Keep your eye on the target: Equations for setting sight pins

Most bow hunters set the sight pins on their bow by simple trial and error.  The top pin is often assigned to a twenty yard range and the entire sight is moved up or down until the shot angle is correct.   Next, the maximum range pin is moved up or down within the sight until the long range shot angle is correct.   Finally, all other pins are set by splitting the difference between the short and long range pins.   This approach works remarkably well except for very short or long shots.  The challenge is that empirical pin adjustment is time consuming and inconvenient if you change any aspect of your shooting setup (arrow weight, drag, or bow draw weight).  In this post I work out the specific geometry for setting up sight pins and provide a free Excel spreadsheet to perform the calculations.  These tools allow the archer to modify their bow properties and predict the effect of setup changes on pin placement with a high degree of confidence.

Lets begin with a few details on sight geometry.

Figure 1.  Side view of bow geometry (photo credit HuntingNet)
Figure 1. Typical shot geometry (photo credit  Notice that the two dashed lines are parallel to each other and the plane of the earth.  The red solid line is the arrow launch angle.  The solid yellow line passes through a hypothetical reference pin that would produce a shot parallel with the earth (you would never take this shot).

Figure 2. Diagram of shot geometry.  Same lines as Figure 1 with additional distance references.

1) The reference level for all of these discussions is the plane of the earth (dashed red and yellow paths, Figure 1.).   An arrow shot parallel to the earth has a shot angle of zero degrees.  In practice we shoot between 1.0 and 3.5 degrees above horizontal to hit targets from zero to 60 yards away.   A picture and diagram of a typical archer is provided in Figures 1 and 2.  

2) While the arrow follows the solid red path (Figure 1), the archer views the target along the dashed yellow path.  Assuming the target is level with the archer, the sight line is almost parallel with the earth.  The purpose of the sight pin is to provide a vertical reference point so that the archer lifts the bow to the correct shot angle above the horizontal earth reference.  This provides the required arc in the arrow flight to compensate for gravity AND the extra vertical distance between the archer's anchor point and their sight line (even a one meter shot must rise 3.5 inches relative to your eye).

3) The angle between the yellow lines (sight angle) and the red lines (arrow angle) are not the same due to optical parallax  (the arrow is not launched from the eye).  Parallax is caused by observing the target 2-3 inches above the plane of the arrow.   Parallax is much more pronounced when the target is close and insignificant to the archer when the target is more than 30 yards away.  

The position of the sight pins accounts for required shot angle and parallax.   We will compute the shot angle first.  The attached Excel spreadsheet (TrajectoryforPinPlacement) does the hard work of computing the required shot angle (see previous posts for full details).   The required angle is a function of arrow weight (mass), bow speed, aerodynamic drag on the arrow, shot distance, and the arrow launch height relative to the eye of the archer.   The spreadsheet incorporates all of these variable into the equation of motion for the arrow and calculates a shot trajectory.  I use the Excel Goal Seek tool to find the correct shot angle for a desired range.   Figure 3 shows a typical arrow trajectory.

Figure 3.  Typical arrow trajectory for a 20 meter shot including the effects of aerodynamic drag.  The launch angle is 1.38 degrees above horizontal.

Notice the arrow starts 9 cm below the eye of the archer, reaches maximum elevation at 12 meters and then hits the target at 20 meters.   The longer the distance the greater the required arc in the shot.   The exception is very close shots.  Notice that the arrow crosses the zero elevation line at 5 and 20 meters.   This means that a 1.38 degree launch angle would produce a perfect shot at both distances.  Shots very close to the archer need steep shot angles   to gain 3.5 inches in elevation in a short distance.  Fortunately, the trajectory calculator takes this geometry into account when computing shot angles.

Parallax is due to the position of the archer's eye above the plane of the arrow as shown in Figure 4.

Figure 4.  Computing the parallax angle for a shot.

The angle can be up to 10% of the total shot angle for close shots (less than 10 meters) and almost zero for long shots and is computed from the arctangent of the launch height relative to the eye divided by the shot distance.

Again, the actual angle of a shot is the sum of both the angle of the arrow and parallax.   The attached Excel spreadsheet adds both of these terms to determine an effective shot angle.   Once the shot angle is known, the pin height is computed from the sine of the shot angle multiplied by the peep to sight distance.    Figure 5. shows the computed pin heights for my bow as a function of distance.

Figure 5.  Computed pin heights as a function of shot distance.  Each line represents different shooting conditions.  The blue, green, and red lines are computed for different eye to arrow distances in inches.  Notice the blue line is almost straight since by definition the arrow is at the plane of the eye.  The purple line shows the pin height calculation for a 20% slower bow.  

The arrow to eye vertical distance does make a difference in pin placement, but only at distances less than 30 inches.   This is a caution if you change your arrow position by moving your anchor point, nocking position, or arrow rest height.  With these changes all pin positions will change by a noticeable amount.   

No surprise, the largest influence on pin position is arrow speed.   A slower bow or heavier arrow will change all the pin positions significantly.   

In practice, I find the calculated distances between pins to agree exceptionally well with empirically defined pin positions.  However, the absolute pin distances below the sight reference height (zero angle height) can be a bit off if measured as the arrow to peep height.   The problem is that the exact reference height is hard to define if your nocking point is slightly above the level of your arrow rest, as is often suggested for optimal rest performance.   In practice I tune my bow sight to get the best groups at 20 yards and then use the calculated distances between pins to set all other pin positions.    The Pin Placement tab on the attached spreadsheet makes these calculations easy.

Probably the most interesting aspect of the pin placements is the distinct arc in the pin placements going from very shot to very long shot distances.  Shots very close to the archer need steep shot angles to gain 3.5 inches (specific to your bow) in elevation in a short distance.  Parallax at short distances also increases the shot angle.   At distances beyond 15 meters the pin angle (and pin distance) increases almost linearly in the 20 to 50 meter range.  This is why a two point empirical pin calibration can work so well and why using 20 yards for the first pin is best (20 yards is the start of the linear region).  Faster bows can extend this linear range to 60 or 70 yards.  However, the slope of pin height versus distance will become nonlinear (curves up) for longer shots and for moderate distance shots from slower bows.  The curvature is due to drag and the downward velocity of the arrow with increasing flight times.  One advantage of the pin position calculator is the ability to test the effect of bow setup on shooting angle and pin positions.

Now that we understand how to calculate the geometry of the sight pins, the next post will look at the effect of bow torque on arrow placement.

Whitney King,
Aug 20, 2013, 8:06 AM