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Perpendicularity

Perpendicularity

Perpendicularity

(OP)
Can anyone please clarify if a perpendicularity call out is controlled by a basic dimension or not?  In other words is the tolerance zone centered on a basic dimension?

RE: Perpendicularity

No, it is sufficient by itself.

It MAY be used together with basic dimensions when used to refine the Position.
 

RE: Perpendicularity

No, the tolerance zone moves with the feature being controlled.

Powerhound, GDTP T-0419
Engineering Technician
Inventor 2010
Mastercam X5
Smartcam 11.1
SSG, U.S. Army
Taji, Iraq OIF II

RE: Perpendicularity

(OP)
The surface is dimensioned from a parallel surface ...would the perp callout control the dimension and if so should it be basic?

RE: Perpendicularity

There is a difference between Perpendicularity and Parallelism.
Even in case of Parallelism the dimension can be toleranced; the Parallelism will refine the tolerance.
If I don't understand something, please provide a picture.
 

RE: Perpendicularity

First, there is an implied basic dim with perpendicuarity. It is the 90 degree relationship to the datum. That's not the kind you were thinking of, but I feel obligated to mention it.

You ask about locating the perp tolerance zone. That should never happen with basic dims. The perp tolerance zone can float freely within whatever other constraints there might be.

IOW, the per tolerance never controls the distance from the parallel surface. There should be something else to do that, such as plus/minus. The perp doesn't get added to that; it must operate within it.

John-Paul Belanger
Certified Sr. GD&T Professional
Geometric Learning Systems
http://www.gdtseminars.com

RE: Perpendicularity

The surface can be dimensioned from a parallel surface and still have a perpendicularity to a datum that is normal to both surfaces.  The dimension will control location and to the extent that the envelope principal applies, form.  The perpendicularity must be a refinement of the dimension tolerance, typically half or less.

----------------------------------------

The Help for this program was created in Windows Help format, which depends on a feature that isn't included in this version of Windows.
 

RE: Perpendicularity

metaldork,

The way I like to look at it is that the Perpendicularity zone can always freely translate - its location is never controlled relative to anything.  So the Perpendicularity zone is always completely independent of any linear dimensions (basic or directly toleranced) connected to the considered feature.

dgallup,

In the scenario you described, the Perpendicularity doesn't have to be a refinement of anything.  The directly toleranced dimension doesn't control the squareness, so the Perpendicularity tolerance doesn't need to refine it.

Evan Janeshewski

Axymetrix Quality Engineering Inc.
www.axymetrix.ca

RE: Perpendicularity

I am with J-P on this, the only basic dimension that applies for perpendicularity is basic 90 degrees to the datum and it is usually not shown on a print due to implied basic 90 deg dimension rule.

dgallup,
You said: "The perpendicularity must be a refinement of the dimension tolerance, typically half or less."
I would be careful with that. If perpendicularity callout is applied only to one "side" of a dimension, its value can be whatever one can imagine (assuming there is no general angular tolerance shown on the print that would control a relationship of the other "side" of the dimension relative to the datum).
The other story is when there are two perpendicularity callouts applied to both "sides" of the dimension. In that case, my vote is their values can be as big as the dimension tolerance (and not half of it).   

RE: Perpendicularity

Evan,
Could you please clarify: Do you believe that perpendicularity tolerance zone can "freely translate" OUTSIDE of linear dimension /position tolerance zone?
 

RE: Perpendicularity

CH, picture a U-shaped bracket -- sort of like field-goal posts.  Now take both vertical posts and bend them 20 degrees to the left.  Are they within the size (width) tolerance?  Yes.  Is the post on the left within its perpendicularity tolerance to the ground?  No.

So there is no relationship between the tolerance number for the size across the posts and the tolerance number chosen for the perpendicularity.

Sorry for jumping in, Evan.  Feel free to modify my explanation  :)

John-Paul Belanger
Certified Sr. GD&T Professional
Geometric Learning Systems
http://www.gdtseminars.com

RE: Perpendicularity

Quote (checkerhater):

"Could you please clarify: Do you believe that perpendicularity tolerance zone can "freely translate" OUTSIDE of linear dimension /position tolerance zone?"

CH,

Quick answer, yes.  But of course there's more to it.

If we think purely in terms of the tolerance zone mechanics, which I always try to do, then the Perpendicularity zone is allowed to translate outside of the linear dimension / Position tolerance zone.

