Theory of General Minories

[Flying Pig]

17 minutes ago, Pacific231G said:

I don't think the trial version of AnyRail includes the curved flextrack function

 

Unless it has changed recently it does, but the section of flextrack has to be unconnected at both ends which can be inconvenient.

 

[Pacific231G]

I have found some anomalies in the way that Anyrail smooths flex track.

The minimum radius produced by replacing a medium radius point is 31.34" and for a large radius point it is 42.59" with the replacement track 10.15 inches long (slightly shorter than the point's linear length because of the relationship between the end points of the diverging and straight tracks)

 

The anomaly is that, if you decrease the gap between the end of the real point and the parallel straight, the minimum radius gradually decreases as you'd expect but it also decreases if you increase the length of the gap. I've been experimenting with this and the largest minimum radius I could get with increments of 0.5 inches was 45.15 when the length of the replacement flex was 9.76cms with the gap half an inch shorter than the long turnout.

 

I'm not going to try smaller increments- life's too short-  but I assume the largest minimum comes when the curve is a constant radius.- not what you actually want when laying track.  If you try to use a larger radius it would overshoot the parallel track and as you increase the length of the gap the software starts adding more of a transition curve that reduces the minimum radius, unfortunately it seem to add te transtion to one end only  (Because AnyRail displays a  minimum radius you can slice the track into very small pieces and then analyse the changing radii through the curve) It would be interesting to know what that minmum radius would be for 50mm centres coming off a 12 degree crossing angle if you used a true transition curve, rather less than 40 inches I suspect.

 

Looking at MRCs from the 1950s, there are fiendishly complex mathematical formulae for calculating and setting out true transition curves and working out the minima needed to avoid buffer locking - the formulae were even more fiendish in Loco-Revue- but you can probably do almost as good a job by eye. I'm wondering though whether something like

a very short tracksetta curve might be useful for avoiding excessively tight spots.  I'm rather intrigued by Peter Denny's tool for laying out curves consisting of a strip of EM gauge wide wood with saw cuts most of the way through at short intervals to make it flexible but with a tendency to return to straight. 

 

Edited December 1, 2020 by Pacific231G

[t-b-g]

18 minutes ago, Pacific231G said:

I have found some anomalies in the way that Anyrail smooths flex track.

The minimum radius produced by replacing a medium radius point is 31.34" and for a large radius point it is 42.59" with the replacement track 10.15 inches long (slightly shorter than the point's linear length because of the relationship between the end points of the diverging and straight tracks)

 

The anomaly is that, if you decrease the gap between the end of the real point and the parallel straight, the minimum radius gradually decreases as you'd expect but it also decreases if you increase the length of the gap. I've been experimenting with this and the largest minimum radius I could get with increments of 0.5 inches was 45.15 when the length of the replacement flex was 9.76cms with the gap half an inch shorter than the long turnout.

 

I'm not going to try smaller increments- life's too short-  but I assume the largest minimum comes when the curve is a constant radius.- not what you actually want when laying track.  If you try to use a larger radius it would overshoot the parallel track and as you increase the length of the gap the software starts adding more of a transition curve that reduces the minimum radius, unfortunately it seem to add te transtion to one end only  (Because AnyRail displays a  minimum radius you can slice the track into very small pieces and then analyse the changing radii through the curve) It would be interesting to know what that minmum radius would be for 50mm centres coming off a 12 degree crossing angle if you used a true transition curve, rather less than 40 inches I suspect.

 

Looking at MRCs from the 1950s, there are fiendishly complex mathematical formulae for calculating and setting out true transition curves and working out the minima needed to avoid buffer locking - the formulae were even more fiendish in Loco-Revue- but you can probably do almost as good a job by eye. I'm wondering though whether something like

a very short tracksetta curve might be useful for avoiding excessively tight spots.  I'm rather intrigued by Peter Denny's tool for laying out curves consisting of a strip of EM gauge wide wood with saw cuts most of the way through at short intervals to make it flexible but with a tendency to return to straight. 

