CANT , and it's effect ?

I don't see any other parts of the problem which we can hash out, do you?

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There is that minor omission of the included angle between the LOS and the target vector to muzzle (angle S1-S2-T1). But as I said earlier, it gets lost in the noise.... tan(0.1 degree) vs sin(0.1 degree)

So nothing more to hash out on the gun cant issue.

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I already fixed the reference to tangent in the calculation. All the formulae in that spread sheet are yours. I will add the calculation of scope cant as a separate page so that one can solve those equations independently and then put the combined result into a graph and table on the third page. Might have that done this evening, or not ...

Thanks for all the help and patience in taking the time to work through that with me. Likes I said, I don't actually speak math, I just pretend to on the Internet.

Well done.

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"...I already fixed the reference to tangent in the calculation...."

I know that was fixed. I was referring to something else that we have not fixed and don't need to. It's involves which vector is assumed to be normal to the target plane. The sight vector or the target vector? In reality, both are going to be slightly off normal. The small contribution of that angle at the target is negligible, so ignore it. They are both normal as far as the target is concerned.



sin(ANG0) = tan(ANG0) = zero

ANG0 is typically way less than 1 degree. About 1/10 degree at typical sight in distances.

"...Likes I said, I don't actually speak math, I just pretend to on the Internet...."

You seem fluent enough to understand/propose the problem/solution.

Ok, I understand what you are saying that neither vector is normal to the target plane. Naturally, that is never going to be the case unless we lived in a flat world. Agreed that "error" can be ignored. 

"... So it seems to me that shooters who dial (change their zero range) rather than use hold off would be mostly immune from scope cant..."

Only true, if you have the opportunity to click the windage turret as well. 

"arctan(v-clicks/h-clicks) "

Normally, I do the longer (though less mentally intensive) route of narrowing in on the correct setting in small increments.

I have used that very technique with holdover to correct a canted scope in one go. Shoot and zero at 25 yards. Shoot a 10 yard shot and measure the horizontal and vertical POI via the reticle. If the horizontal is zero, scope rotation is good. If not, I use that arctan equation to calculate the angle, mark it on the scope tube, and rotate it that amount in the rings. - Done.

I always correct a canted scope. But if you want to know how much error an uncorrected scope would cause - the formulas should look very similar to the gun cant formulas except that drop amount is replaced by the compensation amount. I think they will look like this:

horizontal error = compensation x sin(scope_cant_angle)
vertical error = compensation x (1 - cos (scope_cant_angle))

The more elevation compensation required or the greater the scope cant angle, the greater the errors. A high scope requires more holdover or more click compensation at close range. A low scope requires more compensation at long range.

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Good stuff. I will be checking cant angles on all my optics over the near term. If we were doing this math in MOA it would be pretty obvious that these are significant errors. My earlier assertion that wind and other factors generally swamp the error is true but that doesn't mean we should let errors of this magnitude creep into our shooting platforms.

Well, I'll try to get back to the spread sheet tomorrow maybe. I'm going to have to do a little thinking on how to approach the scope cant problem WRT zero range. I don't see any way around that booger if we want an actual firing solution out of the thing.

You cannot use the same trajectory drop for all LOB angles as you go from 0 to 90 shooting system cant. Each new LOB elevation angle will create a new trajectory form.

Yes you can. We are not computing a firing solution here. We are computing the magnitude and direction of an error. The absolute drop never changes therefore the trajectory never changes. It TRANSLATES around the circle. That impacts the firing solution NOT the amount of error you have to input after you resolve your new aiming point. What I am trying to say is that the displacement of point of aim is a separate problem from the effects of cant. That displacement is NOT dependent upon drop, TOF, or any variable other than a dx, dy which can be calculated and proven to be dependent ONLY upon range to target. It is a different problem.

Right?

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No. We are talking about the relationship of the LOS to the trajectory created by the elevation angle and direction of the LOB. These are interrelated and cannot be separated.

If I elevate the LOB to 15 degrees up, or 30 degrees up, or 45 degrees up..the trajectory form changes. The drop at each distance in each of the trajectory forms changes. A trajectory form is not static.

A LOB elevation angle of 0 degrees gives raw drop. This is meaningful if the shot is actually taken with LOB at 0 degrees. For any other LOB elevation angle it is N/A.

The effect of cant cannot be separated from the drop created by the change in LOB angle as you cant the shooting system and change the elevation angle of the LOB.

The LOB is not fixed in place when the gun is canted.

