OriginsThe Holger Matthes article in RAILBRICKS issue #1

2007

Courtesy : RailBricks

But…It gave a very basic overview of how to do itOnly showed one way & one size

Courtesy : RailBricks

How About Going Even BiggerThe RAILBRICKS article mentions using only 23

tracks but it can be more than that!By understanding the geometry behind the large

curves that will allow us even larger and more gentile curves using 9V straight tracks

Quick Geometry ReviewReview of Triangles Review of Polygons

Triangle Geometry ReviewThe only triangle we need to be most familiar with is

the isosceles triangleA triangle with two equal sides (or legs) & two equal

angles is an isosceles triangleMatthes’ created a ½-8-8 (in studs) triangle and

“wedged” it between the tracksIncrease the length of two legs while keeping the

third side (or base) unchanged decreases the angle between the legs (example: ½-9½ -9½ triangle)

Triangle Geometry Review

A ½ - 8 - 8 triangle has an angle measure of3.58 degrees

A ½ - 9½ - 9½triangle has an angle measure of 3.02degrees

Polygon Geometry ReviewA “regular” polygon has “n” number of sides, each of

equal length and vertices of equal angle measure.The central angle is the angle made at the center of

the polygon by any two adjacent radii of the polygon.By making a full “circle” using only straight tracks we

are in fact making some regular polygon (example: 100-gon or hectogon)

Polygon Geometry Review

An example of a regular polygon and location of the central angle

Visualizing the Central Angle of a Polygon Made of Straight Tracks

Red triangle shows the central angle of the polygon for the inside curve

Blue triangle shows the central angle of the

polygon for the outside curve

The inside curve is made from a 100-gon & the outside curve is from a 108-gon

What to ConsiderThe large triangle created using the radii & central

angle of the polygon is proportional to the small isosceles triangle being wedged between the tracks

The angle measure of the isosceles triangle between the tracks should equal or approximate the central angle of the particular polygon in use

When making 90˚ curves, the regular polygons selected should have values divisible by 4, thus allowing for easy creation of quadrants (example: 128-gon ÷ 4 = 32 tracks per quadrant)

The Number Crunching is Done!Regular Polygons and

their Central AngleIsosceles triangles with

a base length of ½ studSides Central Angle

100 3.6

104 3.4615

108 3.3333

112 3.214286

116 3.103448

120 3.0

124 2.903226

128 2.8125

132 2.727273

136 2.647059

140 2.571429

144 2.5

148 2.432432

152 2.368421

LegLength

Angle Between the Legs

8 3.580403

8.5 3.369854

9 3.18269

9.5 3.015219

10 2.864491

10.5 2.728113

11 2.601429

11.5 2.490925

12 2.387151

Best Matchings of the Central Angles to the Isosceles Triangle

Polygon Sides Central Angle Isosceles Triangle Leg Length

Angle Measure Between Legs

100 3.6 8 3.580403

108 3.3333 8.5 3.369854

112 3.214286 9 3.18269

120 3 9.5 3.015219

124 2.903226 10 2.864491

132 2.727273 10.5 2.728113

140 2.571429 11 2.601429

152 2.368421 12 2.387151

Values were chosen with a less than +/- 0.04 degree error margin

Close Matchings of the Central Angles to the Isosceles Triangle

Polygon Sides Central Angle Isosceles Triangle Leg Length

Angle Measure Between Legs

128 2.8125 10 2.864491

136 2.647059 11 2.601429

148 2.432432 11.5 2.490925

Values were chosen with a greater than +/- 0.04 degree error margin

General Matching of the Central Angle to the Isosceles Triangle

Polygon Sides Central Angle Isosceles Triangle Leg Length

Angle Measure Between Legs

100 3.6 8 3.580403

108 3.3333 8.5 3.369854

112 3.214286 9 3.18269

120 3 9.5 3.015219

124 2.903226 10 2.864491

128 2.8125 10 2.864491

132 2.727273 10.5 2.728113

136 2.647059 11 2.601429

140 2.571429 11 2.601429

148 2.432432 11.5 2.490925

152 2.368421 12 2.387151Those with a “ ” are best matching

So Far…We’ve got the triangles coveredWe’ve got the polygons coveredWe’ve got the combinations of triangle to polygon

covered

But…

HOW BIG ARE THESE THINGS???

Radius Values of Matched Pairs Polygon Sides Straight

Tracks per Quadrant

Approximate Radius Length

in Studs

Approximate Radius Length

in cm./ in.

100 25 255 204 / 80

108 27 275 220 / 96

112 28 285 228 / 90

120 30 306 245 / 96

128 32 326 261 / 103

132 33 336 269 / 106

136 34 346 277 / 109

140 35 357 286 / 112

148 37 377 302 / 119

152 38 387 310 / 122

Important Notes About the RadiusThe radius values given are from the inner most

edge of the curve to the center of the circleWhen planning for layouts, be sure to add 8 studs

for track width and up to 4 studs more depending on the triangle wedge in use example: 128-gon has 326 studs radius + 10 studs (8

for track & 2 extending from wedge) = 336 studsIn layouts using ballast, allow space for the wedge to

rest on and betweenup to 4 plates for depth width and length varies according to positioning and

size of wedge in use

Important Notes About the Radius

Important Notes About the RadiusRunning two distinct (different radius values) curves beside

each other will not produce an 8 stud gap between track

Important Notes About the Radius

To avoid the “It’s not 8 studs!” issue, go back to what you did with regular curved tracks.For your outside curve, use a polygon with the same

radius of the inside curve Adding some straight tracks at the 0˚ & 90˚ marks of

the outer curve will help align the two curves and give the 8 stud gap between the tracks.

Additional NotesReminder: inserting such curves into a layout

requires a lot of space and leaves a big footprint!Matthes noted in his article that there can be

changes in electrical resistance “While electrical continuity is preserved, resistance might

increase with this design, i.e., heavy trains far from the pickup might slow or stop. A simple solution, if such a problem arises, is to use two or more electrical pickups from the same controller, distributed around the track (just be sure to connect them with the same orientation).”

Looking Into the FutureThe creation of gentile uphill/downhill paths that

curveThe creation of an “S” curve as shown below in this

aerial view of an LRV overpass

Courtesy : Google Maps

Looking Into the FutureA table for creating the 60˚ curve

(currently in the works)

Polygon Sides

Tracks per 607

Central Angle

Radius in Studs

Isosceles Triangle to Use

102 17 3.52941 259 8

108 18 3.33333 275 8.5

114 19 3.15789 290 9

120 20 3.00000 306 9.5

126 21 2.85714 321 10

132 22 2.72727 336 10.5

138 23 2.60869 351 11

Thank You&

Leg Godt