SPORT SHOOTING: THE EFFECT OF RIFLE CANT ON THE POINT OF IMPACT
Copyright Jeroen Hogema
email:
August 20, 1999
Translated from the Dutch
document
PCB links made local: January 26, 2000
1. Introduction
2. In what direction does the
point of impact move?
2.1. Large cant angles
2.2. Small
cant errors
3. How
much does the point of impact move?
3.1. The relation between point of
impact movement and drop
3.2. When does the effect become
noticeable?
4. Cant angle in practice
4.1. Shooting with a canted rifle
4.2. Riser blocks
4..3 Try it
5. Discussion and conclusions
References
Appendix A Determining drop
Appendix B Examples
Appendix C Calculating the cant angle effect
The sights of a modern sport rifle typically consist of a diopter and ring sights. The perfect aiming picture is achieved when the target, ring and diopter are all perfectly concentric. However, this ideal picture can be achieved for various cant angles of the weapon. This is illustrated in Figure 1.
Figure 1 Sight picture without (A) and with cant (B).
One of the many elements a spot shooter must pay attention to is that he or she holds the rifle under a constant cant angle. Variation in the cant angle will cause variation in the point of impact, even with perfectly centred sights and with a perfectly still weapon. But in what direction does the point of impact shift for a given cant angle fluctuation? And how far? How much margin is available before the effect becomes visible on the target and in the score? Does it matter under which cant angle the rifle was zeroed? Does the use of riser blocks under the sights increase this effect?
This document answers these questions. The ballistic calculations and mathematics involved are not discussed, but the appendices show a bit more detail. In Chapter 2, the direction of the point of impact displacement is discussed. Chapter 3 continues with the magnitude of the point of impact displacement for a given cant angle variation. Chapter 4 deals with a few practical matters. Finally, Chapter 5 gives the conclusions.
2. In what direction does the point of impact move?
The first question is in what direction the point of impact will move when the weapon is canted. The easiest way to shown this is by regarding extremely large cant angle variations: 90^{o} to the left, 90^{o }to the right, or even 180^{o} cant, i.e. holding the weapon upside down, with the sights under the bore. Once it is clear what happens with the point of impact with these large cant angles, it will also become clear what happens with smaller, more realistic cant angle fluctuations.
For all of the cases that will be discussed below, the following constraints apply:
The initial situation regarded here is a rifle that still must be zeroed. This is done with the sights precisely above the bore axis, i.e. the cant angle is 0^{o}. Figure 2A shows the initial situation. The diopter is set such, that the line through the bore axis is parallel to the visual axis. This makes the line through the bore axis intercept the target a little bit below the bull's eye. So even if the bullet's trajectory were a straight line, the point of impact would be too low. Therefore, the diopter is adjusted, such that the bore axis points exactly at the bull's eye (Figure 2B). However, the bullet trajectory is not a straight line: as soon as the bullet has left the bore, gravity starts to deflect the bullet downwards. Therefore, the point of impact will be too low with these sight setting, straight under the '10'. The distance on the target between the point of impact and the '10' in this situation is named drop.
To make the point of impact end op in the '10', the diopter is adjusted further (Figure 2C). The bore axis line now intercects the target right above the centre, at drop distance above the '10'. This drop will turn out to be of major importance in the cant angle effect.
This finishes the zeroing of the weapon. When held upright, the point of impact will be in the centre of the target. The diopter stays in its current settings. The question is what happens when canting the rifle.
Figure 2 Zeroing the rifle (side view).
A Initial situation: bore axis parallel to the visual axis.
B First correction: bore axis points at centre of target; the point of impact is still too low.
C Rifle correctly zeroed: point of impact is in the centre of the target; bore axis points straight above the '10'.
In Figure 2B and C, the point of impact was a fixed distance below the point where the bore axis line intercects the target. This remains the same when canting the weapon. Therefore, the first step will be to look at the displacement of this intersection point when canting the weapon. After that, the point of impact movement can be found at drop distance below this intersection point.
