RG Contour Explanation
This is an excerpt from the thread "The how and why behind MB/Asymmetrical balls" in the forum where I make an attempt to describe the RG profile of a bowling ball.
The first time I met Mo, I asked him to give me a question that I could work on to better understand bowling ball dynamics and mass properties. He suggested that I find a way to map the RG values of a bowling ball. I already knew that the RG values changed when you drilled into the ball but it took me a while to fully realize that the old man was messing with my mind. How could you map the RG values when they change with all of the various orientations of the core inside the ball, various depths of finger holes, different drilling layouts, etc etc.? That is when I began looking into surfaces, stereographic projections and the idea of a topo map of sorts. Using CAD software, I derived the regression equations that allowed me to map the RG values of any bowling ball. The equations led to a graph of the RG contours which are entirely dependent upon the Low, Intermediate, and High RG values. The differentials translate into the slope of the contours while the actual values of the principal axes define the heights. Let us begin with an un-drilled symmetrically cored ball.
We know that the RG values on the equator of a symmetrically cored ball are equal. Let us assume that the Low RG is 2.500 and the Intermediate and High RG values are both 2.550. This produces a ball with a Total Differential of 0.050 and an Intermediate Differential of 0.000. If we only consider the top half of the ball with the pin at the North Pole, then constant RG values would form concentric circles on the ball with the pin as the center. For example, if we would graph all of the locations that had an RG value of 2.525, then that would form a circle on the ball between the equator and the pin. Once we drill the ball, the RG contours change shape and the circles become ellipses.
Let us assume that a ball had a Low RG of 2.500, Intermediate RG of 2.530 and a High RG of 2.550. This ball has a Total Differential of 0.050 and an Intermediate Differential of 0.020. If we would graph all of the locations that had an RG value of 2.530, the Intermediate RG value, the RG contour would be an ellipse centered about the Pin. Constant RG values less than 2.530 form concentric ellipses centered about the Pin and constant RG values greater than 2.530 form concentric ellipses centered about the PSA. The migratory path of the PAP, or spin axis, retains the RG value of the original PAP. (Thank you Nick Siefers!) If you trace the migratory path of the spin axis on a bowling ball then you will draw an ellipse that will be centered about the Pin or the PSA, depending upon the pin distance used and the relationship between the RG value of the original PAP and the Intermediate RG value. This is true for all bowling balls, even symmetrical balls that appear to migrate in a straight line. Now I must relate this information on ellipses to the Intermediate Differential of the bowling ball.
The Intermediate Differential dictates the eccentricity of the elliptical paths. Smaller Intermediate Differentials produce elliptical contours which are nearly circular with low eccentricity. Remember the un-drilled symmetrical core ball in the earlier example produced perfect concentric circles with zero eccentricity and an Intermediate Differential of 0.000. Larger Intermediate Differentials produce elliptical contours which are less circular which means they are more eccentric. Now how does this apply to the bowling ball?
The spin axis migrates along an elliptical path and the point which intersects the Pin to Spin Line is the endpoint of the minor axis of the ellipse. If we consider all of the RG contours then the Pin to Spin Line represents all of the endpoints of the minor axes and I view this like a “ridge” on the RG profile. We believe that the ball will increase angular velocity as the spin axis passes the Pin to Spin Line. Therefore, smaller drilling angles are used to shorten the length of the 1st transition (skid to hook) because the PAP is closer to the Pin to Spin Line. Larger drilling angles increase the length of the 1st transition because the PAP is farther away from the Pin to Spin Line. Please make note of this in the Drilled MoRich LevRG table above by comparing the drilling angle to the length of the 1st transition.