It is true that the feature might not conform to the Position tolerance if the Perpendicularity zone had to translate partly outside of the Position zone.  It definitely wouldn't conform if the Perpendicularity zone had to translate completely outside of the Position zone.  But that is a different consideration.

The way I like to look at it is that the Position tolerance and the Perpendicularity tolerance are independent requirements, that can be evaluated independently of each other.  This is the only way that I am able to make sense of it all.  Some GD&T books state or imply that the orientation zone must float within the location zone, but to me this is an oversimplification.  It is possible for part of the orientation zone to extend outside of the location zone, and still have the feature conform to both tolerances.

Evan Janeshewski

Axymetrix Quality Engineering Inc.
www.axymetrix.ca

RE: Perpendicularity

Thank you Evan,
Appreciate straight "Yes" - rare thing nowadays.
Unfortunately I am not convinced. Imagine the hypothetical situation:
You are using some sort of measuring machine. You find out that the axis of certain hole (or boss) is laying WITHIN perpendicularity tolerance, but OUTSIDE of position tolerance.
Would you suggest the part to be accepted or rejected? (Another straight Yes or No will be appreciated)
I personally believe that your position and perpendicularity zones should at least partially overlap. (It will create some interesting conditions that probably were never fully documented, but nevertheless).

Joan-Paul,
Your argument appears to be far stretched and borderline cheating (Sorry)
You said it yourself: "Is the post on the left within its perpendicularity tolerance to the ground? No." So what you describe is "bad" part that should never be accepted in the first place. The purpose of GD&T is to describe the parts we will accept, right?
Another stretch: "tolerance number for the size across the posts". Most people would consider "fork" being two features of size, not one. Is space between two holes feature of size?
So your argument basically is: "If we can make bad part from ambiguous drawing, then there is no relationship between the tolerances". Sorry, but I am not buying it.
 

RE: Perpendicularity

CH -- I don't quite understand your concern.  First, yes, the space bewteen the posts is a feature of size; so is the outside of the posts.  But my example was really meant to discuss surface perpendicularity (I just used the posts as a visual example.  Sorry for the confusion.)

"Is the post on the left within its perpendicularity tolerance to the ground? No."

The reason I wrote that is that many people think that a size dimension also controls orientation -- that's not true!  So I'll rephrase it differently:  Two vertical surfaces can lean in the same direction and still be within perpendicularity tolerance. They are also within the size tolerance.  But there is no connection between the tolerance values for the perpendicularity and the size.

John-Paul Belanger
Certified Sr. GD&T Professional
Geometric Learning Systems
http://www.gdtseminars.com

RE: Perpendicularity



Thank you JP, we are getting somewhere.
Let say, we have FOS with perpendicularity requirement added. The size tolerance is .001, the perp. .100. But they both affect VIRTUAL CONDITION, right? Now, what if we add position requirement to our virtual condition?
You see, I never said perpendicularity is a refinement of size. But I still insist on the following conditions: Perp. tolerance zone being outside of position tolerance zone makes no sense. Perp. tolerance zone being larger than position tolerance zone makes no sense. It may be legal though.
When you have 2 FCFs applied to same feature, their requirements have to be met together (I didn't say "simultaneously"). If perp. tolerance is .100 and position is .010 position requirement controls perp. indirectly and makes perp. requirement useless.
So, when used together with position requirement, perpendicularity only makes sense when perp. tolerance zone is smaller than position tolerance zone, and both tolerance zones overlap at least partially. To me it means "refinement".
 
 

RE: Perpendicularity

I believe it is acceptable to use a space between to pins say as a feature of size, right?
Frank

RE: Perpendicularity

pmark,
You are a genius - I think this is exactly what OP was asking: do you use basic dimension to control perpendicularity of parallel elements.
I would use basic dimension with profile to control the whole contraption. That will also make it less ambiguous.
 

RE: Perpendicularity

CH,
You are asking what do I think about my sketch or about your modified version of my sketch? Or is your sketch showing only one of situations that may occur for my tolerancing scheme?  

