PT Innards
Before I finish the subject of pivot triangles, I'll touch on the math behind the method. If you've come this far and are curious about the 'wheels within wheels' of the PT, I'll explain the method with as little math as I can.
By setting the intersection of the B and C lines at 90* it makes a PT a right triangle. This allows using the lengths of B and C to find the angles of the triangle. I use ratios in the description so I'll explain the term before moving on. In this post a ratio is just another name for a fraction. A ratio of 1 : 2 is the same as the fraction of 1/2. It's another way of saying that the short line is half the length of the long line.
Comparing C : B and B : C
When overlaying B and C, a PT is solving a division problem. The division is made when B or C is divided into quarters. Each quarter is a .250 'piece' of the original length. If C is the same length as the first quarter mark on B, it's a .250 length of B. If C is half way between the first and second quarter marks the value will be half way between .250 and .500 or .375 . The result of the division (.375) can be used with a little trig to identify the angle in the triangle.
Let's start by looking at an angle where the ratio of C : B is 1 : 2. The ratio can also be expressed as C/B, 1/2, or .5. The trig term for the ratio of C/B is the tangent (tan). The value of .500 is the tan of the cut angle. Once the tan is known it's simple to look in the trig tables and find the angle that has that value. This method of working 'backwards' is defined as finding the arctan of a value. It's the button labeled tan with a little -1 on calculators if you want to check the results. Solving arctan(.500) the result is about 27*. This is value at the center mark of B after it's divided into quarters.
Using the ratios of 1 : 4, 3 : 4, and 1 : 1 , the results are 1/4 , 3/4 and 1 which are .250 , .750 and 1. Taking the arctan of the values, the angles are 14 , 37 , and 45 to the nearest degree. Knowing that arctan .250 = 14* and etc. , the ratios of 1/4 , 1/2 , 3/4 and 1 can be used to create a small 'lookup table' of the angle values up to 45* on the quarter marks of B. This allows the values of the division C/B to be read as angles along the side of the PT. This is how I set the scale for the line segments on B. If you know the sequence of 14, 27, 37, 45, they can be used to 'compute' the angle.
When C becomes larger than B a switch is made to a different trig function in order to keep comparing a short line to long one. When B is compared to C, (B/C) the cotangent (cot) is used the same way. The logic used to 'set up the table' is the same as the tangent technique above. This gives the sequence: 53, 64, 76, 90 for angles larger than 45* on the C quarter marks.
Comparing C : A
The ratio of C/A is 'trig' for the sine (sin). The value of the sin varies between 0 to 1.000 , 0 for a 0* cut angle and 1.000 for 90*. By using a one inch ball radius it's easier to explain how the sin = CP. Divide the radius into 1,000 pieces. Each .001 step in the sin value would equal a .001 inch step across the ball. A half ball hit would be at 1/2 inch on the ball or .500 inch. At half ball the sin value would be .500. Checking through the arcsin trig table for .500 the result equals 30*, the cut angle for a half ball hit. If .250 or quarter ball is used, arcsin(.250) = 15* and etc. The sin is the 'step' distance from the center of the ball, so is the contact point. The CP = sin.
The A line is divided into quarters to make it easier to visualize the CP's relative position to the OB's quarters. It helps identify which quarter the CP is in and how close it is to the next or last quarter line.
The sin 'scale' is harder to set up for reading angles above 30*. The technique used for the tan and cot works fine to estimate angles until half ball. After that point the angles start to become compressed. When reaching the edge of the ball (getting close to the end of A) they're so tightly packed it's very hard to read individual angles with any accuracy. The first 1/6 of the ball has 10* in it. The last 1/6 ball has 34*. That's over three times as many angles packed into the last 1/6. The scale isn't linear, the .001 steps don't give the same angle ratio past half ball so it's not useful for large angles.
Comparing B : A
The B/A ratio is the cosine (cos). This is used for double checking angles over 60*. It gives the other (or complimentary) angle of the triangle. It's of no other use in the aiming process that I can find.
A PT is a method to graph the division of two legs in a right triangle and 'decode' the result of that division into an angle. It will also show the location of the contact point. It's a 'slide rule' on the table you can use to get shot information if you know it's there and care to use it.