In Figure 1C, with the rifle upright, the intersection point of bore axis and target was at drop distance right above the '10'. Canting the rifle moves this intersection point. Figure 3 shows the situation with the rifle upside down,: the intersection point is now at drop distance below the '10'.
Figure 3 Rifle upside down (side view).
The system can be seen as a compass. The leg with the pin is the visual axis, with the pin placed in the centre of the target. The leg with the pencil is the bore axis line, with the pencil on the target. Rotating the rifle all the way around will make the compass draw a circle on the target with the '10' as its centre and the drop as its radius (see Figure 4). This circle indicates how the intersection point of bore axis and target moves for different rifle cant angles.
Figure 4 intersection of bore axis and target when canting (dashed circle).
Point A: rifle upright
Point B: rifle upside down
Point C: rifle canted a quarter circle to the left
Point D: rifle canted a quarter circle to the right
This describes the effect of rifle cant on the intersection of bore axis and target for all possible cant angles. Now the effect of cant on the point of impact can be found by looking at drop below this intersection point. In other words: the point of impact is found by moving the dashed circle from Figure 4 down by distance drop.
Figure 5 Effect of rifle cant on point of impact
Point A: rifle upright
Point B: rifle upside down
Point C: rifle canted a quarter circle to the left
Point D: rifle canted a quarter circle to the right
This is shown in Figure 5. When canting the rifle, the point of impact moves along a circular pattern: the point of impact circle. The centre of this circle is straight under the '10' and the top point of the circle is in the '10'. The radius of the circle equals the bullet's drop.
Using the findings of Section 2.1, it easy to see where the point of impact moves to for small cant angle variations.
The shooter will (should) try to maintain a constant cant angle. In spite of this effort, there will be some (no matter how small) variation: cant error. As long as the cant error is small, the point of impact will remain near the top of the point of impact circle of Figure 5 (point A). Therefore, the point of impact will appear to move along a horizontal line, to the left or to the right. Only for large cant errors a part of the circle will become noticeable.
3.How much does the point of impact move?
3.1. The relation between point of impact movement and drop
In the final point of impact movement on the target, the size of the point of impact circle plays an important role. This is illustrated with an example where a rifle that was zeroed under 0^{o} cant is now canted 90^{o} to the left. Shooting in this situation will move the point of impact along the point of impact circle on the target from twelve o'clock to nine o'clock. Figure 6 shows for two point of impact circles with different sizes what would happen on the target with this 90^{o} cant. With the small radius, the point of impact moves away from the '10' by a couple of rings. With the large radius, the effect of the same cant angle is much more severs: here, the point of impact moves all the way outside the target.
Figure 6 Effect of 90^{o} cant angle to the left: the larger the circle radius, the larger the point of impact displacement.
So, the larger the circle radius (i.e. the drop), the larger the point of impact displacement for the same cant angle error.
The drop is determined by the time needed by the bullet to fly from the rifle to the target. The more time, the larger the drop. Therefore:
Therefore, a sport shooter has little influence on the drop. The shooting range is fixed for each ISSF discipline. The only way to reduce drop is to use fast ammunition.
To give a general idea of the magnitude of drop: for air rifle (10 m) the drop is approximately 18 mm. For small bore rifle (50 m) this is considerably more, i.e. around 12 cm (see Appendix A).
In conclusion, merely two factors determine the distance of point of impact displacement due to cant. These are the magnitude of the cant angle error and the magnitude of the drop. The mathematical relationship between these variables is given in Appendix B.
3.2. When does the effect become noticeable?
This does not answer the question how much a sport shooter is bothered by this cant angle effect in practice. Or: how much cant angle error is allowed before the point of impact movement becomes noticeable on the target and in the score? It is possible to calculate how much cant angle error can be allowed before a centre'10' becomes a '9justmissingthe10'. This is done in 2 steps. First, based on the dimensions of the '10' and the bullet, it is determined how far the point of impact may move to just miss the '10'. Then (using a known value for the drop) the associated cant angle error is determined.