RE: Perpendicularity

Sure ideally all should be controlled, that is right out of the book. Frank
 

RE: Perpendicularity

Actually both (or all three?).
I feel like using direct dimensioning may create ambiguity here. There is a reason ISO calls it "two-point dimension".
(See my post about using Profile)

RE: Perpendicularity

Frank,
Off-topic, but since you brought it up; have you ever seen round hole dimensioned to the edge rather than center?
I did, I was even forced to do it myself, I still don't like it. smile
 

RE: Perpendicularity

CH,
Are you kidding? I cut my teeth on that. It is what I was looking to GD&T to help me escape from. ;)
This thread has confused me a bit with the whole perpendicularity zone centered, center implies a location. Seems like at least part of the perpendicularity zone must be in the location tolerance zone or you would just move it.
Frank

RE: Perpendicularity

Quote (CheckerHater):

So, when used together with position requirement, perpendicularity only makes sense when perp. tolerance zone is smaller than position tolerance zone, and both tolerance zones overlap at least partially. To me it means "refinement".

When perpendicularity used together with position, isn't the effect same as a composite positional tolerancing with primary datum repeated in lower segment when the toleranced FOS perpendicular to primary datum?

RE: Perpendicularity

To bxbzq:
It is very close to say the least.
There was also discussion(s) here about using position to single datum to control perpendicularity.
 

RE: Perpendicularity

Or as say a refinement of a profile control, like an over all profile.
J-P and Evan are talking about a much more basic level if the are no other controls applied. In theroy those parts are OK, I would not want to walk in court and claim I made a part like CH drew to meet pmarcs ealier print to a judge, he would throw me out of the room until he was instructed in the "letter" of the law.
Frank

RE: Perpendicularity

Somehow this thread got confused -- I might have had something to do with that of course  :)

CH, pmarc's sketch was right on.  The size dimension has absolutely nothing to do with the angle of the sides to the bottom.  That's all we were saying.  

I think Evan was saying earlier that SOME of the perpendicularity zone can go outside of the location zone.  However, since the part must meet both tolerances, that just leaves less of a zone for the actual surface to fall into.

Bxbzq: It's not quite the same as composite position because the lower segment of composite position tolerance also controls the location of the features to each other. A perpendicularity symbol can't do that.

John-Paul Belanger
Certified Sr. GD&T Professional
Geometric Learning Systems
http://www.gdtseminars.com

RE: Perpendicularity

CH,

In your hypothetical situation, I would suggest that the part should be rejected.  In the report, I would list the Perpendicularity tolerance as conforming and the Position tolerance as nonconforming.

I would agree that in order for the part to conform to both tolerances, the Perpendicularity zone and the Position zone must have at least partial overlap.  I just see this as different than the zones' constraints relative to the datum reference frame.  The Perpendicularity zone is allowed to freely translate relative to the DRF, the Position zone is not.

This approach allows me to think about each tolerance individually, without regard to the conformance/nonconformance of other tolerances on the same feature.  Once everything has been evaluated, the part is accepted if every tolerance conformed and rejected if at least one tolerance didn't conform.

Evan Janeshewski

Axymetrix Quality Engineering Inc.
www.axymetrix.ca

RE: Perpendicularity

Is the fig from CH drawn as produced or are you intending to start with a trapazoid. If I choose that it is a produced version of pmarc's original figure; in that case the perpendicularity between the opposing surfaces must be within the limits of size.  Rule #1 applies so at MMB the opposing surface must be perfect form. The most out of parallel would be .2.

There are 3 features of size involved in the figure; width, length, and thickness.

There could be a possible need to "refine" orientation depending on what the default angularity tolerance is. It is difficult to accurately determine or discuss orientation controls and their impact without the rest of the influencing dimensions and controls.

My main point is that the perpendicularity is impacted by size limits and the rule #1 to a degree. I am not sure where the idea that perpendicularity floats outside of the size limits comes from. I would agree that a MMB there would need to be a "refinement" of orientation because there should be at least a default general angularity tolerance. Without an additional perp countrol the orientation tolerance is dictated by the angularity tolerance. So the perpendicularity is controlled by size limits, rule #1, and the angularity tolerance which has not been discussed so far.

Rule #1...

When specifying a form tolerance,
consideration must be given to the control of form
already established through other tolerances such as
size (Rule #1), orientation, runout, and profile controls.

It would be a stretch to say there is positioning in this simple figure....

