1MeshesModeling ObjectsCS148: Intro to CGInstructor: Dan MorrisTA: Sean WalkerJuly 12, 2005Administrative blah-blah{ pp1 was (virtually) handed back{ Exam is coming upz What do you need to know?z Submit questions!{ Submit video-break videos{ Only one late day on pp4Outline for today{ Face culling{ Representing meshes{ Representing surfaces{ Drawing surfacesBeing stingy with our triangles{ When we draw Lego Man, we might draw lots of triangles that end up getting covered up{ It’s not useful to draw the triangles on the other side of Lego ManBackface culling{ For any closed object, it’s not useful to draw any triangles that face away from the viewer{ Often want to eliminate backwards triangles (that face away from the viewer) before rasterization{ This is ‘backface culling’, and OpenGL can do it for youWhich way does a triangle “face”?{ Intuitively, we want the side of the triangle on the outside of our object to be the front{ When I draw a triangle, OpenGL doesn’t know about inside and outside { Need a way to specify front and back of a triangle2Which way does a triangle “face”?{ The order of vertices tells OpenGL which way a triangle faces{ If I look at a triangle, the vertices appear clockwise on one side, counter-clockwise on the other { In GL, the side ordered CCW is the front132132Face normals{ Normal to a face: vector perpendicular to the face, pointing toward the frontHow do we compute the normal to this triangle? 132Which triangles “face the viewer”?In eye coordinates, how do I decide if a triangle faces the viewer (plain English answer)?-z123NReminder: dot productsA • B = |A||B|cosθA • B = 0 → vectors are perpendicularA • B > 0 → vectors “mostly” point the same wayθWhich triangles “face the viewer”?N • (z) > 0 → triangle faces cameraN • (0,0,1) > 0(Nx*0) + (Ny*0) + (Nz*1) > 0Nz> 0 -z123NCulling in OpenGL [culling.cpp]{ Default state listed first:z glDisable(GL_CULL_FACE);glEnable(GL_CULL_FACE);z glCullFace(GL_BACK);glCullFace(GL_FRONT);z glFrontFace(GL_CCW);glFrontFace(GL_CW);When would I want culling disabled?3Front-back in OpenGL{ Frontface/backface determination is also used for lighting{ Backfaces can be a different color / brightness / etc.Illegal Polygons{ Can’t define a normal for a non-planar polygon{ Illegal to send OpenGL non-planar polygons{ Even simple polygons can become non-planar after transformations{ Polygons are generally tessellated by OpenGLOutline for today{ Face culling{ Representing meshes{ Representing surfaces{ Drawing surfacesRepresenting models{ Models so far were made of triangles or simple primitives that we hard-coded in our programs{ “Real” models have too many polygons to create manually{ Need a way to store and represent objectsA mesh data structure: take onestruct vertex {float x,y,z;}struct triangle {vertex a,b,c;}struct object {unsigned int nTriangles;triangle* triangles;}What’s wrong with this approach?How many vertices in a cube?4So what?What are some disadvantages of having redundant vertices?A better mesh data structurestruct vertex {float x,y,z;}struct triangle {unsigned int a,b,c;}struct object {unsigned int nVertices;vertex* vertices;unsigned int nTriangles;triangle* triangles;}Meshes in OpenGL [meshes.cpp]{ Data structures like this are very common{ In fact, a very common way of sending meshes to OpenGL is:// Tell GL where to find verticesglVertexPointer(3,GL_FLOAT,0,myObject.vertices);// Draw “indexed triangles”glDrawElements(GL_TRIANGLES,myObject.nTriangles,GL_UNSIGNED_INT,myObject.triangles);Meshes in OpenGL{ Most mesh file formats store vertices and triangles in a format more or less like thisz Popular formats: .3ds, .obj, .stl{ Most formats add a little more data to each vertex:z Surface normalsz Color informationz Texture coordinatesz These topics make up the next few classes in CS148Outline for today{ Face culling{ Representing meshes{ Representing surfaces{ Drawing surfacesWhen aren’t meshes quite right?Which of these objects could be represented more efficiently with another approach?5Why not always store our objects as meshes?{ Space{ Level of detail{ Modeling{ Non-polygon renderers{ OpenGL (mostly) only knows about triangles and vertices, so we’re leaving the GL universe for a bit...Another approach: object primitives{ Spheres{ Cubes{ Other objects I can easily parameterizeA file format with object primitivesObject LegoMan {// numbers are arbitrarysphere head(0.2,0.3,0.4,0.1);cube body(0.2,0.5,0.3,0.2);triMesh hand {vertices 0.4,0.3,0.5,0.2...triangles 1,2,3,2,3,4...}}{ In fact, VRML – a popular model format –is not all that far from thisStill many objects we can’t compactly represent...{ In particular, we’d like to be able to efficiently represent curves in space{ ...just you can efficiently represent a parabola by writing y = x2, without writing down 100000 points on the parabola.Parametric Surfaces{ Think of an object’s “skin” as a 2D surface{ Use two variables (u,v) to tell me where I am on the surface{ A position function p(u,v) generates points in spacep(u,v) = (X(u,v), Y(u,v), Z(u,v)){ In vector notation:p(u,v) = X(u,v)i + Y(u,v)j + Z(u,v)kParametric Surfaces{ u and v mean different things for different objects{ A parametric surface gives back different points for different (u,v) values{ A cylinder might interpret (u,v) as (h,θ){ A plane might interpret (u,v) as (x,y){ What might (u,v) represent for a sphere?6Defining a Parametric Plane{ I can specify a plane with a point c and any two vectors a,b that live on the plane{ How can I put this in parametric form?{ Define (u,v) = (0,0) as the point c{ Can represent any vector in 2D as a combination of two non-collinear vectors{ So our whole plane is:p(u,v) = ua + vbPlanar Patches{ Could represent the whole plane by letting u and v range from -∞ to +∞{ Often we want to represent a specific chunk of a plane, e.g. the front of Lego Man{ We often define a and b so that letting u and v go from 0 to 1 will give us the object we want{ This restricted piece of a parametric surface is called a patchPlanar Patchesc+a+b11c+a01c+b10c00cornervuOther Patches{ A sphere might interpret (u,v) as (θ,ρ){ A cylinder might interpret (u,v) as (h,θ){ This might be represented as a spherical patch (restricted range of (θ,ρ)) and a cylindrical patch (restricted range of (h,θ) )Other surface types{ Ruled surfaces{ Surfaces of revolution{ Quadric surfacesAside: parametric Line Segment{ If we want a