A first example: air rifle, range 10 m. The '10' is 0.5 mm in diameter, and the bullet 4.5 mm. Therefore, 2.5 mm point of impact displacement is allowed before the diabolo just misses the '10'. As said Section 3.1, the drop for air rifle is about 18 mm. Using these data, one can calculate that there is 8.0^{o} cant angle margin before the '10' is missed (see Appendix B for the calculations).
A second example: small bore rifle, 50 m. The drop is about 11.9 cm. The '10' ring has a 10.4 mm diameter, and the bullet (0.22 inch) 5.59 mm. Therefore, a 8.0 mm point of impact movement will cause the '10' being missed. Appendix B shows that a cant error of 3.9^{o} will make this happen.
It is also possible to calculate how much rifle cant is allowed before the point of impact movement equals that of one dipoter click. When for air rifle 1 ring on the target equals four diopter clicks, then 3.9^{o} cant angle error equals 1 diopter click (see Appendix B).
Figure 7 shows the aiming picture for various cant angles.
Figure 7 Aiming picture for different cant angles.
4.1. Shooting with a canted rifle
The previous discussions were based on a rifle that was zeroed under 0 degrees cant, and then the effect of cant on point of impact were regarded. However, there are reasons for canting the weapon while shooting. Using a nonzero cant angle can allow the shooter to maintain an upright head position, resulting in a more relaxed posture and creating an optimal situation for the vestibular system.
The effects of cant angle errors remain the same, as long as the cant angle error is regarded with respect to the cant angle under which the rifle was zeroed.
Take for example a rifle that was zeroed upright (0^{o} cant). When this rifle is used by a (right handed) shooter who cants the rifle 45^{o} towards his face, the point of impact will be left and low. By correcting this on the dipoter, the point of impact can be moved back to the '10'. From this moment on, the same situation exists. A small cant angle error to the left (i.e. error with respect to the now normal 45^{o}) moves the point of impact to the left. A small cant error to the right (again with respect to the now normal 45^{o}) moves the point of impact to the right.
When this rifle is now fired in the upright position again, the point of impact will move right and low, since a cant angle of 0^{o }now corresponds to a cant angle error of +45^{o}.
A shooter can equip his or her with riser blocks. These enlarge the distance between dipoter and bore, and ring sights and bore. These accesoires enlarge the distance between the visual axis and the bore axis. The end of the bore describes a larger circle with respect to the visual axis when canting the rifle. Therefore, at first sight it may seem that riser blocks will enlarge the effect of a given cant angle. However, this is not the case.
Figure 8 The use of riser blocks.
A: normal sights
B: with riser blocks
The riser blocks do not affect the time needed by the bullet to fly from the rifle to the target. Therefore, the drop remains the same. After zeroing the rifle with the riser blocks, the intersection point of bore axis line and target must again be at drop distance right above the '10' (see Figure 8). This is the only way the '10' can be hit, irrespective of the line of sight height. Returning to the compass comparison: both legs of the compass are shorter when using riser blocks (Figure 8B) compared to the situation without riser blocks (Figure 8A), but the circle drawn remains the same.
So once again: a bit of extra cant to the left will move the point of impact to the left and a bit of cant extra to the right will move the point of impact to the right. The effect of a given cant angle error on the point of impact movement is not affected by the riser blocks.
Ultimately, there is a simple way of experiencing the effect of cant error: just try it. For example:
Comparing the resulting shot groups will show where the point of impact moves to (direction as well as distance).
One of the many issues a rifle shooter must take into consideration is to cant the weapon under the same angle during each subsequent shot. Variation in the cant angle yields variation in the point of impact, even when holding the rifle perfectly still and having the sights perfectly aligned with the target.
