Polygon Meshes and in Computer Graphics? Graded: Implicit - PowerPoint PPT Presentation

Announcements What do we need from shapes Polygon Meshes and in Computer Graphics? • Graded: Implicit Surfaces – Programming Assignment 1 – Ian or Michael • Local control of shape for modeling » Grades in file in your turnin directory • Ability to model what we need – Written Assignment – Michael Polygon Meshes • Smoothness and continuity – Derivation for Assignment 2 – Ian Polygon Meshes Implicit Surfaces Implicit Surfaces • Programming Assignment 2 due on • Ability to evaluate derivatives Constructive Solid Geometry Constructive Solid Geometry Thursday – questions? • Ability to do collision detection Watt: Chapter 2 • Written Assignment 2 out on Thursday • Ease of rendering No one technique solves all problems 10/01/02 1 3 Computer Graphics 15-462 Computer Graphics 15-462 Two Ways to Define a Circle Surface Representations Curve Representations • Explicit: y = f(x) • Parametric surface — x(u,v), y(u,v), z(u,v) y = = + 2 x y mx b Parametric Implicit – e.g. plane, sphere, cylinder, torus, bicubic surface, swept surface – must be a function (single-valued): – parametric functions let you iterate over the surface by incrementing u – big limitation—vertical lines? and v in nested loops F>0 • Parametric: (x,y) = (f(u),g(u)) – great for making polygon meshes, etc = F=0 ( x , y ) (cos u , sin u ) – terrible for intersections: ray/surface, point-inside-boundary, etc. u • Implicit surface: F(x,y,z) = 0 + easy to specify, modify, control – extra “hidden” variable u, the parameter – e.g. plane, sphere, cylinder, quadric, torus, blobby models F<0 – terrible for iterating over the surface – great for intersections, morphing • Implicit: f(x,y) = 0 • Subdivision surfaces + − = 2 2 2 x y r 0 x = f(u) = r cos (u) – defined by a control mesh and a recursive subdivision procedure F(x,y) = x² + y² - r² + y can be multiple valued function of x y = g(u) = r sin (u) – good for interactive design – hard to specify, modify, control 4 5 6 Computer Graphics 15-462 Computer Graphics 15-462 Computer Graphics 15-462 1

Frontfacing / Backfacing Modeling Complex Shapes Polygon Meshes •A polygon has two sides, of course. • We want to build models of very complicated • Any shape can be modeled out of •Customary in CG to use the right hand rule to pick one polygons objects side to call the front face . – if you use enough of them… • An equation for a sphere is possible, but how •Counterclockwise = front, clockwise = back • Polygons with how many sides? about an equation for a telephone, or a face, or a – Can use triangles, quadrilaterals, pentagons, … n- •Important for: cloud? gons –lighting – Triangles are most common. • Complexity is achieved using simple pieces –backface culling – When > 3 sides are used, ambiguity about what to do – polygons, parametric surfaces, or implicit surfaces when polygon nonplanar, or concave, or self- –for the triangle ABC below, the front face is up . intersecting. • Goals N • Polygon meshes are built out of – Model anything with arbitrary precision (in principle) – vertices (points) C – Easy to build and modify – edges (line segments between vertices) – Efficient computations (for rendering, collisions, etc.) A – faces (polygons bounded by edges) – Easy to implement (a minor consideration...) B 7 8 9 Computer Graphics 15-462 Computer Graphics 15-462 Computer Graphics 15-462 Normals and Plane Equations Polygon Models in OpenGL Data Structures for Polygon Meshes • Need normals for shading, plane eqns for intersection tests • Simplest (but dumb) • A normal to a plane is a vector that is perpendicular to that plane – float triangle[n][3][3]; (each triangle stores 3 (x,y,z) points) (two possible choices) • for faceted shading • for smooth shading – redundant: each vertex stored multiple times • A plane is specified by a point P and a normal vector N < calculate face normal n glBegin(GL_POLYGONS); • Vertex List, Face List • N•(X-P) = 0 if and only if X lies in the plane; this is an implicit using cross product rule > glNormal3fv(normal1); equation for the plane – List of vertices, each vertex consists of (x,y,z) geometric (shape) glNormal3fv(n); glVertex3fv(vert1); info only – Expand this out: 0 = N•X - N•P = ax + by + cz + d glBegin(GL_POLYGONS); glNormal3fv(normal2); – List of triangles, each a triple of vertex id’s (or pointers) topological • 3 vertices define a plane, its normal is: N=(B-A) x (C-A) (connectivity, adjacency) info only glVertex3fv(vert1); glVertex3fv(vert2); • Unit normal Fine for many purposes, but finding the faces adjacent to a vertex glVertex3fv(vert2); glNormal3fv(normal3); takes O(F) time for a model with F faces. Such queries are N glVertex3fv(vert3); glVertex3fv(vert3); important for topological editing. ˆ = glEnd(); • Fancier schemes: N N / N glEnd(); C Store more topological info so adjacency queries can be answered in O(1) time. A X P Winged-edge data structure – edge structures contain all topological info (pointers to adjacent vertices, edges, and faces). B 10 11 12 Computer Graphics 15-462 Computer Graphics 15-462 Computer Graphics 15-462 2