 

 

The Denny wooden gauge was used in track construction rather than laying out the plan on a board. Peter made a selection of card curves, with the radius marked on them, at 2" intervals. To make a transition curve, he would use the 3ft, then the 3ft 2ins, then 3ft 4ins etc.

 

He would then lay the sleepers, complete with pins already inserted, solder one rail and use the wooden gauge to add the second rail. The slots in the timber would allow the gauge to go to any appropriate curve and the straight bits, between the saw cuts, created a bit of gauge widening on tighter curves.

 

 

[Chimer]

1 hour ago, Pacific231G said:

I have found some anomalies in the way that Anyrail smooths flex track.

 

 

X-TrackCad does something similar in that it usually fills the gap with a curve and a tiny piece of straight.  But when you actually get round to laying the track, when presumably you'll be using the same bit of flexi for the reverse curve and the start of the parallel track, I'm sure your eye will tell you all you need to know to get it right.  I waste so much design time trying for perfect alignment down to 3 decimal places on the PC, when I'll count myself lucky if I follow a plan to within a quarter of an inch on the baseboard ..... 

[Pacific231G]

1 hour ago, t-b-g said:

 

The Denny wooden gauge was used in track construction rather than laying out the plan on a board. Peter made a selection of card curves, with the radius marked on them, at 2" intervals. To make a transition curve, he would use the 3ft, then the 3ft 2ins, then 3ft 4ins etc.

 

He would then lay the sleepers, complete with pins already inserted, solder one rail and use the wooden gauge to add the second rail. The slots in the timber would allow the gauge to go to any appropriate curve and the straight bits, between the saw cuts, created a bit of gauge widening on tighter curves.

 

 

Thanks Tony

It was never entirely clear to me from his books and articles exactly how he did it and, since he'd already widened the gauge from 18 to 18.2mm (I've just been looking at the 1950 BRMSB standards and both EM and EEM are still 18mm) the gauge widening aspect is interesting. I don't recall any mention by him of the  card curves so had assumed he used the wooden gauge - which I saw at ExpoEM- for laying out the track as well.

[Pacific231G]

1 hour ago, Chimer said:

 

X-TrackCad does something similar in that it usually fills the gap with a curve and a tiny piece of straight.  But when you actually get round to laying the track, when presumably you'll be using the same bit of flexi for the reverse curve and the start of the parallel track, I'm sure your eye will tell you all you need to know to get it right.  I waste so much design time trying for perfect alignment down to 3 decimal places on the PC, when I'll count myself lucky if I follow a plan to within a quarter of an inch on the baseboard ..... 

I used to use X-TrkCad for all my track planning and yes I'd noticed the way it could only analyse a curve as a single radius arc and straights though it does produce proper easements (transition curves)  . I often found it refusing to join two ends of track a metre apart because they were parallel with a small displacement . Trying to get it to insert the short track in Minories between the point at the entrance to platform one and the loco siding point was a regular frustration. The trouble was that if you couldn't quite get track to join you couldn't try out ideas by operating trains on the plan which is one of XtkCad's great virtues- especially as you can set up a file with yout own rolling stock (appearing only as geomettic shapes but very good for testing the operational practicality of plans. I think youcan make it a bit less picky but I've noticed that all the plans I'ce drawn up with Xtkcd tend to be a bit straight line based and definitely change when I actually build the layout.

That said it's completely free and is a very useful programme. I tend to use AnyRail for "sketching" track plans but XtkCad for when the planning gets serious.

Edited December 1, 2020 by Pacific231G

[Keith Addenbrooke]

1 hour ago, Chimer said:

 

X-TrackCad does something similar in that it usually fills the gap with a curve and a tiny piece of straight.  But when you actually get round to laying the track, when presumably you'll be using the same bit of flexi for the reverse curve and the start of the parallel track, I'm sure your eye will tell you all you need to know to get it right.  I waste so much design time trying for perfect alignment down to 3 decimal places on the PC, when I'll count myself lucky if I follow a plan to within a quarter of an inch on the baseboard ..... 

