Computer Graphics. This Week. Meshes. Meshes. What is a Polygon Mesh? Meshes. Modelling Shapes with Polygon Meshes.

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1 1 This Week Computer Graphics Modeling Shapes Modelling Shapes with Polygon Solid Modelling Extruded Shapes Mesh Approximations 2 What is a Polygon Mesh? A surface made up of a collection of polygon faces. 3 Ultimately 3D objects are represented and rendered by a set of polygons triangles, quadrilaterals polygonal mesh data structure containing information concerning object surfaces entered manually, ok for simple objects generated from a volume (isosurface( isosurface) computed analytically (function evaluation) 4 Polygonal Mesh collection of polygons representing the surface of an object polygons can be linear approximations to an underlying surface faces, facets Fully specified models verses approximate natural to use polygons to approximate it cylinder is approximated by a series of faces we can use other OpenGL capabilities to make it appear smooth 5 6

2 7 Modelling Solids can be used to create the skin of a solid object. Simple solids are really just polygon meshes as their faces are polygons (with the exception of a real sphere or cone!) When a mesh encloses a space it is said to have created a solid. Not a Solid A Solid Modelling Solids -meshsolid1.exe 8 Modelling Solids Each polygon is coloured so you can see the mesh in action. Note: Some meshes can represent solid or non solid shapes depending on which polygons are shaded in. Defining a Polygon Mesh There are several ways of defining a mesh. For the cube you could list each polygon: the vertices location and the normals for each vertex (24 vertices and 24 normals for the 6 faces) This can create redundant data. vertices and normals can be listed more than once Defining a Polygon Mesh A better method Create 3 lists one for vertices one for faces one for normals For the cube there are: 8 vertices 6 faces 6 normals Polygonal meshes contain vertices of each facet vector normal to the facet (used for shading) surface orientation is important in rendering realistic views 11 12

3 13 Associate normal vectors with each vertex of each facet simplifies some processing OpenGL is implemented this way a location in space may (does not have to, but can) have more than one normal vector associated with it Basic Barn seven faces 10 unique vertices flat walls, one unique normal vector per facet or in this case seven 14 Basic Barn: data structure list of faces each face contains all its vertices redundant data and could be significant for the basic barn we would store 30 vertices and 30 normal vectors instead of 7 simple to render worse for larger more primitive (dinosaur) models 15 Basic Barn: data structure more efficient (storage wise) to store unique vertices and normal vectors each face would then contain a set of indices for the corresponding vertices and normals Number of vertices: n Vertex list V1(x,y,z) V2(x,y,z) V3(x,y,z)... Vn(x,y,z) Number of normals: m Normal list N1(x,y,z) N2(x,y,z) N3(x,y,z)... Nm(x,y,z) Number of facets: l Face list V1, N1 V8, N2 V4, N2... V1, Nl 16 Basic Barn: data structure 17 Basic Barn: data structure vertex list: geometry information normal list: orientation information face list: topological information unique vertices and normal vectors unit length normal vectors for shading Vertices of each polygon face are listed in counter-clockwise clockwise order 18

4 19 Basic Barn: data structure Vertices of each polygon face are listed in counter-clockwise clockwise order as seen from the outside normal points from inside to outside traverse face vertex by vertex and the interior of the face is to the left allows us to determine the front and back face of a polygon Defining a Polygon Mesh The Vertex List vertex x, y, z 0 1.0,-1.0, ,-1.0, , 1.0, , 1.0, ,-1.0, 1.0, , 1.0, , 1.0, ,-1.0, 1.0, Defining a Polygon Mesh The Normal List normal nx, ny, nz 0 0.0, 0.0, , 0.0, , 1.0, ,-1.0, , 0.0, , 0.0, 0.0 Defining a Polygon Mesh The Face List face vertices, normals 0(front) 0,1,2,3 0,0,0,0 1(back) 4,5,6,7 1,1,1,1 2(top) 5,3,2,6 2,2,2,2 3(bottom) 4,7,1,0 3,3,3,3 4(right) 7,6,2,1 4,4,4,4 5(left) 4,0,3,5 5,5,5, Defining a Polygon Mesh Traversing a face: to determine the interior of the polygon to determine the outside of the solid to determine where the normal is pointing Traverse a face counterclockwise as seen from the outside of the object. The inside of the polygon will always be on your left!! Calculating the Normals Why to we need the normals? The normal tells us which is the outside of the face. The normal is used for calculating how much light falls on the outside surface. The normal determines how smoothly textures are rendered on surfaces

