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12. Polyhedra. 2005. 6. Polyhedra. All possible polyhedra are defined by only the three simplest of geometric elements (points, lines, and planes) Contents Regular polyhedra Semi-Regular polyhedra Dual polyhedra Star Polyhedra Nets The convex Hull of a polyhedron Euler’s Formula
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12. Polyhedra 2005. 6
Polyhedra • All possible polyhedra are defined by only the three simplest of geometric elements (points, lines, and planes) • Contents • Regular polyhedra • Semi-Regular polyhedra • Dual polyhedra • Star Polyhedra • Nets • The convex Hull of a polyhedron • Euler’s Formula • The Connectivity Matrix
Definition • Polyhedron • A multifaceted 3D solid bounded by a finite connected set of plane polygons • Every edge of each polygon belongs to one other polygon • The polygon faces form a closed surface, dividing space into two regions • The interior of the polyhedron and the exterior • All face of a polyhedron are plane polygons • All its edges are straight line segments • Each polyhedral edge is shared by exactly two polygonal faces • Simplest possible polyhedron (fig. 12.1) • tetrahedron(사면체) with 4 faces
Definition • 3 geometric elements define all polyhedra in space • Vertices(V), edges(E), and faces(F) • Each vertex is surrounded by an equal number of edges and faces • Each edge is bounded by two vertices and two faces • Each face is bounded by a closed loop of coplanar edges that form a polygon • Half-planes: any straight line in the plane divides the plane into two half planes • Dihedral angle: Angle b/w faces that intersect at a common edge • Two half planes extending from a common line form a dihedral angle
Definition • Polyhedral angle • Three or more planes intersecting at a common point form a polyhedral angle • The common point is the vertex of this angle • The intersection of the planes are the edges of the angle • The parts of the planes lying b/w the edge are the faces of the angle • face angle of the polyhedral angle • The angle formed by adjacent edges • For any polyhedral angle • There is an same number of edges, faces, face angles, and dihedral angles • Ex) Cube: trihedral angle a polyhedral angle with 3 faces
Definition • Comparison • An angle of 360o surrounds a point in the plane • The sum of the face angles around a vertex of a polyhedron • Angular deficit (결손): defined as difference b/w the sum of the face angles surrounding the vertex and 360o • Total angular deficit : the sum of the angular deficits over all the vertices of a polyhedron • The smaller the angular deficit, the more sphere-like the polyhedron • Regular polyhedron (=simple polyhedron) is homeomorphic to a sphere • Homeomorphics • If their bounding surfaces can be deformed into one another without cutting or gluing (= They are topologically equivalent)
The regular Polyhedra • The regular Polyhedra • A convex polyhedron is a regular polyhedron if the following condition are true • All face polygons are regular • Equal edge and interior angles • All face polygons are congruent(=identical) • All vertices are identical • All dihedral angles are equal Ex) Cube: All its face are identical All its edge are of equal length • In 3D space we can construct only 5 regular polyhedra • Tetrahedron(4면체), hexahedron(=cube) (6면체), octahedron(8면체), dodecahedron,(12면체) icosahedron(20면체)
The regular Polyhedra • The sum of all face angles • The sum of all face angles at a vertex of a convex polygon is always less than 2PI • Otherwise • If the sum of the angles = 2PI, then the edges meeting at the vertex are coplanar • If the sum of the angles > 2PI, then some of the edges at vertex are reentrant(오목한) and the polyhedron is concave • Characteristic properties of the 5 regular polyhedra (Table 12.1-2) • e: the length of the edge, RI: the radius of the inscribed sphere • RC: the radius of the circumscribed sphere • Theta: dihedral angle • Vertex coordinate for each of the regular polyhedra (Table 12.3-7)
