**EFFECT OF CANT ANGLE
VARIATION ON POINT-OF-IMPACT IN AIR-RIFLE SHOOTING**

Copyright Jeroen Hogema

e-mail:

May 1999

**1 INTRODUCTION**

This document describes an experiment that was carried out to study the effects of cant angle variation on the point of impact in air rifle shooting. Only a typical ISSF-style competition shooting situation is regarded:

- it is assumed that the rifle is properly zeroed at the range under consideration
- the target is always at this range.

The question is how the point-of-impact on the target moves when the rifle is canted (i.e. rotated around the visual axis), all other conditions remaining constant. First, ballistic calculations were carried out with the computer program PCB (version 1.8). This program was written by Odd Håvard Skevik, who distributes it via his web site for free.

When first a rifle is zeroed at 0^{o}
cant angle and then cant angle is varied, PCB predicts that the
point of impact will move along a circular pattern. The top of
this circle is located in the centre of the target, and its
radius equals the drop of the bullet. The sidemost points of the
circle would be reached by using plus or minus 90^{o} and
the lowest point by using 180^{o} cant angle (i.e.
holding the rifle upside down, with the barrel above the line of
sight).

The experiment described below was carried out to verify this pattern qualitatively and quantitatively.

**2 METHOD**

The experiment was carried out on April 23, 1996 at the indoor shooting range of S.V. Prins Hendrik, Amersfoort, The Netherlands. Standard ISSF/UIT 10 m air rifle targets were used, positioned at a range of 10 m, at the standard height of 1.4 m. To eliminate as much random variation as possible, the rifles were fired from a table, supported by cushions. All together, this yielded a height approximately equal to the targets, i.e. the line of sight was about horizontal.

Two experienced air rifle shooters participated, both using their own Feinwerkbau (FWB) model 601 air rifle:

- JH (right handed)
- MH (left handed)

Both rifles were used with the diopter settings as obtained from zeroing during normal shooting. Both shooters maintained a non-zero cant angle in their normal shooting position.

Selected RWS R10 diabolos were used in both
rifles. The weight of this bullet is 0.53 grams according to the
package. The ballistic coefficient is 0.011 (Eichstädt, 1995).
Both rifles had the standard front aperture and diopter. The
initial speeds V_{0} could not be measured in the
experiment.

JH used 4 sets of targets, with each set consisting of 3 targets, marked '0', '45' and '90' respectively. MH used 3 such sets. The targets were shuffled, and 2 shots were fired on each target, applying approximately the cant angle indicated on the target. MH canted his rifle clockwise and JH counter-clockwise. No special equipment was used to check the cant angle.

Afterwards, the location of the points of
impact were measured in the following manner. One by one, the
targets used in the test were put exactly over an unused new
target and the location of the centre of the holes were marked on
the new target by means of a pin. Next, the offset of each pin
mark with respect to the centre of the target was measured in
horizontal and in vertical direction by means of a marking gauge
(readings in 0.1 mm). This yielded a pair of co-ordinates (x,y)
for each point of impact: x being the horizontal offset or *drift*
(to the right of the centre is positive) and y the vertical
co-ordinate or *drop* (high of centre positive).

**3 RESULTS**

The following table shows the averages and standard deviations (s.d.) of x and y for all occurring levels of shooters and cant angle.

Table I Mean and s.d. of drift (x) and drop (y) as a function of shooter and cant angle.

shooter | cant (degr) |
x (mm) | y (mm) | ||

mean | s.d. | means | s.d. | ||

JH | 0 |
-0.5 |
1.4 |
-0.2 |
1.5 |

JH | 45 |
-9.5 |
2.2 |
-3.4 |
1.2 |

JH | 90 |
-17.5 |
1.4 |
-12.6 |
1.2 |

MH | 0 |
1.9 |
0.7 |
-1.4 |
1.0 |

MH | 45 |
9.8 |
1.9 |
-4.6 |
2.0 |

MH | 90 |
19.0 |
1.7 |
-18.9 |
1.5 |

The s.d. of x and y within each cell is typically between 1 and 2 mm, i.e. smaller than half the diameter of the diabolo. This shows that the test results at each cant angle for each shooter were fairly consistent.

A plot of the means of x vs. means of y from Table I is shown in Figure 1. This seems to reveal roughly the expected semi-circle.

Figure 1 Mean drift (x) and drop (y) as a function of shooter and cant angle.

As a next step, the best fitting circle through
all points of impact was determined for JH's and MH's results
separately. The radius as well as the co-ordinates of the centre
were estimated by means of a hill-climbing routine. Most relevant
of these three parameters is the circle's radius, as this equals
the drop of the diabolo during its flight, and it is also equal
to the drift created by using a 90^{o} cant angle instead
of 0^{o}. The advantage of this approach is that it is
insensitive to systematic errors in the realisation of the three
separate cant angles, and also to differences between the diopter
settings of both rifles.

A Sum of Squared Errors (SSE) criterion was used, where the error was defined for each shot as the distance between the centre of the point of impact and circle. Hence, the cost function to be minimised was:

J = SUM [r – {(x

_{i}– x_{c})^{2}+ (y_{i}– y_{c})^{2}}^{0.5}]^{2}

with

- J = cost (summed from i=1 to i=n shots)
- r = radius of the circle (mm)
- x
_{c}, y_{c}= co-ordinates of centre of the circle (mm) - x
_{i}, y_{i}= co-ordinates of i^{th}point of impact (mm)

Parameters that were varied in the procedure
were x_{c}, y_{c}, and r. Using different initial
search values, the procedure converged to the same optimum which
shows that the search procedure was not disturbed by local
minima. The results are shown in Table II.

Table II Best-fitting circles through individual shot data

Shooter | x_{c}(mm) |
y_{c}(mm) |
r(mm) |

JH | -0.7 | -18.0 | 17.5 |

MH | 3.5 | -23.7 | 23.1 |

The values found for y_{c} and r are
approximately equal to each other for both shooters. The x_{c}
values are small compared to y_{c} and r. Both these
findings are in line with the predictions from PCB.

Next, the results are to be compared qualitatively with the output data from the ballistics computer program PCB (version 1.8.)

The radius of the circles in Table II should
equal the drop of the diabolo over its 10 m flight as predicted
by PCB. When using a bullet weight of 0.53 gram and a ballistic
coefficient of 0.011, PCB V1.8 gives the following drop as a
function of initial velocity V_{0}:

Figure 2 Calculated drop as a function of initial velocity

Values from the last column of Tables II (1.7 to 2.3 cm), according to Figure 2, correspond to an initial velocity of 150 to 180 m/s, which are plausible values. This shows that the quantitative results of PCB are plausible.

Later on, the muzzle velocity of JH´s rifle was measured with a Weinlich VM25. The average was found to be 170 m/s. According to PCB, this would yield a drop of 1.8 cm at 10 m (see Figure 2). This matches the results shown in Table II (17.5 mm) very well.

**4 CONCLUSIONS**

The experimental results are in line with the PCB computer calculations: canting the rifle causes the point of impact to move along a circle. The radius of this circle equals the drop of the bullet at the range under consideration.

**REFERENCE**

Eichstädt, U. (1995). *Visier, *10,
42-48.