5 25 26 Normal Vectors Given N unique vertices of a polygon, compute an average normal vector Newell s s method N 1 m x = ( yi ynext( i) )( zi + znext( i) ) i= 0 N 1 m y = ( zi znext( i) )( xi + xnext( i) ) i= 0 N 1 m z = ( xi xnext( i) )( yi + ynext( i) ) i= 0 next ( i) = ( i + 1)% N must insure m points toward the outside of the face Calculating the Normals Two ways to determine the normal 1) Find the cross product using 3 points on the surface of the face. However, if the vectors are almost parallel the cross product will be small and inaccuracies will occur!! the polygon may not be a perfect planar n Calculating the Normals Two ways to determine the normal 2) Use Newell s s method. m m x = y = m = N 1 ( y y )( zi + ) i next i znext( i ), ( ) i= 0 N 1 ( zi znext( i ))( xi + xnext( i )), i= 0 N 1 ( )( y + y ) x z i i= 0 x next( i ) i next( i ) Example For the three vertices (6, 1, 4), (7, 0, 9), and (1, 1, 2), using the Newell Method : mx = (1-0)(4+9) + (0-1)(9+2) + (1-1)(2+4) 1)(2+4) = 2 my = (4-9)(6+7) + (9-2)(7+1) + (2-4)(1+6) = -23 mz = (6-7)(1+0) + (7-1)(0+1) + (1-6)(1+1) =

6 31 Calculating the Normals Now that I have the normals,, what am I going to do with them? OpenGL uses the normals to determine where the outside face of your mesh is and how to light it. Lets look at an example: Calculating the Normals Lets plot the same cube as before, but turn the lights on it. The cube is bathed in the same amount of light on each face and therefore appears flat. -meshsolid2.exe 32 Calculating the Normals glbegin(gl_quads); // Start Drawing Quads // Front Face glnormal3f( 0.0f, 0.0f, 1.0f); // Normal pointing towards viewer glvertex3f(-1.0f, -1.0f, 1.0f); glvertex3f( 1.0f, -1.0f, 1.0f); glvertex3f( 1.0f, 1.0f, 1.0f); glvertex3f(-1.0f, 1.0f, 1.0f); // Back Face glnormal3f( 0.0f, 0.0f,-1.0f); // Normal pointing away from viewer glvertex3f(-1.0f, -1.0f, -1.0f); glvertex3f(-1.0f, 1.0f, -1.0f); glvertex3f( 1.0f, 1.0f, -1.0f); glvertex3f( 1.0f, -1.0f, -1.0f); glend(); Solidity Properties of represents a solid object if faces enclose a positive and finite space Connectedness connected if an unbroken path along polygon edges exists (if not connected it is more than one object) Simplicity simple, if it is a solid with no holes Planarity planar if every face of the object represents a plane Convexity convex and concave (with dents ) -meshsolid3.exe in a Program in a Program 35 36

7 37 Faces with holes Mesh Class Access the vth vertex in the fth face pt[face[f].vert[v].vertindex] norm [face[f].vert[v].normindex[ face[f].vert[v].normindex] 38 Mesh Class Simple to render a Mesh object A Polyhedron is a connected mesh of simple planar polygons enclosing a finite amount of space A Pyramid is a polyhedra A donut is not 41 42

8 43 Euler s s Formula a simple polyhedra satisfies Euler s equation. V-E+F = 2 V=4 E=6 F=4 V=8 E=12 F=6 Euler s s for complex polyhedra with holes V-E+F-H H = 2(C-G) V=24 E=36 F=15 H=3 (number of holes in faces C=1 (number of parts) G=1 (number of through holes) 44 Schlegel Diagram A 2D diagram of a 3D polyhedra as seen in perspective. A Model A Model is the unfolded representation of a solid Prisms and Antiprisms A prism is a polyhedra that embodies certain symmetries and therefore is quite simple to describe. a prism is defined as a sweep or extrusion of a polygon along a straight line. Prisms and Antiprisms An antiprism has a top and bottom of the same polygon, however the bottom polygon is rotated through n/180 degrees

9 49 Solids Platonic Solids We briefly looked at these last week as wireframe models. tetrahedron dodecahedron isosahedron Extruded Shapes Creating Prisms We take a basic polygon and use it for the top. Then extrude the vertexes in some direction for some length. 50 Extruded Shapes Creating Prisms For example, we have an arrow shape and we want to create an arrow prism. Extruded Shapes Creating Prisms For example, we have an arrow shape and we want to create an arrow prism. Both the front and back faces are the same polygon. However their normals will be in opposite directions Extruded Shapes Creating Prisms For example, we have an arrow shape and we want to create an arrow prism. Both the front and back faces are the same polygon. However their normals will be in opposite directions. Tubes A tube can be created by extruding a polyhedra along a spine. For example, we take a simple sphere and drawing it as it moves around a helix. -extrude_mesh.bpr 53 54