Large cant errors will result in a circular pattern of point of impact displacement. Smaller cant errors, as may occur unintendedly, will mainly result in a horizontal displacement of the point of impact, in the same direction as the cant error. Cant error can become manifest in once score: what should have been a clear '10' becomes a '9' merely due to rifle cant. The margin available before this happens depends on the shooting discipline. For small bore rifle at 50m, there is a margin of merely 4^{o}.
In Section 4.3, it was stated that riser blocks do not increase the effect of cant error on the point of impact displacement. This is in line with the results from the computer program PCB (see Appendix C). Nevertheless, training manuals do warn for the risks of riser blocks in relation to cant (Reinkemeier, 1994; Reinkemeier & Bühlmann, 1994; Bühlmann, Reinkemeier & Eckhardt 1998). Unfortunately, these authors did not indicate what the occurring problem is. Possibly they erroneously think that the line of sight height is of influence on the point of impact movement. Another possibility is that the effect of a given cant error is the unaffected, but that the magnitude of the cant error itself is increased by using riser blocks, e.g. because the cant angle is more difficult to perceive.
Bühlmann, G., Reinkemeier, H., & Eckhardt, M. (1997). Wege des Gewehrs. Band 1: Die Technik. Münster: Eigenverlag.
Eichstädt, U. (1995). Visier, 10, 4248.
Reinkemeier, H. (1994). Vom Training des Schützen  Gesamtausgabe. Westfälischen Schützenbund.
Reinkemeier, H., & Bühlmann, G. (1994). Trainingsplan Luftgewehr. Münster: Eigenverlag.
Appendix A Determining drop
For the qualitative examples from this document, typical values for the drop were determined for 2 disciplines:
This was done using the computer programme PCB (version 1.8). Dit programme was written by Odd Håvard Skevik, who distributes it via the World Wide Web as freeware.
The major inputs needed for PCB (or a similar programme) to calculate the drop are the muzzle velocity of the bullet (V0) and the ballistic coefficient (BC).
Air rifle 10 m
Calculations are based on my own Feinwerkbau 601 in combination with RWS R10 diabolo's.
Using these input parameters, PCB can calculate the drop at 10 m:
PCB 1.8 Trajectory chart
Bullet name.....................
RWS R10 diabolo
Bullet weight...................
0.53 gram
Bullet diameter.................
4.50 mm
Muzzle velocity.................
170 m/s
Ballistic coefficient...........
0.011
Zero............................
0.0 cm at 10.0 meters
Crosswind.......................
0.00 m/s from 90 degs
Line of sight above bore axis...
2.0 cm
Temperature.....................
15.0 grad C
Altitude........................
0 meters
Sights zeroed at ...............
0 degs
Firing angle....................
0 degs
Cant angle......................
0 degs
Range 
Velocity 
Energy 
Flight 
Drop 
Max 
Path 
Drift 
Click 
Click 
time 
height 
up 
side 

[m] 
[m/s] 
[J] 
[s] 
[cm] 
[cm] 
[cm] 
[cm] 

0 
170 
8 
0 
0 
2 
2 
0 
0 
0 
5 
160 
7 
0.03 
0.4 
0.7 
0.5 
0 
10.5 
0 
10 
150 
6 
0.063 
1.8 
0.3 
0 
0 
0 
0 
15 
141 
5 
0.097 
4.3 
0.4 
0.6 
0 
3.8 
0 
The drop of the diabolo at 10 m is 1.8 cm.
Small bore 50 m
Eley gives the following specifications of .22 rifle ammunution on the WWW (http://www.eley.co.uk/rflfst.htm):
Rifle cartridges:
 Tenex
 Match Extra
 Club Xtra
 Target rifle
 Standard
All Eley competition rifle cartridges are loaded to the same nominal ballistic level allowing shooters to move up the quality scale for important competitions or as their performance improves with minimum sight adjustment.