 

My baseboards are usually 1/4” out to start with.

[t-b-g]

1 hour ago, Pacific231G said:

Thanks Tony

It was never entirely clear to me from his books and articles exactly how he did it and, since he'd already widened the gauge from 18 to 18.2mm (I've just been looking at the 1950 BRMSB standards and both EM and EEM are still 18mm) the gauge widening aspect is interesting. I don't recall any mention by him of the  card curves so had assumed he used the wooden gauge - which I saw at ExpoEM- for laying out the track as well.

 

I don't think he ever really measured the gauge and he certainly wasn't to a decimal point of a millimetre!

 

He found that the Romford axles giving a 16.5mm back to back plus thick flanges on the wheels of the day meant that the wheels were a bit tight in 18mm, so he made the track "a bit wider". He never specified the "bit" but later wrote that it was "about 18.25mm".

 

He was quite pragmatic about track and if there was a place where he had trouble, he would tinker with until it ran well. I have measured, with my Vernier, a minimum of 17.5mm in one place up to a maximum of 19.5mm. It certainly isn't a railway where every bit of stock will run everywhere without trouble but once I got my head around the idea that the sequence/timetable prevents certain trains going to places where they would fall off, it all works!  

[Pacific231G]

56 minutes ago, t-b-g said:

 

I don't think he ever really measured the gauge and he certainly wasn't to a decimal point of a millimetre!

 

He found that the Romford axles giving a 16.5mm back to back plus thick flanges on the wheels of the day meant that the wheels were a bit tight in 18mm, so he made the track "a bit wider". He never specified the "bit" but later wrote that it was "about 18.25mm".

 

He was quite pragmatic about track and if there was a place where he had trouble, he would tinker with until it ran well. I have measured, with my Vernier, a minimum of 17.5mm in one place up to a maximum of 19.5mm. It certainly isn't a railway where every bit of stock will run everywhere without trouble but once I got my head around the idea that the sequence/timetable prevents certain trains going to places where they would fall off, it all works!  

Isn't that how we ended up with four feet eight and a half inches from George and seven feet and a quarter inch from Isambard Kingdom?    though I've long suspected that Spooner's one foot eleven and a half inches was because he was really working in metric and it was sixty centimetres to the nearest quarter inch- (ISTR that the ffestiniog's gauge  was originally given as 1ft 11 5/8 inches which is 600mm to the nearest hundredth of a millimetre) 

[RJS1977]

1 hour ago, t-b-g said:

He was quite pragmatic about track and if there was a place where he had trouble, he would tinker with until it ran well. I have measured, with my Vernier, a minimum of 17.5mm in one place up to a maximum of 19.5mm. It certainly isn't a railway where every bit of stock will run everywhere without trouble but once I got my head around the idea that the sequence/timetable prevents certain trains going to places where they would fall off, it all works!  

 

That's good to know! Whilst it doesn't make Denny any less of a modeller or Buckingham any less of a layout, it's something to remember next time I get a derailment....

[Pacific231G]

11 hours ago, Flying Pig said:

 

Unless it has changed recently it does, but the section of flextrack has to be unconnected at both ends which can be inconvenient.

 

Thanks Simon

I'd not spotted that feature before. I've been experimenting and you can indeed use a 48 inch 12 degree curve to connect a single set of points to a parallel curve at Peco's standard 2 inch  separation though I don't think you'd actually want to use a simple arc with no transition at all in that situation.

 

Using the easement curve function I found that, for a true transition curve (but only at the point end) the minimum radius  actually comes down to 24 inches.

What that also tells me is that Peco couldn't possibly have produced a turnout with a 12  degree/ 1 inch  divergence with a radius more than four foot and actually rather less than that given the degree of transition inherent in a normal set of points  so excluding setrack or Decauville/industrial/mine type portable track with a fixed radius curve and wagons lurching over them at slow speed like this.