"Position is the location of one or more features of size
relative to one another or to one or more datums. A positional
tolerance defines either of the following:
(a) a zone within which the center, axis, or center
plane of a feature of size is permitted to vary from a true
(theoretically exact) position
(b) (where specified on an MMC or LMC basis) a
boundary, defined as the virtual condition, located at the
true (theoretically exact) position, that may not be violated
by the surface or surfaces of the considered feature
of size.
Basic dimensions establish the true position from
specified datums and between interrelated features. A
positional tolerance is indicated by the position symbol,
a tolerance value, applicable material condition modifiers,
and appropriate datum references placed in a feature
control frame."

 

RE: Perpendicularity

dtmbtz,
I think the AME adds a new element of uncertainty to FOS features now.
Frank

RE: Perpendicularity

I am referring to my sketch again.
The values of perpendicularity callouts specified on the print can be whatever - even infinity - because size dimension 9.9-10.1 has no influence on orientation of block's width relative to the bottom of the part.

However the measured (actual) values of those perpendicularity errors cannot differ more than 0.2, otherwise I see no chance that limits of size for the width (two-point measurements and rule #1) will be met together with both perp requirements.

Side note: I agree that using basic width plus profile callouts on both sides would make it easier to read, though I do not think my dimensioning scheme creates ambiguity - it is simply less common.   

RE: Perpendicularity

pmarc,
"ambiguity" is not a swear word - it just means different interpretations are possible smile

RE: Perpendicularity

pmarc,
Your method is still much more common around where I work.sad
Frank

RE: Perpendicularity

6.3.3 Perpendicularity
Perpendicularity is the condition of a surface, feature's
center plane, or feature's axis at a right angle to a
datum plane or datum axis. See Fig. 3-1.

To say that a perpendicularity control can be a tolerance zone that can be infinite is to lose sight of the meaning of perpendicular. When the tolerance zone approaches the point that the right angle becomes something other than 90 degrees then it is no longer perpendicular. At best a tolerance zone could only result in an angle that approaches 90 degrees. At some point it would then be considered parallel to the referenced datum and it certainly wouldn't be perpendicular.

Much earlier than  when a tolerance zone allows the surface to approach 90 degrees,  the part's design intent and mating condition would need to be evaluated as to whether or not the perpendicularity callout is proper or whether and a angularity control would be sufficient. Does the part really need to be a rectangle?

2. The dimension shown across the width of a simple rectangle is to the opposing surfaces not two corner points. Basic drafting 101.

3. The introduction of Rule #1 into the thread was to refocus on the design intent. To focus  on one dimension and one control without looking at the whole picture is not productive. Appling the GD&T fundamentals of  function and relationship is an exercise in considering the importance and  relationship of all features and their dimension's and geometric controls that capture the design intent.  

Will the part function and be cost effective?
 

RE: Perpendicularity

Sorry,
I never had a chance to take Drafting 101; so could someone direct me to the place in standard where distinction between point-to point, edge-to-edge and plane-to-plane dimensions is made.
(Genuinely curious, only tiny bit of trolling smile)
 

RE: Perpendicularity

CH, I think it boils down to "actual local size" vs. "actual mating envelope."   So check out these paragraphs: 1.3.25,  1.3.54,  along with all of 2.7.

John-Paul Belanger
Certified Sr. GD&T Professional
Geometric Learning Systems
http://www.gdtseminars.com

RE: Perpendicularity

Let's start with the fundamental rules...

(i) A 90" angle applies where center lines and
lines depicting features are shown on a drawing at
right angles and no angle is specified. See para.
2.1.1.2.

Notice the word lines and not points?
Can you have a point at a right angle?

(j) A 90" basic angle applies where center lines
of features in a pattern or surfaces shown at right
angles on the drawing are located or defined by basic
dimensions and no angle is specified.

Notice the word "surfaces" and not "points"...

or how about ASME Y14.2

2.3 Visible lines
Visible lines consist of solid lines and shall be used for representing edges or contours.

Can you show me the mathematical standard were 2+2=4?

or how about the mathematical standard for definition for a square, rectangle, cylinder, diameter, radius,  right triangle etc;  geometric relationships in general...

How does one create a mechanical engineering drawing without having basic drafting 101.... you realize that "101"  is a  
colloquialism for fundamental drafting....

Actually did have an engineering college student in a CAD class that asked me what a radius was.

Problem is that many folks never had drafting 101...
and don't forget logic and common sense... which aint too common anymore....