Bullet weight  (grams)  2.59 
(grains)  40 
muzzle 
50m (50yrds) 
100m (100yrds) 

Velocity(m/sec) (ft/sec) 
331 (1085) 
305 (1006) 
284 (941) 
Energy (Kg. m) (ft. lb) 
14.5 (105) 
12.3 (90) 
10.6 (79) 
First, PCB can be used to estimate the BC from these data:
The average of these values (0.153) was used in PCB to determine the gebruikt drop at 50 m given the V0 of 331 m/s:
PCB 1.8 Trajectory chart
Bullet name.....................
Eley LR
Bullet weight...................
2.59 gram
Bullet diameter.................
5.59 mm
Muzzle velocity.................
331 m/s
Ballistic coefficient...........
0.153
Zero............................
0.0 cm at 50.0 meters
Crosswind.......................
0.00 m/s from 90 degs
Line of sight above bore axis...
3.0 cm
Temperature.....................
15.0 grad C
Altitude........................
0 meters
Sights zeroed at ...............
0 degs
Firing angle....................
0 degs
Cant angle......................
0 degs
Range 
Velocity 
Energy 
Flight 
Drop 
Max 
Path 
Drift 
Click 
Click 
time 
height 
up 
side 

[m] 
[m/s] 
[J] 
[s] 
[cm] 
[cm] 
[cm] 
[cm] 

0 
331 
142 
0 
0 
3 
3 
0 
0 
0 
50 
305 
120 
0.158 
11.9 
1.9 
0 
0 
0 
0 
100 
285 
105 
0.328 
50.2 
12.1 
23.4 
0 
23.4 
0 
These output data (velocity as a function of range) are in line with the Eley data. The drop of the bullet at 50 m is 11.9 cm.
Appendix B Examples
When the weapon is canted with angle ß, the point of impact moves along a circle with radius r (equal to the drop). This is shown in Figure B1: by canting, the point of impact moves from the centre of the ‘10’ (point P1) to point P2.
Figure B1 Point of impact movement from P1 to P2 due to cant angle ß.
The distance D between P1 and P2 can be calculated as follows. The coordinates from P1 in the indicated xyaxes are (0,r) and (r*sin ß, r*cos ß), respectively. Using Pythagoras leads to:
D = [(r*sin ß  0)^{2} + ( r*cos ß  r)^{2}]^{1/2}
= r * [sin^{2} ß + cos^{2} ß + 1  2* cos ß]^{1/2}
= r * [ 2  2* cos ß]^{1/2}
= r * [2*(1  cos ß)]^{1/2}^{...............................................................}(1)
From this equation ß can be solved:
ß = arccos [1  0.5 * (D/r)^{2} ] ....................... .............(2)
Example I
How much cant is allowed for air rifle shooting before a ‘10’ becomes a ‘9’?
For air rifle, diameter of the bullet is 4.5 mm and of the ‘10’ 0.5 mm. The distance D over whith the point of impact must move to get to the edge of the ‘10’ is D = (4.5+0.5)/2 = 2.5 mm. With a drop of r=18 mm (see Appendix A), Equation (2) gives: ß = 7.964^{o}.
Example II
How much cant is allowed for small bore rifle at 50 m before a ‘10’ becomes a ‘9’? For this disciplene, the diametre of the bullet (0.22 inch) is 5.59 mm and of the ‘10’ it is 10.4 mm. The distance D needed to move from the centre of the '10' to the edge of the '10' is D = (5.59+10.4)/2 = 7.994 mm. With a drop of r=119 mm (see Appendix A), Equation (2) gives: ß = 3.85^{o}.
Example III
How much degrees cant angle equals one click on the diopter, in terms of point of impact movement?
For an air rifle with a stander diopter, the point of impact movement over 1 ring on the target (2.5 mm) euqals 4 diopter klicks. One click thus gives 2.5/4 = 0.63 mm point of impact movement.
