2003CS Hons RW778 Graphics1 Chapter 6: Polygonal Meshes 6.2 Introduction 6.2 Introduction –Polygonal mesh: collection of polygons (faces) –List of polygons, each with direction (normal vector) –Vertex normals vs. face normals »Vertex normal facilitates clipping and shading. –6.2.1 Defining a polygonal mesh »Vertex list (geometric information) »Normal list (orientation information) »Face list (topological information)

2003CS Hons RW778 Graphics2 Chapter 6: Polygonal Meshes Finding the Normal Vectors Finding the Normal Vectors –Newell’s method. –N is number of vertices in face, (x i,y i,z i ) is position of i- th vertex, and next(j)=(j+1) mod N is index of ‘next’ vertex after vertex j. –Calculate m x, m y, m z of normal m as:

2003CS Hons RW778 Graphics3 Chapter 6: Polygonal Meshes Properties of meshes Properties of meshes –Solid : faces enclose positive finite amount of space –Connected : unbroken path along polygon edges exists between any two vertices –Simple : solid object, no holes (object can be deformed into sphere without tearing) –Planar : every face is planar polygon –Convex : line connecting any two point in object lies wholly inside object.

2003CS Hons RW778 Graphics4 Chapter 6: Polygonal Meshes Mesh Models for Nonsolid Objects Mesh Models for Nonsolid Objects –surfaces

2003CS Hons RW778 Graphics5 Chapter 6: Polygonal Meshes Working with Meshes in a Program Working with Meshes in a Program –Class Mesh, with vertex list, normal list, face list –Using SDL to draw mesh: Selfstudy.

2003CS Hons RW778 Graphics6 Chapter 6: Polygonal Meshes 6.3 Polyhedra 6.3 Polyhedra –Polyhedron is connected mesh of simple planar polygons that encloses finite amount of space. »Every edge shared by exactly two faces »At least three edges meet at each vertex »Faces do not interpenetrate: Either don’t touch, or only touch along common edge. –Euler’s formula: V + F – E = 2 (simple polyhedron) V + F - E = 2 + H - 2G (non-simple polyhedron)

2003CS Hons RW778 Graphics7 Chapter 6: Polygonal Meshes –Schlegel diagram: View from outside center of a given face Prisms and antiprisms Prisms and antiprisms –Prism defined by sweeping polygon along straight line –Regular prism has regular polygon base, and squares for side faces

2003CS Hons RW778 Graphics8 Chapter 6: Polygonal Meshes –Antiprism : top n-gon rotated 180/n degrees; connected to bottom n-gon to form faces which are equilateral triangles Platonic solids Platonic solids –Polyhedron; identical faces; each face regular polygon  regular polyhedron

2003CS Hons RW778 Graphics9 Chapter 6: Polygonal Meshes –Only 5 such objects! Platonic solids –Schäfli symbol (p,q) ; each face is p-gon, q faces meet at each vertex.

2003CS Hons RW778 Graphics10 Chapter 6: Polygonal Meshes –Dual polyhedra: Each Platonic solid has dual D –Vertices of D is centers of faces of P  edges of D connect midpoints of adjacent faces of P –Dual can be constructed directly from P »Model to keep track of vertex and face numbering »Detail : Selfstudy

2003CS Hons RW778 Graphics11 Chapter 6: Polygonal Meshes Normal vectors for Platonic solids Normal vectors for Platonic solids –Assume solid centered at origin, then normal to face is vector from origin to center of face, which is average of vertices. –m = (V 1 +V 2 +V 3 )/3 –Selfstudy: Tetrahedron, icosahedron, dodecahedron

2003CS Hons RW778 Graphics12 Chapter 6: Polygonal Meshes Other Interesting Polyhedra Other Interesting Polyhedra –Archimedean (semi-regular) solids »More than one kind of face »Face still regular polygon »Every vertex surrounded by same collection of polygons in same order. –Only 13 possible Archimedean solids –Normal vector still found using center of face. –Examples: truncated cube, Buckyball

2003CS Hons RW778 Graphics13 Chapter 6: Polygonal Meshes

2003CS Hons RW778 Graphics14 Chapter 6: Polygonal Meshes Geodesic domes Geodesic domes –Approximate sphere by faces, usually triangles; cut in half –Faces? Each edge divided into 3F equal parts; result projected outward onto sphere.

