Figure 3 shows a matrix of α-β plots at three distances (*r*_{1}=3.218 ft, *r*_{2}=5.905 ft, and *r*_{3}=11.15 ft) and three polar angles (θ_{1}=10°, θ_{2}=50° and θ_{3}=80°). The ranges of α and β at launch are shown as solid gray regions for direct shots and as unfilled regions for bank shots. The centered shot is indicated by a dot and the perimeters of the unfilled regions are red in the upper left quadrant, blue in the upper right, green in the lower left, and black in the lower right. The colors remind the reader of the quadrant from which the ball is launched.

Let’s now compare the bank shot (unfilled regions) and the direct shot (gray regions), beginning with the range of pitch angles α. For the bank shot the range of α does not change significantly with polar angle θ and distance *r* to the basket (except at *r*=3.281 ft, *θ*=10°). On the other hand, the range of α tends to be smaller for the direct shot than for the bank shot at smaller and moderate distances *r* and then about the same for larger distances. This means that the shooter will find the bank shot to be more forgiving (wider range) in α at small and moderate *r* and to have about the same forgiveness in α at larger *r*. Turning to the side angle β, the range of β tends to be the same for different polar angles θ and decreases with *r* for both the bank shot and the direct shot. However, the range of β tends to be smaller for the bank shot than for the direct shot at smaller *r*. This means that the shooter will find the direct shot to be more forgiving in β at small *r* and to have about the same forgiveness in β at moderate and larger *r*.

This figure shows that at smaller distances the direct shot can be preferred over the bank shot, that at moderate distances the bank shot is preferred over the direct shot, and that at larger distances neither is preferred. Pitch angle α is always more forgiving than side angle so these observations are influenced greatly by the shooter’s ability to prescribe side angle.

Figure 4 shows a matrix of backboard imprint plots. The inner (red) boxes are the regulation rectangles. The color bands on the perimeters of the imprints correspond to the color bands on the quadrants of the *α*-*β* plots. The same blue and green colors on the left (red and black on the right) quadrants of the imprint are on the right (left) quadrants of the *α*-*β* plots and the same green and black colors on the top (bottom) quadrants of the imprint are on the top (bottom) quadrants of the *α*-*β* plots except for the imprint at θ=10° and *r=*3.281 ft. The switching of this imprint occurs because the distance that a ball travels decreases when the pitch angle increases in this case for which the pitch angles are largest.

The backboard imprint shows the region where the shooter must aim the ball for a successful bank shot. Unlike the *α*-*β* plot and the hoop and reflection hoop described later, during play the shooter sees where the ball strikes the backboard. The shooter then makes future adjustments, gaining a sense of the backboard imprint.

On the other hand, note that the overall size of the imprint does not necessarily correlate with the size of the *α* plot, which does correspond to the level of forgiveness in the aiming parameters. An *α* plot of relatively small area can correspond to a small or large backboard.

Keeping in mind that backboard imprint size does not necessarily correlate with shooter forgiveness, observe that the overall imprint size increases with both polar angle θ and distance *r.* Like imprint size, the increase in the imprint’s thickness with increasing θ should not be interpreted as corresponding to an increase in the range of β with increasing θ because the imprint acts on the backboard – a plane that is *not* perpendicular to the vertical plane of the trajectory. For example, imagine that the shooter shines a flashlight onto the backboard from the launch position. The flashlight will shine a spot on the backboard that increases in width as the polar angle increases (holding *r* constant) even though its range of β is constant. Indeed, notice that the range of β did not increase with θ in Figure 3.

Figure 5 shows a matrix of hoops and reflection hoops. Each element of the matrix shows solid circular regions of direct swish shots (surrounded by the hoops) and solid reflection hoop regions of swish bank shots corresponding to the same launch *r* and *θ*. As shown, the reflection hoops allow you to visually compare the forgiveness of the bank shot and the direct shot. As shown, the reflection hoops are elliptical-like in shape and their long axes are aligned with the directions of the trajectories. Also, note that the distances that the centered shots travel are near-maximal so deviations in α, whether positive or negative, can cause the ball to travel a shorter distance. For example, if α=55° is the optimal centered pitch angle, releasing the ball at either α=50° or α=60° will cause the ball to travel a shorter distance in the *x-y* plane. This effect becomes more pronounced as the distance increases. The result is the colorless regions on the boundaries of the reflection hoops and the blue and red colors on the top (green and black on the bottom) to switch to the bottom (top) of the reflection hoops.

First observe that the reflection hoops are considerably longer than wide and the lengths increase with θ. This figure, more than the α-β plot and the backboard imprint, shows that the bank shot is extremely forgiving in pitch angle. The widths of the reflection hoops are about the same as the widths of the hoops, so the bank shot and the direct shot have similar levels of forgiveness in side angle. Reinforcing the trends found in the α-β plot and the backboard imprint, the reflection hoop shows that the advantages of the bank shot over the direct shot are significant in α but highly dependent on the shooter’s accurate launch of β.

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