 

 

A set of points for 500mm from Decauville's portable track range.

 

 

 

Edited December 1, 2020 by Pacific231G

[Keith Addenbrooke]

9 hours ago, Pacific231G said:

I've been experimenting and you can indeed use a 48 inch 12 degree curve to connect a single set of points to a parallel curve at Peco's standard 2 inch  separation though I don't think you'd actually want to use a simple arc with no transition at all in that situation.

 

Good point - the 48” Rad. 12 Deg. curve I use for sketching is effectively a piece of super-sized Setrack to help with track spacing.  When track laying, using a spare point as a template to plot the reverse easement should work (either that, or the eye would naturally produce an easement?)

 

Applying this to Minories proper, the relevant curve is the one for departures from P1:

 

 

The easement would be that of a right hand facing point, not a left hand trailing point.  I think that’s right?

[Flying Pig]

If my trig is correct, the radius for the return curve works out as 45.76", assuming an offset of 1" and an arc of 12°.  This is without any easement of course.  

 

[Pacific231G]

6 hours ago, Keith Addenbrooke said:

Good point - the 48” Rad. 12 Deg. curve I use for sketching is effectively a piece of super-sized Setrack to help with track spacing.  When track laying, using a spare point as a template to plot the reverse easement should work (either that, or the eye would naturally produce an easement?)

 

Applying this to Minories proper, the relevant curve is the one for departures from P1:

 

 

The easement would be that of a right hand facing point, not a left hand trailing point.  I think that’s right?

Yes. If you don't include the loco layover siding. Replacing that point with a smoothed flex AnyRail gave me a minimum radius of 37 inches -with most of the curve at the outer end (which is where you need it to avoid an S curve with the exit point from plaftom one ) With the original CJF throat you end up having to place the loco spur point some way between between the two main line points if you want the track between it and them to be straight. It's fiendishly difficult to get that spot on in track planning software though easier in reality. I ended up with 1.08 inches of tracks between the final exit point and the toe end  of the loco siding point and 10.05 inches between the heel ends of the other two. 

This track is causing me some grief in my attempt to improve the excessive throwover across the two back to back right hand points by subsitutung the outer with a Y and the final exit point with a large radius point.Having solved that problem  I'm in danger of getting a reverse curve on the exit route from platform one which could defeat the entire object of the exercise which is to avoid a situation where the corridor connections are totally displaced.

 

Edited December 2, 2020 by Pacific231G

[Flying Pig]

1 hour ago, Pacific231G said:

It's fiendishly difficult to get that spot on in track planning software though easier in reality.

 

As I recall, XTrackCAD will connect two intersecting pieces of flexi just by dropping a suitable turnout onto the intersection, provided angles and alignments are within tolerance.  It's an extremely useful feature.

[Keith Addenbrooke]

1 hour ago, Pacific231G said:

Yes. If you don't include the loco layover siding. Replacing that point with a smoothed flex AnyRail gave me a minimum radius of 37 inches -with most of the curve at the outer end (which is where you need it to avoid an S curve with the exit point from plaftom one ) With the original CJF throat you end up having to place the loco spur point some way between between the two main line points if you want the track between it and them to be straight. It's fiendishly difficult to get that spot on in track planning software though easier in reality. I ended up with 1.08 inches of tracks between the final exit point and the toe end  of the loco siding point and 10.05 inches between the heel ends of the other two. 

This track is causing me some grief in my attempt to improve the excessive throwover across the two back to back right hand points by subsitutung the outer with a Y and the final exit point with a large radius point.Having solved that problem  I'm in danger of getting a reverse curve on the exit route from platform one which could defeat the entire object of the exercise which is to avoid a situation where the corridor connections are totally displaced.