Forget the poopzfeld Paper. It is a bunch of hooey masquerading as "myth busting". The shooting system does not rotate around the LOB and some fixed drop. Cant error is all about the trajectory! The LOS must intersect the trajectory. Period. End of story.

When the shooting system is canted, we don't lock the bore of the gun in a vise and shoot. When the shooting system is canted, it tips. LOS and LOB are locked together. As the system tips and the sight is realigned to point at the target, the LOB changes in elevation angle. Once the system tips, there is no way to put the LOS back on the trajectory by simply clicking vertically. That's impossible. When the shooting system is canted, the LOB points in the direction of the cant. The NEW trajectory points low relative to LOS and in the direction of cant. The trajectory has moved in TWO directions..horizontal and vertical..away from the LOS. Is the trajectory the same as in the original zero canted shooting system? No. It is a completely NEW trajectory. Yes, the change in LOB is minimal. That does not matter. The change creates a NEW trajectory. New drop values. New deviations fro LOS.

https://www.nssf.org/articles/how-rifle-canting-affects-long-range-accuracy/

The guy in the video explains with his hand how this works. If it helps, make this simple model to visualize. You can do it with a piece of cardboard and two straws. Like this..










At close ranges, cant error might be minimal, but it still exists. At medium to long airgun hunting ranges it matters.

It seems everyone is trying to make this more difficult than it is.

You cannot use the same trajectory drop for all LOB angles as you go from 0 to 90 shooting system cant. Each new LOB elevation angle will create a new trajectory form.

Yes you can. We are not computing a firing solution here. We are computing the magnitude and direction of an error. The absolute drop never changes therefore the trajectory never changes. It TRANSLATES around the circle. That impacts the firing solution NOT the amount of error you have to input after you resolve your new aiming point. What I am trying to say is that the displacement of point of aim is a separate problem from the effects of cant. That displacement is NOT dependent upon drop, TOF, or any variable other than a dx, dy which can be calculated and proven to be dependent ONLY upon range to target. It is a different problem.

Right?

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No. We are talking about the relationship of the LOS to the trajectory created by the elevation angle and direction of the LOB. These are interrelated and cannot be separated.

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No we are not talking about that relationship. Until we get past this point there is nothing more we can do.

If I elevate the LOB to 15 degrees up, or 30 degrees up, or 45 degrees up..the trajectory form changes. The drop at each distance in each of the trajectory forms changes. A trajectory form is not static.

Sure, but the DROP does not change. This is drop from LINE OF DEPARTURE. Line of Bore is the same thing. The projectile NEVER rises above LOB, not ever. At T=0, when the projectile exits the bore it instantly drops below LOB and that drop is a simple Newtonian equation. The vertical acceleration component of the complex vector defining the projectile's flight DOES change, but it still starts to drop instantly and the magnitude of the drop is always the same between T=0 and any later Time of Flight no matter what the angle of the bore is relative to the plane of the earth. The equation depends purely upon that drop. Scotchmo's math is right. He does GROK the problem (in the traditional sense of that word).

There really are two different problems here if we lump reticle cant with sight cant, which I think should be done because the solution to both is the procedure just described above. Shoot the group, move in closer and shoot another, calculate the scope cant and remove it. When you do that you have solved both scope cant and reticle cant in one process, so that part of the problem is only one problem. The other part of the problem is drop below LOB (not trajectory) relative, not zero dependent and not sight height dependent. That part is a simple physics problem which is independent of Point of Aim and therefore independent of LOS.

Go back over the drawings and last several posts, it will become clearer if you look over the spread sheet discussion and maybe DL it and play with it a bit? You have to be willing to be wrong to see it. I don't mean that as an insult either. It just is...

Sure, but the DROP does not change. This is drop from LINE OF DEPARTURE. Line of Bore is the same thing. The projectile NEVER rises above LOB, not ever. At T=0, when the projectile exits the bore it instantly drops below LOB and that drop is a simple Newtonian equation.

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I did not say the trajectory rises above LOB. Duh! Watch the video link and try to understand what the guy is saying.

No we are not talking about that relationship. Until we get past this point there is nothing more we can do.

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Yes we absolutely are. That is the only thing that matters. The relationship to where LOS points relative the the actual trajectory form.

There really are two different problems here if we lump reticle cant with sight cant, which I think should be done because the solution to both is the procedure just described above.

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No there isn't, just one. We aren't talking about sight cant or reticle cant. We're talking shooting system cant..gun and scope together with the reticle properly aligned..canting together.