How Many Polygons to Use? A File Format for Polygon Models: OBJ Why Level of Detail? • Different models for near and far objects # OBJ file for a 2x2x2 cube v -1.0 1.0 1.0 - vertex 1 • Different models for rendering and collision detection v -1.0 -1.0 1.0 - vertex 2 v 1.0 -1.0 1.0 - vertex 3 • Compression of data recorded from the real world v 1.0 1.0 1.0 - … v -1.0 1.0 -1.0 v -1.0 -1.0 -1.0 We need automatic algorithms for reducing the polygon Syntax: v 1.0 -1.0 -1.0 v 1.0 1.0 -1.0 count without f 1 2 3 4 •losing key features v x y z - a vertex at (x,y,z) f 8 7 6 5 •getting artifacts in the silhouette f 4 3 7 8 •popping f v 1 v 2 … v n f 5 1 4 8 a face with vertices v 1 , v 2 , … v n f 5 6 2 1 f 2 6 7 3 # anything - comment 13 14 15 Computer Graphics 15-462 Computer Graphics 15-462 Computer Graphics 15-462 What Implicit Functions are Good For Surface Representations Sets of Points, Surfaces and Solids • Implicit surface: set of all points that satisfy F(x,y,z)=0 • Parametric surface — x(u,v), y(u,v), z(u,v) • The points that satisfy F(x,y,z)<0 define a solid (or – e.g. plane, cylinder, bicubic surface, swept surface solids) bounded by the surface F < 0 ? – parametric functions let you iterate over the surface by F = 0 ? X + kV • The solid is directly defined (unlike definitions using incrementing u and v in nested loops F > 0 ? parametric surfaces) – great for making polygon meshes, etc • Example X – terrible for intersections: ray/surface, point-inside- – An infinitely long (solid) cylinder with radius r: boundary, etc. F = x 2 + y 2 - r 2 F(X + kV) = 0 – To limit cylinder to length L, abs(z) < L/2 and keep the function implicit use • Implicit surface: F(x,y,z) = 0 max: ( ) Ray - Surface Intersection Test Inside/Outside Test F = max abs(z)-L/2,x 2 + y 2 -r 2 – e.g. plane, sphere, cylinder, quadric, torus, blobby models • Implicit functions for a cube? Any convex polyhedron? – terrible for iterating over the surface – great for intersections, morphing 16 17 18 Computer Graphics 15-462 Computer Graphics 15-462 Computer Graphics 15-462 3

Surfaces from Implicit Functions Blobby Models How to draw implicit surfaces? • Constant Value Surfaces are called • It’s easy to ray trace implicit surfaces (depending on whom you ask): – because of that easy intersection test – constant value surfaces • Volume Rendering can display them – level sets – isosurfaces • Convert to polygons: the Marching Cubes algorithm • Implicit function is the sum of Gaussians centered at • Nice Feature: you can add them! (and other – Divide space into cubes several points in space, minus a threshold tricks) – Evaluate implicit function at each cube vertex – this merges the shapes – Do root finding or linear interpolation along each • varying the standard deviations of the Gaussians – When you use this with spherical exponential potentials, it’s edge called Blobs , Metaballs, or Soft Objects . Great for modeling makes each blob bigger animals. – Polygonize on a cube-by-cube basis • varying the threshold makes blobs merge or separate 19 20 21 Computer Graphics 15-462 Computer Graphics 15-462 Computer Graphics 15-462 Isosurfaces of Simulated Tornado Constructive Solid Geometry (CSG) A CSG Train Generate complex shapes with basic building blocks machine an object - saw parts off, drill holes glue pieces together This is sensible for objects that are actually made that way (human-made, particularly machined objects) Brian Wyvill & students, Univ. of Calgary 22 23 24 Computer Graphics 15-462 Computer Graphics 15-462 Computer Graphics 15-462 4