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12. Polyhedra

2005. 6

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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

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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

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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

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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

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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 wit

hout cutting or gluing (= They are topologically equivalent)

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The regular Polyhedra

• The regular Polyhedra – A convex polyhedron is a regular polyhedron if the follow

ing condition are true1. All face polygons are regular

– Equal edge and interior angles

2. All face polygons are congruent(=identical)3. All vertices are identical4. All dihedral angles are equalEx) Cube: All its face are identical All its edge are of equal length• In 3D space we can construct only 5 regular polyhedra

– Tetrahedron(4 면체 ), hexahedron(=cube) (6 면체 ), octahedron(8 면체 ), dodecahedron,(12 면체 ) icosahedron(20 면체 )

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The regular Polyhedra

– The sum of all face angles• The sum of all face angles at a vertex of a convex polygon is alwa

ys 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 r

eentrant( 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)

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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 (13 개 )

• 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)

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Semiregular Polyhedra (Examples)

Archimedean semiregular polyhedra

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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

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Regular polygon and star polygon

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Regular polygon and star polygon

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Star Polyhedra

• Star Polyhedra– If we extend the edges of a regular polygon with five or m

ore 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

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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

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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)

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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

1. All faces must be bounded by a single ring of edges, with no holes in the faces

2. The polyhedron must have no holes through it

3. Each edge is shared by exactly two faces and is terminated by a vertex at each end

4. At least three edges must meet at each vertex

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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 = 24. Four-dimensional polytope satisfies N0 – N1 + N2 – N3 = 25. 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}

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The Connectivity Matrix

• The Connectivity Matrix– A two-dimensional list or table that describes how vertice

s 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 compr

ised 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)

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Halfspace Representation of Polyhedra

• Halfspace Representation of Polyhedra– Represent a convex polyhedron with n faces by a consiste

nt 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

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Halfspace Representation of Polyhedra

• Definition of the polyhedron P

– Ex) four-side polyhedron (figure 12.15)

where),,( 2111

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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