The cant error has no relation to the raw drop except when the shooting system is canted 90 degrees. The LOS does not point to a trajectory form made of raw drop values. It is related to the shape the trajectory takes after leaving the barrel and the relation to where the LOS is pointing to that specific trajectory form. The trajectory form depends on LOS distance above the LOB and the elevation angle of LOB at launch. Nothing more. And yes, it is that simple.

No there isn't, just one. We aren't talking about sight cant or reticle cant. We're talking shooting system cant..gun and scope together with the reticle properly aligned..canting together.

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Ok, yes, that defines the parameters of ONE type of cant. "System Cant" or "Gun Cant". That's one problem directly dependent upon drop from line of departure.

If it does not depend upon drop then feel free to SHOW YOUR MATH. If I don't understand it, I am sure you will explain it to me like Scotchmo has. If you have another way to calculate it then show me the calculation. Once you have done that you can explain to me what you call the situation shown below.. Surely it is some form of "cant".

... That is the only thing that matters. The relationship to where LOS points relative the the actual trajectory form. ....

... We're talking shooting system cant..gun and scope together with the reticle properly aligned..canting together.

The cant error has no relation to the raw drop except when the shooting system is canted 90 degrees. The LOS does not point to a trajectory form made of raw drop values. It is related to the shape the trajectory takes after leaving the barrel and the relation to where the LOS is pointing to that specific trajectory form. The trajectory form depends on LOS distance above the LOB and the elevation angle of LOB at launch. Nothing more. And yes, it is that simple.

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... That is the only thing that matters. The relationship to where LOS points relative the the actual trajectory form. ....

For a properly sighted gun, LOS is the only vector that does NOT matter. It does not enter into the equation.

"...The cant error has no relation to the raw drop except when the shooting system is canted 90 degrees..."

Not true. Gun cant ("system cant") error is directly related to drop from the bore-line. At any specific cant angle, a 500fps gun will have approx 4x the cant error as a 1000fps gun. The 500fps gun has approx 2x the time of flight and therefore 4x the drop at any given distance.

"...The trajectory form depends on LOS distance above the LOB and the elevation angle of LOB at launch...."

Not true. LOS has no affect on the trajectory form. For any given gun, the trajectory form from muzzle to target is set by the projectile path, not by the scope. High scope, low scope, or no scope - the trajectory to the target remains the same.



When you introduce system cant, the bore-line vector (LOB) is canted around the target vector. For various cant angles, the LOB traces out a circular path.at the target. The POI follows along directly below that path by an amount equal to drop.

 

you can explain to me what you call the situation shown below.. Surely it is some form of "cant".

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Uh...someone that doesn't know how to set up their shooting system.

If it does not depend upon drop then feel free to SHOW YOUR MATH.

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It doesn't require math to understand. Just the basic understanding that if you don't point the LOB at the target you're going to miss. Lowering the LOB elevation angle from what the shooting system is sighted in for will result in a low miss.

...

It doesn't require math to understand. Just the basic understanding that if you don't point the LOB at the target you're going to miss. ...

...

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You mean above the target, correct?

"...if you don't point the LOB at above the target you're going to miss...."

If you point it "at" the target, you will hit low by an amount equal to drop.

In your picture of the canted misses, the errors are because you were not sighted in correctly for the 5yd target. You are sighted in 1.225" low. You are canting a 1.225" aiming error. If you were to use the correct holdover point on the reticle (mil-dot or other stadia) or adjust your elevation turret for the 5yd zero, the cant errors would not be noticeable at 5yds.

When canting an aim error, the magnitude of the error can be calculated. The formulas are similar to cant equations for the horizontal error, but the vertical error is a little different as it actually becomes less until you reach 90 degrees where it then matches the vertical elevation of your aim point:

horizontal error = aim_error x sin(cant_angle) = 1.225" x sin(cant_angle)

vertical error = aim_error x cos(cant_angle) = 1.225" x cos(cant_angle)

You mean above the target, correct?

"...if you don't point the LOB at above the target you're going to miss...."

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If you don't point LOB so that it intersects the target.. and yes, that assumes above, too

You mean above the target, correct?

"...if you don't point the LOB at above the target you're going to miss...."

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If you don't point LOB so that it intersects the target.. and yes, that assumes above, too

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For a properly sighted gun, the bore-line vector (LOB) will not intersect the target. The LOS and the target vector (trajectory endpoints) intersect the target.

"Angle between Scope1 and LOB is GREATER than the Angle between Scope2 and LOB"

That is true, but those angles are irrelevant for a properly sighted gun. When properly sighted they all intersect at the target bullseye. And that is the point about which the gun is canted.

.

Never mind, never mind...

Never mind, don't want any part of this.