The same amount of point of impact movement can be obtained (Equation 2, with D = 0.63 mm and again r = 18 mm:) by cant angle ß = 3.9^{o}.
Figure 7 shows the aiming picture for various cant angles.
Appendix C Calculating the cant angle effect
Using the computer programme PCB, it is also possible to calculate the effect of cant angle and LineofSight height on the trajectory, and thus on the point of impact.
Cant angle variation
Cant angle is in input parameter of PCB. The next example shows (following the 50 m small bore example of Appendix A) what happens with 10^{o} cant:
PCB 1.8 Trajectory chart
Bullet name..................... Eley LR (various types)
Bullet weight................... 2.59 gram
Bullet diameter................. 5.59 mm
Muzzle velocity................. 331 m/s
Ballistic coefficient........... 0.153
Zero............................ 0.0 cm at 50.0 meters
Crosswind....................... 0.00 m/s from 90 degs
Line of sight above bore axis... 3.0 cm
Temperature..................... 15.0 grad C
Altitude........................ 0 meters
Sights zeroed at ............... 0 degs
Firing angle.................... 0 degs
Cant angle...................... 10 degs
Range 
Velocity 
Energy 
Flight 
Drop 
Max 
Path 
Drift 
Click 
Click 
time 
height 
up 
side 

[m] 
[m/s] 
[J] 
[s] 
[cm] 
[cm] 
[cm] 
[cm] 

0 
331 
142 
0 
0 
3 
3 
0 
0 
0 
50 
305 
120 
0.158 
11.9 
1.9 
0.2 
2.1 
0.4 
4.1 
100 
285 
105 
0.328 
50.2 
12.1 
23.8 
4.7 
23.8 
4.7 
Due to this 10^{o} cant angle, the point of impact moves 2.1 cm sideways (Drift) and 0.2 cm down (Drop).
Thus, for each cant angle entered, the point of impact can be determined. Figure C1 shows the results for 0, 10, 20 etc. degrees cant angle.
Figure C1 Effect of cant angle on point of impact as calculated by PCB (small bore rifle, 50 m)
The results are in line with the findings of Section 2.1. For 0^{o} cant, the point of impact is in the centre of the target: Path=0 and Drift=0. When canting the weapon, the point of impact moves along a circle with radius equal to the drop on the range under consideration (here 11.9 cm).
Variation of line of sight height
In PCB, the line of sight height can also be varied (Line of sight above bore axis); see the next example.
PCB 1.8 Trajectory chart
Bullet name..................... Eley LR (various types)
Bullet weight................... 2.59 gram
Bullet diameter................. 5.59 mm
Muzzle velocity................. 331 m/s
Ballistic coefficient........... 0.153
Zero............................ 0.0 cm at 50.0 meters
Crosswind....................... 0.00 m/s from 90 degs
Line of sight above bore axis... 6.0 cm
Temperature..................... 15.0 grad C
Altitude........................ 0 meters
Sights zeroed at ............... 0 degs
Firing angle.................... 0 degs
Cant angle...................... 10 degs
Range 
Velocity 
Energy 
Flight 
Drop 
Max 
Path 
Drift 
Click 
Click 
time 
height 
up 
side 

[m] 
[m/s] 
[J] 
[s] 
[cm] 
[cm] 
[cm] 
[cm] 

0 
331 
142 
0 
0 
6 
6 
0 
0 
0 
50 
305 
120 
0.158 
11.9 
0.7 
0.2 
2.1 
0.4 
4.1 
100 
285 
105 
0.328 
50.2 
10.9 
20.8 
5.2 
20.8 
5.2 
Using the came cant angle as in the previous example, (10^{o}) the line of sight was enlarged from 3 cm to 6 cm. Conform the findings of Section 4.2, the point of impact displacement remains the same; here 2.1 cm sideways (Drift) and 0.2 cm down. (Path).