 

I don’t know if this has been covered earlier, but presumably the prototype has rules governing installation of crossovers to manage this kind of situation, both in terms of track and - I assume - the degree of throwover corridor connections need to ‘stretch’ to.

[ejstubbs]

1 hour ago, Pacific231G said:

It's fiendishly difficult to get that spot on in track planning software

 

I have a small Excel worksheet that will tell me what length of straight track I need at a given angle in order to link two points (both in the sense of turnouts, and x,y coordinates).  In the case of the Minories loco spur point, you just connect the toe end of the loco spur turnout to the trailing turnout on the departure road, and use the y coordinate of the end of the diverging track at the heel end, and that of the point you're going to connect it to.  Feed those two coordinates into the worksheet, plus the 12° angle, and make (and freeze) a straight piece of flexi to that length.  Connect that to the lower point, disconnect the loco spur point from the trailing point on the departure road and re-connect it to the free end of the piece of frozen flexi.  Measure the resulting gap between the loco spur point and the departure road point (by selecting one of the free track ends, and hovering the mouse pointer over the other free end - the distance between the two is shown at the bottom of the screen) and make & freeze another piece of straight flexi to that length.  Plug it in to the gap and Robert's your mother's brother.

 

The steps above are as for AnyRail, by the way (I've tried but never got on with XTrackCAD).  The process sounds a bit complicated when described in the cack-handed way I have above but it takes seconds to do when you have a simple tool to hand on screen to do the trig.

 

I have another worksheet in the same workbook that does the curve radius calculation outlined by Flying Pig further up, and another one that does the fixed-radius-curve+short-straight calculation that Chimer indicated XTrackCAD uses.

[Pacific231G]

2 hours ago, Keith Addenbrooke said:

I don’t know if this has been covered earlier, but presumably the prototype has rules governing installation of crossovers to manage this kind of situation, both in terms of track and - I assume - the degree of throwover corridor connections need to ‘stretch’ to.

Yes they do. For straight crossovers it's really a matter of using an appropriate crossing angle to give enough straight track between the crosssings and, for points set toe to toe, leaving a suitable gap between them. I've hopefully attached a pdf of the relevant page from the Network Rail Track Design Handbook from 1999 and maximum throwover of buffers is 300mm or about a foot. The formula is actially a bit simpler than those I remember seeing in 1950s Model Railway Constructors ans is one we could probably use- probably horrifying ourselves in the process!

nr40_track design handbook_2007 buffer locking.pdf

 

The most extreme case I'm personally familiar with was on the Greenford branch when its trains used to go to and from Paddington. Coming from Paddington, after leaving West Ealing,  the two car train crossed from the down relief to the up relief on a facing crossover then immediately took a right hand point onto the down branch.  You can see

in the photo below the throwover that entailed. Reflections in the door glass of the seats in my carriage make it quite hard to make out the side of the displaced door in the other car so I've drawn in a white bar to indicate the degree of throwover. I reckon it was about eight or nine inches.

The service was and is operated by a two car class 165 DMU with a car length of 75ft (22.91m) which is faitly long and about the same as the Mk.3 coaches used in the HST sets that,  used to make regular ECS movements on the branch and would have encountered the same crossover. 

Edited December 2, 2020 by Pacific231G
Addition of page from NR handbook

[Chimer]

2 hours ago, Flying Pig said:

 

As I recall, XTrackCAD will connect two intersecting pieces of flexi just by dropping a suitable turnout onto the intersection, provided angles and alignments are within tolerance.  It's an extremely useful feature.

 

I used to be able to make that work, but I've lost the knack  - I think there's a tutorial about ladder sidings in the wiki that explains it.

 

A new(ish) feature of XTrackCad is that it can do joins using what it calls "Cornu" curves, which are curves where the radius varies along the length to provide the best possible transitions.  I don't pretend to understand the maths, and sometimes it produces funny results, but it will join the divergent leg of a turnout to a track parallel to the straight leg, as far along the other track as you want, as a single piece of track.  It's an option under "easement" from version 5.1 (I think) onward.  In my experience, best only switched on when really wanted, for example if you want a big sweeping turn somewhere which you'd normally have to fiddle with two or three sections of curve at different radii - it'll just do it for you in one smooth curve (and tell you the tightest radius it's used anywhere along the length).