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Uh huh, the drawings you are posting are Copyright "Pinwheel Software". So I guess the first question is do you own them? They aren't very good, are they? Can you link the document you are pulling them from because a visit to the site simply shows me they do software for archery "ballistics". I'd like to see what other misinformation the company is putting out.

It seems to me the surest way to simplify this problem is to introduce another two or three optical systems before we start to simply DO THE MATH? Because there is NOTHING like half a dozen extraneous variables to simplify the problem, right?

There is a reason the KISS principle is not the KIS principle. "Duh" right back at ya.

you can explain to me what you call the situation shown below.. Surely it is some form of "cant".

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Uh...someone that doesn't know how to set up their shooting system.

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https://badlandstactical.org/rifle-scope-reticle-cant-the-disease/

Reticle cant... Scope cant... sight cant... Cute though.

lbc_PSI,

This might help explain:



At your 5yds shooting distance, drop is almost insignificant, so no significant gun cant errors. We are left with your 1.225" aiming error which is being rotated about the LOS. Your error never got worse as you increased the cant angle. It always remained at 1.225" from the bullseye. You just changed the direction of the error. An actual gun cant error gets worse the more you cant the gun. And as you move further out, drop increases, and gun cant errors increase more. And gun cant error shifts the POI in the same horizontal direction of the cant, not opposite like yours did. Aiming errors are combined with any actual gun errors. Sometimes making them greater (vertical), and sometimes making them less (horizontal).

The scope that is off the bullseye by the most MOA will have the the greatest aiming error. That could be a high scope or a low scope. Aiming errors can be corrected regardless of any of gun cant errors. Either zero for the distance you are shooting at, and place the cross-hair on the bullseye, or use the correct holdover point on the reticle, and place that on the bullseye. Best to have a scope with enough vertical turret adjustment, or a reticle with sufficient vertical compensation. That way the POA always coincides with the intended POI.

When you start mixing multiple types of errors (gun_cant_errors, scope_cant_errors, aiming_errors), it gets tougher to sort them out.

If we know the trajectory (distance_to_target, velocity, BC), gun_cant_angle+direction, scope_cant_angle+direction, and aiming error (horizontal & vertical), it is possible to calculate the amount of horizontal and vertical dispersion from the bullseye.

But if all you know is the horizontal and vertical dispersion, you need to go through a process to discover/eliminate the actual errors (gun_cant, scope_cant, or aiming).

By the way, Scotchmo, the spread sheet is still wrong in a major way.

We did not translate the axies. Trig holds up as 90 degrees not 0 degrees and ccw as the direction of positive rotation. I did not take that into consideration when I calculated the tables. It is kind of odd that you did not notice that but anyway. I looked at it a bit more last night and realized that translation needed doing. I must be getting really rusty as that was one of the problems I constantly ran into when I was doing software for the Navy ... It was 20 years ago... =( I also realized that the errors we have calculated in the spread sheet would be translated from the LOB to the POA to get POI. In other words we calculated (wrongly) the errors at the point of intersection of LOB with the target plane. Those translate directly from that point to the POA. No math involved there just a logical understanding of the problem.

Oh yeah and then there is the use of the sine and cosine in the solution of the error triangle. That's going to depend upon which triangle you are solving. A check would be to calculate the solution more than one way. We didn't do that and get agreement. That could because I got the math wrong or because you got the math wrong. One of us did. I thought it was me, until I started working the problem on paper today. Now not so sure. One thing is for sure, this problem requires the undistracted application of trig and logic to solve. Debating the solution will get us no closer to the correct understanding than we are right now.

Anyway that basically destroyed my confidence in the math we have done so far. WRT using it as a firing solution. So I sat down and started working on a white paper because that sort of structure lets me think about the problem without a bunch of extraneous noise. Frankly the arguments here are too distracting from the math. I need to go and work the math.

IF I have the inclination, I'll keep working on the white paper and IF it bears fruit, I'll add it to the downloads here on the site.

Like I said, I don't actually speak math ... but I don't see anyone here that speaks it any better than I do so I'll take a time out and maybe work the problem for myself ...

Here is the take away for anyone reading the thread. Cant is a problem you should take careful measures to eliminate. It is not sufficient to just "eye ball" your optic into place and call it good UNLESS you find that gives you sufficient accuracy to accomplish whatever it is you want to do with the rifle. The BR or FT or long range shooter should not ignore cant when setting up his platform.

I leave you gentlemen to solve this problem among yourselves. I need to spend a lot more time thinking about it before I talk about it.

CARRY ON