Large Curves Using Straight Track

Origins The Holger Matthes article in RAILBRICKS issue #1 2007 Courtesy : RailBricks

But… It gave a very basic overview of how to do it Only showed one way & one size Courtesy : RailBricks

How About Going Even Bigger The 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 Review Review of Triangles Review of Polygons

Triangle Geometry Review The only triangle we need to be most familiar with is the isosceles triangle A triangle with two equal sides (or legs) & two equal angles is an isosceles triangle Matthes’ created a ½-8-8 (in studs) triangle and “wedged” it between the tracks Increase 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 of 3.58 degrees A ½ - 9½ - 9½ triangle has an angle measure of 3.02 degrees

Polygon Geometry Review A “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 Consider The 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 Angle Isosceles triangles with a base length of ½ stud Sides 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 Leg Length 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 132 2.727273 10.5 2.728113 136 2.647059 11 2.601429 140 2.571429 148 2.432432 11.5 2.490925 152 2.368421 12 2.387151 Those with a “ ” are best matching

HOW BIG ARE THESE THINGS??? So Far… We’ve got the triangles covered We’ve got the polygons covered We’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 Radius The radius values given are from the inner most edge of the curve to the center of the circle When 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 studs In layouts using ballast, allow space for the wedge to rest on and between up 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 Radius Running 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 Notes Reminder: 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 Future The creation of gentile uphill/downhill paths that curve The creation of an “S” curve as shown below in this aerial view of an LRV overpass Courtesy : Google Maps

Looking Into the Future A table for creating the 60˚ curve (currently in the works) Polygon Sides Tracks per 60̊ 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