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Author: Dirk Bertels

Hobart, November 12, 2004

Proportions in a Geodesic Dome

Calculations of the geodesic dome based on the icosahedron (frequency 3)

Proportions in a Geodesic Dome Dirk Bertels

Index

INTRODUCTION ....................................................................................................................6

DEFINITIONS..........................................................................................................................7

STANDARDS USED IN THIS PAPER..................................................................................9 VECTORS .................................................................................................................................9 VERTICES AND POINTS .............................................................................................................9

Example ............................................................................................................................10 LINES.....................................................................................................................................10

THE ICOSAHEDRON ..........................................................................................................11 LENGTH OF THE ICOSAHEDRON EDGE....................................................................................13 LENGTH OF THE ICOSAHEDRON RADIUS.................................................................................13

APPROXIMATING THE SPHERE.....................................................................................14

FREQUENCY...........................................................................................................................14 CONVERTING THE ICOSAHEDRON TO A GEODESIC DOME ........................................................14

First step...........................................................................................................................14 Second step .......................................................................................................................15 Second step .......................................................................................................................16

DETAILED CALCULATIONS OF THE GEODESIC DOME ........................................17 STEP 1 - CALCULATING SECONDARY AND TERTIARY VERTEX COORDINATES AND RADII ON THE SURFACE OF AN ICOSAHEDRON .......................................................................................17

Locate the new vertices onto the 3-dimensional grid (calculate the coordinates). ..........17 STEP 2 - CALCULATING VERTICES ON THE GEODESIC DOME.................................................18 GEODESIC SPHERE COORDINATES FOR TRIANGLE AIJ...........................................................19 LENGTH OF GEODESIC DOME EDGES.....................................................................................20

ANGULAR INFORMATION ...............................................................................................21 LOCAL AXES ..........................................................................................................................21

DIHEDRAL ANGLES ...........................................................................................................22

ANGLE_DI_BB1 ANGLE BETWEEN 2 'B' EDGES INSIDE A ICOSAHEDRON TRIANGLE............22 ANGLE_DI_BB2 ANGLE BETWEEN 2 'B' EDGES JOINING 2 ICOSAHEDRON TRIANGLES ........24 ANGLE_DI_CC ANGLE BETWEEN 2 MEETING C EDGES......................................................25

Interlude ...........................................................................................................................25 ANGLE_DI_AA ANGLE BETWEEN 2 MEETING A EDGES ......................................................27 SUMMARY .............................................................................................................................28 ILLUSTRATIONS .....................................................................................................................29

DESIGNING ANIMATION OBJECTS...............................................................................30

APPLYING AN EVEN THICKNESS - BACKGROUND .................................................31 THE 'RUSSIAN DOLL' METHOD ..............................................................................................31 CENTERS OF A TRIANGLE .......................................................................................................32

1. Gravitational center .................................................................................................32

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Proportions in a Geodesic Dome Dirk Bertels

2. Orthocenter...............................................................................................................32 3. Circumradius ............................................................................................................32 4. In-center ...................................................................................................................33 5. Inner-radius system ..................................................................................................33

APPLYING AN EVEN THICKNESS ..................................................................................35 TRANSLATING THE PLANE......................................................................................................35 FINDING THE COORDINATES...................................................................................................35 EXAMPLE TRIANGLE BLOCK...................................................................................................35 PLANE EQUATIONS - GENERAL ..............................................................................................37

vector equation .................................................................................................................37 scalar equation .................................................................................................................37

THE ACF PLANE ...................................................................................................................38 Calculate the normal vector .............................................................................................38 Adding unit normal to plane ACF to get projected plane coordinates ............................39

EQUATION FOR THE AR - CL - FL PLANE................................................................................40 THE CBE PLANE....................................................................................................................41

Plane equation for Bl........................................................................................................41 THE BAD PLANE ...................................................................................................................42

Plane equation for Al........................................................................................................42 THE BCA PLANE ...................................................................................................................42 THE BCA PLANE ...................................................................................................................43

The plane equation for point Ab.......................................................................................43 THE EDF PLANE ....................................................................................................................43 THE EDF PLANE ....................................................................................................................44

The plane equation for point Dt .......................................................................................44 PLANE INTERSECTIONS ..........................................................................................................45

Point 1: Intersection of planes Ar - Cl - Fl, Br - Al - Dl, and Bb - Cb - Ab .....................45 ILLUSTRATIONS .....................................................................................................................46 CALCULATING THE RADIAL ANGLE BETWEEN 2 VERTICES - GENERAL EQUATION ..................48

angle_A_Yaxis ..................................................................................................................48 angle_AIJ_Yaxis = 0.364864 radians ..............................................................................49 angle_A_AIJ = 0.652358 radians ....................................................................................49 angle_A_DJI = 1.38209 radians......................................................................................49 angle_A_DGJ = 1.75951 radians ....................................................................................49 angle_A_CGD = 2.48923 radians ...................................................................................49

EDGE-RADIAL ANGLES ....................................................................................................50

SUPPLEMENT 1 - CLASSIFICATION / NOMENCLATURE ........................................51

Edges ................................................................................................................................51 Chords[over edge]............................................................................................................51 Angle[chord,edge]............................................................................................................51

SUPPLEMENT 2 - EQUATION CELLS...................