2003CS Hons RW778 Graphics15 Chapter 6: Polygonal Meshes 6.4 Extruded shapes 6.4 Extruded shapes –2D polygon swept through space. –6.4.1 Creating Prisms »Polygon swept in straight line »Flat face  same normal vector with every vertex of face (normal vector to face itself) –Building mesh for prism : Selfstudy –6.4.2 Arrays of extruded prisms (“Bricklaying”) »Some software (OpenGL) draw only convex polygons »Decompose polygon into sets of convex polygons

2003CS Hons RW778 Graphics16 Chapter 6: Polygonal Meshes –Special case: Extruded Quad-Strips »Quad-strip: array of quadrilaterals connected in chain »Described by sequence of vertices {p 0,p 1,...,p m-1 } »Vertices taken in pairs »When mesh formed as extruded quad-strip, only 2M vertices in vertex list; only “outside walls” included in face list  no redundant walls drawn.

2003CS Hons RW778 Graphics17 Chapter 6: Polygonal Meshes Extrusions with Twisting Extrusions with Twisting –Base polygon P = {p 0,p 1,..., p N-1 } –Cap polygon P’ = {Mp 0,Mp 1,..., Mp N-1 } –M is 4x4 matrix representing affine transformation Segmented Extrusions: Tubes and Snakes Segmented Extrusions: Tubes and Snakes –Sequence of extrusions, each with own transformation.

2003CS Hons RW778 Graphics18 Chapter 6: Polygonal Meshes Designing Tubes based on 3D Curves Designing Tubes based on 3D Curves –Wrap tube round curve (called spine C(t)), that undulates through space in organized fashion. –Form waist polygon? Sample C(t) at {t 0,t 1,...} and build transformed polygon in plane perpendicular to curve at each C(t i ).

2003CS Hons RW778 Graphics19 Chapter 6: Polygonal Meshes –Frenet frame at each point along spline »Calculate T(t i ) tangent to curve; N(t i ), B(t i ) perpendicular to T and each other. –M i = ( N(t i ) B(t i ) T(t i ) C(t i ) ) –Forming Frenet Frame (for C differentiable) » (t), normalize, get unit tangent vector T(t). »Unit binormal vector B(t): »N(t) = C(t) x B(t)

2003CS Hons RW778 Graphics20 Chapter 6: Polygonal Meshes –Finding Frenet Frame Numerically (C not diff.) »Approximations for derivates; see Hill p. 317.

2003CS Hons RW778 Graphics21 Chapter 6: Polygonal Meshes Discretely Swept Surfaces of Revolution Discretely Swept Surfaces of Revolution –Place all spline points at origin, and use rotation for affine transformation. –Base polygon called profile –Operation equivalent to circularly sweeping shape about axis –Resulting shape called surface of revolution. –Note: only discrete approximation!

2003CS Hons RW778 Graphics22 Chapter 6: Polygonal Meshes 6.5 Mesh Approximations to Smooth Objects 6.5 Mesh Approximations to Smooth Objects –Previous: Mesh with each face polygon shape specified by listing data of vertices. –Now: Polygon approximations of object, but with vertices calculated using formulas (evaluate parametric representation of surface at discrete points). –Shading smooth – individual faces invisible. Compute normal to surface. –Create mesh by building vertex list and face list, but vertices computed.

2003CS Hons RW778 Graphics23 Chapter 6: Polygonal Meshes Representation of Surfaces Representation of Surfaces –Similar to planar patch P(u,v) = C + au +bv –Generalize: P(u,v) = (X(u,v), Y(u,v), Z(u,v)) (point form). –If v constant, u varies: v-contour –If u constant, v varies: u-contour Implicit Form of Surface Implicit Form of Surface –F(x,y,z)=0 iff (x,y,z) is on surface. –F(x,y,z)<0 iff (x,y,z) is inside surface –F(x,y,z)>0 iff (x,y,z) is outside surface

2003CS Hons RW778 Graphics24 Chapter 6: Polygonal Meshes The Normal Vector to a Surface The Normal Vector to a Surface –Case 1: Parametric equation –Planar patch near (u 0,v 0 ) essentially flat –Note that partial derivates exist if surface smooth enough. –Also, derivative of vector is vector of derivatives.