What would be the logic of that dimension being to the "corner points"? If one really needed to identify the dimension "points"  they could be identified as points. This would be the exception not the norm.

Maybe this is a parallelogram or trapezoid in disguise and you just fooled me...
wink
 

RE: Perpendicularity

Thank you JP,
This is not exactly what I asked, but good start.
Does any inside/outside dimension automatically define FOS, or is there something else to it?
And if it is not in/out, then how to measure "non-envelope" dimension?

RE: Perpendicularity

dtmbiz,
You said it yourself, "Visible lines consist of solid lines and shall be used for representing edges or contours". On the drawing we dimension between lines, not surfaces.
Also, I have nothing against angles. Never mentioned angle between points.
And guess what, there are definitions for square and the rest of the gang. Taking Math 101 before drafting actually helps a lot.
And when it comes to college students; worked with one who didn't know how many millimeters are in one inch.
This is why I am leaning towards exact definitions rather than common sense. If common sense existed, nobody would ask for it. smile
 

RE: Perpendicularity


Not sure how you are supporting your "point" assertion... just dont get your point ponder

What is "is" ? shadessad

Good luck CH... thumbsup2

RE: Perpendicularity

CH,

Since I do agree with much that you write, I still have hope for you....

1.3.22 Dimension
dimension: a numerical value(s) or mathematical
expression in appropriate units of measure used to
define the form, size, orientation or location, of a part
or feature.


1.3.27 Feature
feature: a physical portion of a part such as a surface,
pin, hole, or slot or its representation on drawings, models,
or digital data files.

No points though.... I will concede that points do get dimensions at times...

however not in normal dimensioning of a rectangular block...


 

RE: Perpendicularity

A feature of size must have directly opposed points. So yes, pretty much any inside/outside idea can be a FOS, if the surfaces are directly opposed.

But also, for something to be a FOS, it must associated with a directly toleranced dimension.  So if we use Fig. 4-9 in the standard as a random example, the slot in the left side of that part is not a FOS.

John-Paul Belanger
Certified Sr. GD&T Professional
Geometric Learning Systems
http://www.gdtseminars.com

RE: Perpendicularity

Are we still talking about Perpendicularity?

RE: Perpendicularity

We are talking about your sketch.

RE: Perpendicularity

pmarc,

I agree with all of your statements about the Perpendicularity callouts.

CH,

Ambiguity may not be a swear word, but in Quality I would say that it's at least a dirty word.  If different interpretations of a spec are possible, then different assessments of conformance to the spec are also possible.  We all know how nasty this can get.

dtmbiz,

If the as-designed relationship is 90 degrees, then Perpendicularity is the appropriate orientation tolerance to specify.  The Perpendicularity tolerance zone is always at exactly 90 degrees to the datum, that never changes.  

When pmarc said that the Perpendicularity tolerance could be infinite, this was just to emphasize that the Perpendicularity tolerance is completely independent of the size tolerance in this case.  I don't think he's suggesting that an arbitrarily large Perpendicularity tolerance should be specified.  That said, there are cases in which the as-produced tilt angle of the feature can get very large, perhaps even close to parallel, and still conform to the Perpendicularity zone.  This can occur when Perpendicularity is specified on features in very thin material.

Evan Janeshewski

Axymetrix Quality Engineering Inc.
www.axymetrix.ca

RE: Perpendicularity

CH,

That drawing is a good demonstration of the disadvantages of dimensional tolerancing compared to geometric tolerancing.  Dimensions are easy to define on perfect CAD geometry, but are ambiguous on imperfect real geometry.

What exactly is the "F" measurement?  Good question.  It's the distance between two surfaces that are not necessarily flat and not necessarily parallel.  The F dimension could be defined in more than one way on an as-produced part.

Is the groove a "feature of size" ?  If that drawing is all we have to work with I believe the answer according to Y14.5 would be no, because the F dimension is not directly toleranced.

Evan Janeshewski

Axymetrix Quality Engineering Inc.
www.axymetrix.ca

RE: Perpendicularity

Axym,

I think you are missing it. Depending on the size of the tolerance zone, the angle from the bottom of the part to the top of the part across the tolerance zone will result in an angle. Once the tolerance zone is large enough to allow an angle greater than 1 degree lets say, and then maybe up to 5 degees for example; that surface would no longer be a right angle to datum A.
Of course the tolerance zone is perpendicular to Datum zone within the limits of datum simulation. However if the tolerance zone is large enough you will not have perpendicularity.
Forget infinity, lets look at a .25" tolerance zone yielding over 7 degrees of angle. That result is not perpendicular to me.
Basic geometry.

