.0,0.0,0.0}, {1.0,0.0,0.0}, {1.0,1.0,0.0}, {0.0,1.0,0.0}, {0.0,0.0,1.0}, {1.0,0.0,1.0}, {1.0,1.0,1.0}, {0.0,1.0,1.0}}; Model a color cube for rotating cube program Define global arrays for vertices and colors

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11 Drawing a polygon from a list of indices Draw a quadrilateral from a list of indices into the array vertices and use color corresponding to first index void polygon(int a, int b, int c, int d) { glBegin(GL_POLYGON); glColor3fv(colors[a]); glVertex3fv(vertices[a]); glVertex3fv(vertices[b]); glVertex3fv(vertices[c]); glVertex3fv(vertices[d]); glEnd(); } polygon(0,3,2,1);

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12 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Draw cube from faces void colorcube( ) { polygon(0,3,2,1); //Front polygon(2,3,7,6); //Right polygon(0,4,7,3); //Bottom polygon(1,2,6,5); //Top polygon(4,5,6,7); //Back polygon(0,1,5,4); //Left } 0 56 2 4 7 1 3 Note that vertices are ordered counter-clockwise on the outside so that we obtain correct outward facing normals

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13 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Efficiency The weakness of our approach is that we are building the model in the application and must do many function calls to draw the cube Drawing a cube by its faces in the most straight forward way requires 6 glBegin, 6 glEnd 6 glColor 24 glVertex More if we use texture and lighting

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14 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Vertex Arrays OpenGL provides a facility called vertex arrays that allows us to store array data in the implementation Six types of arrays supported Vertices Colors Color indices Normals Texture coordinates Edge flags We will need only colors and vertices

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15 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Initialization Using the same color and vertex data, first we enable glEnableClientState(GL_COLOR_ARRAY); glEnableClientState(GL_VERTEX_ARRAY); Identify location of arrays glVertexPointer(3, GL_FLOAT, 0, vertices); glColorPointer(3, GL_FLOAT, 0, colors); 3d arraysstored as floats data contiguous data array

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16 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Mapping indices to faces Form an array of face indices Each successive four indices describe a face of the cube Draw through glDrawElements which replaces all glVertex and glColor calls in the display callback GLubyte cubeIndices[24] = {0,3,2,1,2,3,7,6 0,4,7,3,1,2,6,5,4,5,6,7,0,1,5,4};

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17 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Drawing the cube Method 1: Method 2: for(i=0; i<6; i++) glDrawElements(GL_POLYGON, 4, GL_UNSIGNED_BYTE, &cubeIndices[4*i]); format of index data start of index data what to draw number of indices glDrawElements(GL_QUADS, 24, GL_UNSIGNED_BYTE, cubeIndices); Draws cube with 1 function call!!

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Unit-4 Geometric Objects and Transformations- I

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