Chapter 6: Polygonal Meshes

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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)

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Chapter 6 Polygonal Meshes

• 6.2.2 Finding the Normal Vectors
• Newells method.
• N is number of vertices in face, (xi,yi,zi) is
  position of i-th vertex, and next(j)(j1) mod N
  is index of next vertex after vertex j.
• Calculate mx, my, mz of normal m as

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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.

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Chapter 6 Polygonal Meshes

• 6.2.4 Mesh Models for Nonsolid Objects
• surfaces

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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.

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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 dont touch,
  or only touch along common edge.
• Eulers formula V F E 2 (simple
  polyhedron) V F - E 2 H - 2G (non-simple
  polyhedron)

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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

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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

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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.

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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

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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 (V1V2V3)/3
• Selfstudy Tetrahedron,icosahedron, dodecahedron

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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

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Chapter 6 Polygonal Meshes

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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.

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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

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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.

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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.

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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).

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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)

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Chapter 6 Polygonal Meshes

• Finding Frenet Frame Numerically (C not diff.)
• Approximations for derivates see Hill p. 317.

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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!

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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.

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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)lt0 iff (x,y,z) is inside surface
• F(x,y,z)gt0 iff (x,y,z) is outside surface

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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.

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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.

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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.

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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).

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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

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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

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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

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Chapter 6 Polygonal Meshes

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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

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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.

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Chapter 6 Poygonal Meshes

• Programming Task 5 Taper, twist, bend and
  squash it. Case Study 6.14, pp. 355-356, Hill.