 

RE: Perpendicularity

CH,

Don't see how you compare the original sketch and an o-ring groove.

Your example shows a 90deg angle with a max 5deg.

The sketch pmarc offered uses a perpendicular control which suggests to me that something very close to a right angle is required for design function.

So we can now call perpendicularity achieved on a thin part that has almost a parallel relationship?



 

RE: Perpendicularity

dtmbiz,
First of all let me say that Evan exactly described my intentions of posting the sketch - it was only a theoretical excercise, without going into considerations about functional requirements and all that stuff. And in the light of that, "infinity" is absolutely correct answer, though of course no one will ever specify infinity on a print. But like I said - it was only a theoretical question.

Now, to your last post. No offence, but I am afraid the sketch you posted and all your assertions stem from the simple fact that you are mixing "perpendicularity" with "perpendicularity error". You already cited what "perpendicularity" is. I will just emphasize it is a perfect condition in opposite to "perpendicularity error" which tells how far from this perfect condition the actual surface is. Of course in order to have the as-produced surface within the spec., this error has to be less than or equal to perpendicularity tolerance value specified in perpendicularity FCF. But the point is the actual surface of the part can be at any angle to datum, even very close to 0 degrees and still be inspected for conformity with perpendicularity requirement, and not angularity or parallelism.

RE: Perpendicularity

dtmbiz,

I was thinking along the same lines as pmarc and was about to write a response, but he beat me to it.

Y14.5 defines several characteristics in terms of the perfect condition, which many people (including me) find a bit counter-intuitive.  Perpendicularity is the condition in which the surface or axis is at a right angle to the datum plane or datum axis.  Perpendicularity is only achieved on idealized entities in the drawing or model.  No real feature is ever Perpendicular, by the Y14.5 definition.  In reality, As pmarc mentioned, all real parts have "Perpendicularity error" and so there must be a Perpendicularity tolerance.

But the Perpendicularity tolerance does not directly control the angle.  It will indirectly control the angle at which a perfectly flat as-produced feature could be tilted, but this control will depend on the ratio of the height of the feature to the width of the zone.  The taller the feature, the less the feature could tilt and still conform to a given Perpendicularity tolerance.  For very thin parts and thus very short features, a seemingly small Perpendicularity tolerance may allow a significant (even extreme) angular tilt.  This is especially true for Perpendicularity tolerances on cylindrical holes referenced at MMC.

So yes, the way Y14.5 defines things, the as-produced part surface could be almost parallel to the datum and still conform to the Perpendicularity tolerance.   

Evan Janeshewski

Axymetrix Quality Engineering Inc.
www.axymetrix.ca

RE: Perpendicularity

Thank you guys,
I apologize for confusing posts; as I was trying to communicate with 3 people at once, it was hard to present my point in concise manner.
pmarc's sketch just triggered some old controversies and never fully satisfied doubts.
It is popular opinion among GD&T crowd, that regular dimensions are no good for anything without geometrical control. And yet it looks like dimensions are here to stay.
Different standards have different way to deal with it. ISO adds missing requirements by means of 2768 (ISO has its own drawbacks, this is not the place for another discussion); ASME has Envelope req't (Rule 1) that indirectly controls some geometrical requirements for features of size.
The problem is: not every feature is a feature of size, and we are not always sure it's the FOS we are dealing with.
Is diameter symbol indication of FOS?
Is chamfer FOS?
What if we dimension chamfer with diameter?
When we apply dimensions to the drawing, do they always mean what we think they mean?
I had my first discussion with manufacturing guy about checking O-ring groove back in '85 or '86. The picture in the standard did not become any more clear ever since. BTW, the dimensions are toleranced thru general notes, so if that's the only obstacle to calling groove feature of size, no worries.
So the sidewalls of the groove are square by design (but may be not) and dimension "F" is probably measured between edges, which, in turn, are rounded, so we have to resort to "virtual sharps".
Now, Is pair of imaginary features FOS?
This could be continued but I think I better break the chain of thoughts started by pmarc's sketch and keep quiet for a while.
Sorry for hijacking the thread.
 

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