2003CS Hons RW778 Graphics25 Chapter 6: Polygonal Meshes –Case 2: Implicit equation –The Effect of an Affine Transformation: Selfstudy Generic Shapes: Sphere, Cylinder,Cone Generic Shapes: Sphere, Cylinder,Cone –For each shape: Implicit form, parametric form, normal.

2003CS Hons RW778 Graphics26 Chapter 6: Polygonal Meshes Polygon Mesh for a Curved Surface Polygon Mesh for a Curved Surface –Tesselation – replace surface by collection of triangles and quadrilaterals –Vertices lie in surface, joined by straight edges (not in surface) –Obtain vertices by sampling values of u and v in parametric form of surface; place in vertex list. –Face list from vertices –Associate with each vertex normal to surface. –Selfstudy: rest of pp. 329, 330.

2003CS Hons RW778 Graphics27 Chapter 6: Polygonal Meshes Rules Surfaces Rules Surfaces –Surface is ruled if, through every one of tis points, there passes at least one line that lies entirely on the surface. –Rules surfaces are swept out by moving a straight line along a particular trajectory. –Parametric form: P(u,v) = (1-v)P 0 (u) +vP 1 (u). –P 0 (u) and P 1 (u) define curves in 3D space, defined by components P 0 (u)=(X 0 (u),Y 0 (u),Z 0 (u)). –P 0 (u) and P 1 (u) defined on same interval in u. –Ruled surface consists of one straight line joining each pair of points P 0 (u’) and P 1 (u’).

2003CS Hons RW778 Graphics28 Chapter 6: Polygonal Meshes Cones Cones –Ruled surface for which P 0 (u) is a single point (apex) Cylinders Cylinders –Ruled surface for which P 1 (u) is a translated version of P 0 (u): P 1 (u) = P 0 (u) + d

2003CS Hons RW778 Graphics29 Chapter 6: Polygonal Meshes Bilinear Patches Bilinear Patches –P 0 (u) and P 1 (u) are straight line segments defined over same interval in u. Bilinear Blended Surfaces (Coons Patches) Bilinear Blended Surfaces (Coons Patches) –Rules surface that interpolates between four boundary curves

2003CS Hons RW778 Graphics30 Chapter 6: Polygonal Meshes –Formula for patch: add and then subtract, otherwise not affine Surfaces of Revolution Surfaces of Revolution The Quadric Surfaces The Quadric Surfaces –3D analogs of conic sections

2003CS Hons RW778 Graphics31 Chapter 6: Polygonal Meshes

2003CS Hons RW778 Graphics32 Chapter 6: Polygonal Meshes Properties of Quadric Surfaces Properties of Quadric Surfaces –Trace is curve formed when surface is cut by plane »All traces of quadric surfaces are conic sections. –Principal traces are curves generated when cutting planes aligned with axes. –Selfstudy: Ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, elliptic cone, elliptic paraboloid, hyperbolic paraboloid –Selfstudy: Normal vectors to quadric surfaces

2003CS Hons RW778 Graphics33 Chapter 6: Polygonal Meshes Superquadrics : Selfstudy Superquadrics : Selfstudy Tubes based on 3D Curves Tubes based on 3D Curves –As before, but normals to surface instead of normals to face – allows smooth shading Surfaces based on Explicit Functions of Two Variables Surfaces based on Explicit Functions of Two Variables –If surface shape single valued in one dimension, position can be represented as single function of 2 independent variables. –Example: Single value of ‘height’ of surface above xz- plane for each point (x,z). Known as height field.

2003CS Hons RW778 Graphics34 Chapter 6: Poygonal Meshes Programming Task 5 : Taper, twist, bend and squash it. Case Study 6.14, pp , Hill. Programming Task 5 : Taper, twist, bend and squash it. Case Study 6.14, pp , Hill.