[thegreenhowards]

This is my first post on this fascinating thread having skim read much of it over the last few days. I am designing an 0 gauge layout for our club's new venture into the senior scale loosely based on the Minories plan. This is what we've settled on at the moment.

 

It's designed to fit into a space in our clubroom with a wall which projects out at the right end (as shown) but still allows room for a narrow shelf in front of it. The scenic length is 19' with the shelf being another 10' which can support cassettes for a fiddle yard. This area is where members sit and socialise (Covid permitting) on our main club night, so the layout will have to pack down easily and just be put up on '0' gauge nights. 

 

The layout is designed to be exhibitable and for exhibitions can have a wider fiddle yard board with traverser fiddle yard and a narrow scenic section in front which we plan will be an additional goods facility. It will be viewed and operated from the bottom of the plan in the clubrooms, but viewed from the top at exhibitions.

 

I started with a classic Minories but then was asked to add in more goods interest. I have done this in two ways:

1. A line to a hidden goods cassette (bottom left) which I envisage as through goods to Billingsgate fish market or similar. The line would descend and enter a tunnel rather like platform 15 at the old King's Cross. The 4' cassette can hold a J50 + 6 vans and a brake van in 0 gauge.

2. Rather more goods on the kick back (top right) than the classic Minories with a line which can extend in future 

 

The Peco pointwork in O gauge has an 8 degree angle which makes for gentler double slips, so getting round some of the problems listed over the last few pages.

 

I'd be interested in any comments the 'Minories experts' on here may have on the plan.

 

Thanks

 

Andy

[Pacific231G]

3 hours ago, Chimer said:

 

I used to be able to make that work, but I've lost the knack  - I think there's a tutorial about ladder sidings in the wiki that explains it.

 

A new(ish) feature of XTrackCad is that it can do joins using what it calls "Cornu" curves, which are curves where the radius varies along the length to provide the best possible transitions.  I don't pretend to understand the maths, and sometimes it produces funny results, but it will join the divergent leg of a turnout to a track parallel to the straight leg, as far along the other track as you want, as a single piece of track.  It's an option under "easement" from version 5.1 (I think) onward.  In my experience, best only switched on when really wanted, for example if you want a big sweeping turn somewhere which you'd normally have to fiddle with two or three sections of curve at different radii - it'll just do it for you in one smooth curve (and tell you the tightest radius it's used anywhere along the length).

I've just downloaded it for my current computer and noticed the "Cornu" curves. I knew them as Euler spirals. The principle is very simple (though the maths isn't) . When joining two fixed radius curves (or a straight and a curve)  the curvature (1/R) of the Euler curve changes gradually from one to the other at a constant rate. It means that a pony truck say doesn't suddenly slam over when a loco meets a curve but moves smoothly from straight ahead to the position coresponding to the final radius of the curve.

[Pacific231G]

16 minutes ago, thegreenhowards said:

This is my first post on this fascinating thread having skim read much of it over the last few days. I am designing an 0 gauge layout for our club's new venture into the senior scale loosely based on the Minories plan. This is what we've settled on at the moment.

 

It's designed to fit into a space in our clubroom with a wall which projects out at the right end (as shown) but still allows room for a narrow shelf in front of it. The scenic length is 19' with the shelf being another 10' which can support cassettes for a fiddle yard. This area is where members sit and socialise (Covid permitting) on our main club night, so the layout will have to pack down easily and just be put up on '0' gauge nights. 

 

The layout is designed to be exhibitable and for exhibitions can have a wider fiddle yard board with traverser fiddle yard and a narrow scenic section in front which we plan will be an additional goods facility. It will be viewed and operated from the bottom of the plan in the clubrooms, but viewed from the top at exhibitions.