Semiregular Polyhedra • Semiregular Polyhedra • If we relax condition 2 and 4 • All face polygons are congruent(=identical) • All dihedral angles are equal • Infinite number of polyhedra is possible • Archimedean polyhedron (13개) • Faces are regular polygons and equilateral angle • If we relax condition 1 and 3 • All face polygons are regular • All vertices are identical • Another Infinite set of polyhedra is possible • If we appropriately truncate the five regular polyhedra • Generate all the semiregular polyhedra exept 2 snub form • (Figure 12.4-5)
Semiregular Polyhedra (Examples) Archimedean semiregular polyhedra
Dual Polyhedra • Dual Polyhedra • Two polyhedra are dual • If the vertices of one can be put into a 1-to-1 correspondence with the center of the faces of the other • If we connect the centers of the faces of one of them with line segments, we obtain the edges of the other • The number of faces of one becomes the number of vertives of the other • Total number of edges does not change Ex) The octahedron and cube are dual. (Table 12.9) icoshedron and dodecahedron, tetrahedron is self dual
Star Polyhedra • Star Polyhedra • If we extend the edges of a regular polygon with five or more edge • it will enclose additional region of the plane and form a star or stellar polygon • This does not work for cubes • Their faces interpenetrate • They are not topologically simple • Euler’s Formula dose not apply
Nets • Nets (fig. 12.6-7) • By careful cutting and unfolding, we can open up and flatten out a polyhedron a net of the polyhedron • It lies in a plane • No single, unique net for a particular polyhedron
The convex Hull of a Polyhedron • A polyhedral convex hull is a 3D analog of the convex hull for a polygon • The convex hull of a convex polyhedron • Identical to the polyhedron itself • The convex hull of a concave polyhedron • By wrapping it in a rubber sheet • (Figure 12.8)
Euler’s Formula for Simple Polyhedra • Euler’s Formula (fig. 12.9) • V – E + F = 2 • Vertices (V), Edges (E), Faces (F) Ex) a cube 8 – 12 + 6 =2 a octahedron 6 – 12 + 8 = 2 • All faces must be bounded by a single ring of edges, with no holes in the faces • The polyhedron must have no holes through it • Each edge is shared by exactly two faces and is terminated by a vertex at each end • At least three edges must meet at each vertex
Euler’s Formula for Simple Polyhedra • Ludwig Schlafi’s formula • Euler’s formula is only a special case of this formula 1. An edge, or one-dimensional polytope, has a vertex at each end: N0 = 2 2. A polygonal face, or two-dimensional polytope, has as many vertices as edges: N0 – N1 = 0 3. A polyhedron, or three-dimensional polytope, satisfies Euler formula: N0 – N1 + N2 = 2 4. Four-dimensional polytope satisfies N0 – N1 + N2 – N3 = 2 5. Any simply-connected n-dimensional polytope satisfies N0 – N1 + … + (-1)n-1Nn-1 = 1 – (-1)n polytope is the general term of the sequence-point, segment, polygon, polyhedron, and so on. • Also he invented the symbol {p, q} for the regular polyhedron whose face are p-gons, q meeting at each vertex, or the polyhedron with face {p} and veretx figure {q}
The Connectivity Matrix • The Connectivity Matrix • A two-dimensional list or table that describes how vertices are connected by edges to form a polyhedron • square matrix with as many rows and columns as vertices • Symmetric matrix about its main diagonal, which is comprised of all zeros (twice as much information as necessary) • If element aij = 1, then vertices I and j are connected by an edge • If element aij = 0, then vertices I and j are not connected • Ex) fig 12.13-14 What do we do about the faces?? Form a matrix with each row containing the vertex sequence bounding a face (counterclockwise order outward from the interior of the polyhedron)
Halfspace Representation of Polyhedra • Halfspace Representation of Polyhedra • Represent a convex polyhedron with n faces by a consistent system of n equations constructed as follows Aix+ Biy + Ciz + Di >0 • Any point that satisfies all n inequalities lies inside the polyhedron • Ex) a cube : x > 0 -x + 4 > 0 y > 0 -y + 4 > 0 z > 0 -z + 4 > 0
Halfspace Representation of Polyhedra • Definition of the polyhedron P • Ex) four-side polyhedron (figure 12.15)
Maps of Polyhedra • Schlegel diagram or map • A special two-dimensional image of a polyhedron • Projecting its edges onto a plane from a point directly above the center of one of its face Tetrahedron Hexahedron (=cube) octahedron dodecahedron • icosahedron