1 / 34

Chapter 6: Polygonal Meshes

Chapter 6: Polygonal Meshes. 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

race
Download Presentation

Chapter 6: Polygonal Meshes

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 6: Polygonal Meshes • 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) CS Hons RW778 Graphics

  2. Chapter 6: Polygonal Meshes • 6.2.2 Finding the Normal Vectors • Newell’s method. • N is number of vertices in face, (xi,yi,zi) is position of i-th vertex, and next(j)=(j+1) mod N is index of ‘next’ vertex after vertex j. • Calculate mx, my, mz of normal m as: CS Hons RW778 Graphics

  3. Chapter 6: Polygonal Meshes • 6.2.3 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. CS Hons RW778 Graphics

  4. Chapter 6: Polygonal Meshes • 6.2.4 Mesh Models for Nonsolid Objects • surfaces CS Hons RW778 Graphics

  5. Chapter 6: Polygonal Meshes • 6.2.5 Working with Meshes in a Program • Class Mesh, with vertex list, normal list, face list • Using SDL to draw mesh: Selfstudy. CS Hons RW778 Graphics

  6. Chapter 6: Polygonal Meshes • 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) CS Hons RW778 Graphics

  7. Chapter 6: Polygonal Meshes • Schlegel diagram: View from outside center of a given face • 6.3.1 Prisms and antiprisms • Prism defined by sweeping polygon along straight line • Regular prism has regular polygon base, and squares for side faces CS Hons RW778 Graphics

  8. Chapter 6: Polygonal Meshes • Antiprism : top n-gon rotated 180/n degrees; connected to bottom n-gon to form faces which are equilateral triangles • 6.3.2 Platonic solids • Polyhedron; identical faces; each face regular polygon  regular polyhedron CS Hons RW778 Graphics

  9. 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. CS Hons RW778 Graphics

  10. 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 CS Hons RW778 Graphics

  11. Chapter 6: Polygonal Meshes • 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 = (V1+V2+V3)/3 • Selfstudy: Tetrahedron,icosahedron, dodecahedron CS Hons RW778 Graphics

  12. Chapter 6: Polygonal Meshes • 6.3.3 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 CS Hons RW778 Graphics

  13. Chapter 6: Polygonal Meshes CS Hons RW778 Graphics

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

  15. Chapter 6: Polygonal Meshes • 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 CS Hons RW778 Graphics

  16. Chapter 6: Polygonal Meshes • Special case: Extruded Quad-Strips • Quad-strip: array of quadrilaterals connected in chain • Described by sequence of vertices {p0,p1,...,pm-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. CS Hons RW778 Graphics

  17. Chapter 6: Polygonal Meshes • 6.4.3 Extrusions with Twisting • Base polygon P = {p0,p1,..., pN-1} • Cap polygon P’ = {Mp0,Mp1,..., MpN-1} • M is 4x4 matrix representing affine transformation. • 6.4.4 Segmented Extrusions: Tubes and Snakes • Sequence of extrusions, each with own transformation. CS Hons RW778 Graphics

  18. Chapter 6: Polygonal Meshes • 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 {t0,t1,...} and build transformed polygon in plane perpendicular to curve at each C(ti). CS Hons RW778 Graphics

  19. Chapter 6: Polygonal Meshes • Frenet frame at each point along spline • Calculate T(ti) tangent to curve; N(ti), B(ti) perpendicular to T and each other. • Mi = ( N(ti) B(ti) T(ti) C(ti) ) • 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) CS Hons RW778 Graphics

  20. Chapter 6: Polygonal Meshes • Finding Frenet Frame Numerically (C not diff.) • Approximations for derivates; see Hill p. 317. CS Hons RW778 Graphics

  21. Chapter 6: Polygonal Meshes • 6.4.5 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! CS Hons RW778 Graphics

  22. Chapter 6: Polygonal Meshes • 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. CS Hons RW778 Graphics

  23. Chapter 6: Polygonal Meshes • 6.5.1 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 • 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 CS Hons RW778 Graphics

  24. Chapter 6: Polygonal Meshes • 6.5.2 The Normal Vector to a Surface • Case 1: Parametric equation • Planar patch near (u0,v0)essentially flat • Note that partial derivates exist if surface smooth enough. • Also, derivative of vector is vector of derivatives. CS Hons RW778 Graphics

  25. Chapter 6: Polygonal Meshes • Case 2: Implicit equation • The Effect of an Affine Transformation: Selfstudy. • 6.5.4 Generic Shapes: Sphere, Cylinder,Cone • For each shape: Implicit form, parametric form, normal. CS Hons RW778 Graphics

  26. Chapter 6: Polygonal Meshes • 6.5.5 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. CS Hons RW778 Graphics

  27. Chapter 6: Polygonal Meshes • 6.5.6 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)P0(u) +vP1(u). • P0(u) and P1(u) define curves in 3D space, defined by components P0(u)=(X0(u),Y0(u),Z0(u)). • P0(u) and P1(u) defined on same interval in u. • Ruled surface consists of one straight line joining each pair of points P0(u’) and P1(u’). CS Hons RW778 Graphics

  28. Chapter 6: Polygonal Meshes • Cones • Ruled surface for which P0(u) is a single point (apex) • Cylinders • Ruled surface for which P1(u) is a translated version of P0(u): P1(u) = P0(u) + d CS Hons RW778 Graphics

  29. Chapter 6: Polygonal Meshes • Bilinear Patches • P0(u) and P1(u) are straight line segments defined over same interval in u. • Bilinear Blended Surfaces (Coons Patches) • Rules surface that interpolates between four boundary curves CS Hons RW778 Graphics

  30. Chapter 6: Polygonal Meshes • Formula for patch: add and then subtract, otherwise not affine. • 6.5.7 Surfaces of Revolution • 6.5.8 The Quadric Surfaces • 3D analogs of conic sections CS Hons RW778 Graphics

  31. Chapter 6: Polygonal Meshes CS Hons RW778 Graphics

  32. Chapter 6: Polygonal Meshes • 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 CS Hons RW778 Graphics

  33. Chapter 6: Polygonal Meshes • 6.5.9 Superquadrics : Selfstudy. • 6.5.10 Tubes based on 3D Curves • As before, but normals to surface instead of normals to face – allows smooth shading. • 6.5.11 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. CS Hons RW778 Graphics

  34. Chapter 6: Poygonal Meshes • Programming Task 5 : Taper, twist, bend and squash it. Case Study 6.14, pp. 355-356, Hill. CS Hons RW778 Graphics

More Related