 

I started with a classic Minories but then was asked to add in more goods interest. I have done this in two ways:

1. A line to a hidden goods cassette (bottom left) which I envisage as through goods to Billingsgate fish market or similar. The line would descend and enter a tunnel rather like platform 15 at the old King's Cross. The 4' cassette can hold a J50 + 6 vans and a brake van in 0 gauge.

2. Rather more goods on the kick back (top right) than the classic Minories with a line which can extend in future 

 

The Peco pointwork in O gauge has an 8 degree angle which makes for gentler double slips, so getting round some of the problems listed over the last few pages.

 

I'd be interested in any comments the 'Minories experts' on here may have on the plan.

 

Thanks

 

Andy

Hi Andy

It looks fine to me and a layout I'll definitely want to go and see when it gets exhibited (Coulsdon should make it accessible for me.)  I like the idea of it being both an exhibition layout and one you can enjoy operating on club nights and not just to prepare for exhibition. (I sometimes think the exhibition tail tends to wag the operating the layout just to enjoy it dog) 

Looking at the plan. I can't see any reverse curves through pointwork for any normal routes, the loco spur is in a logical place  and I like the idea of the goods line disappearing behind the terminus (from an exhibtion visitor's point of view)  That's not disimilar to what Geoff Ashdown did with Tower Pier - though his goods line to St. Katherine's dock and the corresponding sidings were a separate line from the passenger terminus - something I wasn't that keen on- but it worked well with a hinged road in front of the terminus covering the hidden sidings representing St. Katherine's dock.

Do you know what sort of stock you'll be running on it and what sort of train lengths the platforms can handle?

I think my only caveat would be that if you're not using automatic couplers (or even if you are and they need help) the good sidings may be a bit of a stretch- over two foot- across the station platforms and their canopies)

 

 

 

[St Enodoc]

1 hour ago, Pacific231G said:

I've just downloaded it for my current computer and noticed the "Cornu" curves. I knew them as Euler spirals. The principle is very simple (though the maths isn't) . When joining two fixed radius curves (or a straight and a curve)  the curvature (1/R) of the Euler curve changes gradually from one to the other at a constant rate. It means that a pony truck say doesn't suddenly slam over when a loco meets a curve but moves smoothly from straight ahead to the position coresponding to the final radius of the curve.

Also known, probably more widely in the railway world, as clothoids. They are commonly used as transition curves on full-sized railways. Another form of transition curve is the cubic parabola.

 

The curvature of a clothoid varies according to the distance from the tangent point along the curve, whereas the curvature of a cubic parabola varies according to the distance from the tangent point in a straight line along the tangent.

 

Over short lengths and small angles of divergence these are virtually indistinguishable, certainly in model form.

[Compound2632]

The Peco large radius points don't actually conform to their standard streamline geometry - the crossing angle is less than 12°; that angle is only achieved at the end of the unit, the curve continuing through the crossing. This results in an unprototypical S-bend if one uses a pair as an ordinary crossover and even more problems if you try to trim them to achieve a nearer-scale four-foot (i.e. track centres of 45 mm rather than 2") and use them in conjunction with a slip or crossing.

[Harlequin]

23 minutes ago, Compound2632 said:

The Peco large radius points don't actually conform to their standard streamline geometry - the crossing angle is less than 12°; that angle is only achieved at the end of the unit, the curve continuing through the crossing. This results in an unprototypical S-bend if one uses a pair as an ordinary crossover and even more problems if you try to trim them to achieve a nearer-scale four-foot (i.e. track centres of 45 mm rather than 2") and use them in conjunction with a slip or crossing.

 

Peco don't make any claims about the crossing angle, which as you say, is not 12° on the Large and the Curved points. All they say is that where the centre lines are 1 inch apart the angle will be 12° and in that respect those turnouts do meet the standard.