10 55 But each time the sphere moves, the old sphere is left in the image. Tubes Tubes By varying the x, y and z values for the center of the sphere along spiral formulae, you can come up with all kinds of interesting shapes. - polyspine.exe - toroidal.exe 56 A swept surface takes a simple (or complex) 2D polygon and rotates it. Swept Surfaces Swept Surfaces A swept surface takes a simple (or complex) 2D polygon and rotates it. - swept_glass.exe Smooth Surfaces A polygon mesh can also be used to create smooth surfaces. How is it done? Step One: Program a data structure to hold the mesh coordinates. GLfloat Mesh[Height][Width][3]; Vector3 Normals[Height][Width]; ]; 59 60

11 61 Step One: GLfloat Mesh[Height][Width][3]; (Mesh[3][0][0], Mesh[3][0][1],Mesh[3][0][2]) (Mesh[0][0][0], Mesh[0][0][1],Mesh[0][0][2]) Step Two: Populate the Mesh with Coordinates for(int i = 0; i < Height; i++) { for(int j = 0; j < Width; j++) { Mesh[i][j][0] = j; Mesh[i][j][1] = i; Mesh[i][j][2] = 1; } } 62 Step Three: Draw the Mesh Step Three: Draw the Mesh void drawmesh() { for(int i = 0; i < MH-1; i++) { for(int j = 0; j < MW-1; j++) { glbegin(gl_line_strip); glvertex3f(mesh[i][j][0],mesh[i][j][1],mesh[i][j][2]); glvertex3f(mesh[i+1][j][0],mesh[i+1][j][1],mesh[i+1][j][2]); glvertex3f(mesh[i+1][j+1][0],mesh[i+1][j+1][1],mesh[i+1][j+1][2]); ][2]); glvertex3f(mesh[i][j+1][0],mesh[i][j+1][1],mesh[i][j+1][2]); [j+1][2]); glvertex3f(mesh[i][j][0],mesh[i][j][1],mesh[i][j][2]); glend(); } } } -surface_mesh1.exe Creating a Height Map Modify the value of z. Currently it is set to 1 so each vertex in the mesh is at the same z location. Creating a Height Map Mesh[i][j][2] = pow(x,2) + pow(y,2); //for x and y between -11 and 1 e.g. Mesh[i][j][2] = cos(j/2.0)*2; -surface_mesh.bpr -surface_mesh2.bpr 65 66

12 67 Creating a Height Map Mesh[i][j][2] = (sin(3.14*x)/3.14*x + sin(3.14*y)/3.14*y); //for x and y between -11 and 1 Ingredience 1 mesh 1 bitmap (preferable the same size as the mesh dimension wise) -surface_mesh3.bpr 68 1 mesh (200x200) 1 bitmap (200x200) Step One: Apply the bitmap to the mesh. glbegin(gl_quads); glnormal3f(normals[i][j].x, Normals[i][j].y, Normals[i][j].z); gltexcoord2f(j/(float)width, i/(float)height); glvertex3f(mesh[i][j][0],mes h[i][j][1],mesh[i][j][2]); Step One: Apply the bitmap to the mesh. glbegin(gl_quads); glnormal3f(normals[i][j].x, Normals[i][j].y, Normals[i][j].z); gltexcoord2f(j/(float)width, i/(float)height); glvertex3f(mesh[i][j][0],mes h[i][j][1],mesh[i][j][2]); 71 72

13 73 Step Two: Shape the table cloth. Ripple: y = cos(z); Step Two: Shape the table cloth. Ripple: y = cos(z); Drape: y = 1 - pow(x,4) - pow(z,4); 74 Step Two: Shape the table cloth. Ripple: y = cos(z); Drape: y = 1 - pow(x,4) - pow(z,4); Bump: y = cos(z) ) + sin(y); Step Three: Flatten the Top if (Mesh[i][j][1] > 0.5) Mesh[i][j][1] = 0.5; else Mesh[i][j][1] = Mesh[i][j][1]; Step Four: Add accessories The End Next Week: Viewing in 3 Dimensions Positioning the Camera Projections 77 78