Aiming & Shot Making

01 — The question

You make balls. This is about the ones you don't.

The tournament layer asks a sharper question than league night: what does each shot demand of your aim, and what breaks when the ball rattles out anyway?

This episode is about the aiming problem, not the stroke problem. The two are coupled, and the sources rank them: Dr. Dave Alciatore's ranked list of why players miss puts failure to visualize and aim the shot first, and poor stroke fifth instructional¹¹. Both matter. The aiming layer gets the coverage here because it is where a league player's habits are most invisible and least practiced.

The shape of the answer, in order: what the pocket geometrically demands (02), the reference method every system is measured against (03), the named systems and the arithmetic any fixed menu of aims faces (04 to 05), what the efficacy fight actually shows (06), how documented pros aim (07), why players miss (08), the three forces that bend a correctly aimed line after the tip hits the ball: throw, squirt, and swerve (09 to 11), the stroke and cue-ball rules that aiming presumes (12), what the documented pro numbers show (13), and what changes when you cross from league night to tournament conditions (14).

02 — What a pot demands

Before any system: the size of the target

A pot has a named target and a measured tolerance. Every aiming argument in this episode is an argument about fitting inside that tolerance.

A pot is a geometry problem with a named target. The ghost ball is "the imaginary position the CB must be, at contact with the OB, to make a shot" instructional¹. Two lines matter, and they are different lines: the aiming line runs from cue ball through the ghost-ball center; the impact line runs between the ghost-ball and object-ball centers, and that is the line the object ball travels ¹. In advanced use, the ghost-ball position is adjusted for cut- and spin-induced throw, which is where sections 09 and 10 are headed ¹.

Cut angle and ball-hit fraction are two views of the same decision. An exact half-ball hit is a 30-degree cut instructional². Three-quarters ball is a cut slightly under 15 degrees, and one-quarter ball is about a 48-degree cut ². Dr. Dave's 30-degree-rule page rounds the same endpoints as 14 and 49 degrees ¹⁹. The old classroom roundings, 15 and 45, are handy and inexact. Of the standard fractions, only the half-ball 30 is exact ³.

The target itself is tight. A corner pocket accepts about 4 degrees of object-ball angle on a typical approach instructional³. The tolerance is angular and fixed, but the margin it buys at the object ball is linear and shrinks with distance. Long cut shots are the accuracy bottleneck for that reason synthesis³.

03 — Ghost ball

The reference method

Ghost-ball aiming is the method every named system gets compared to, including by the systems.

The method is one visualization: put a ball-shaped ghost at the contact position and drive the cue ball through its center. Published practice methods make it concrete. A helper places a real ball at the ghost-ball position and pulls it away. Striped practice balls line up both lines at once. A printed cut-shot template trains the ghost position, the contact point, and the amount of overlap for any cut, and the source calls striped balls lined up on both lines "the most useful advice in the video" instructional¹¹².

A practitioner's walk-up, quoted on the ghost-ball page under the name CreeDo: read the object-ball-to-pocket line for the contact point, rest the tip on that line half a ball width behind the object ball, then walk into the shot so the cue runs through the cue ball's center anecdotal¹.

Where it sits in the system fight, from the same host: cut-shot aiming systems "can most definitely initially help people who have a lot of trouble aiming, but ghost-ball aiming is a better teaching and learning approach that helps players develop faster" instructional⁴. His metaphor: ghost-ball approaches are "more like 'training wheels'", while limited-lines systems like fractional-ball, SAM, and CTE "are more like 'crutches'" ⁴. The named variants, contact-point-to-contact-point, double-the-distance, double-the-overlap, and bisect-pivot-and-shift, are listed as ghost-ball derivatives ⁴.

04 — The named systems

Six named systems, all pivot or fraction, all with eyes

The systems in circulation are a small, well documented family. None of them removes the ball-speed and throw adjustments every shot still needs.

The canonical definition covers the whole pivot family. CTE is an align-and-pivot method: align the cue with the shot's center-channel or outside-edge reference, then pivot the cue to center ball before delivery. Documented variants are Hal Houle's original three alignments, Ronnie Allen's early version, Stan Shuffett's CTE Pro One with six alignments and its 15/25/35/45-degree sweep additions, and David Segal's simplified two-alignment version instructional⁷.

90/90, taught by Ron Vitello, is described as a system for aiming cut shots within the 9-to-90-degree range, via 15, 30, and 45 degree cut-angle slots and visualizing the cue ball's 360-degree band. Its method asks you to pivot the cue through center ball, "last 10% of the cue ball and object ball", and the discipline rule is stated plainly: a pivot moved before center ball or aligned incorrectly ruins accuracy instructional¹⁰.

Fractional-ball aiming, "aiming from the line of centers", teaches the standard fractions: straight full, 3/4, 5/8, 1/2, 3/8, 1/4, 1/8, and thin 1/8, with visual references for each instructional⁶. A 2009 Billiards Digest article tested the popular claim that three angles cover the table. Its TP A.11 finding: the 15- and 45-degree aims "are not exact", and it closes the section with a call for "more discussion of why this is a bad idea" instructional³.

The Pro One reference numbers are printed by its author. Stan Shuffett, describing CTE's source of truth: "CTE is a VISUAL system. VISION does not have language... your goal is to SEE, not to think words" secondary²⁶. On gearing as the central skill: "The most important CTE skill that you can develop to the hilt is GEARING... Professional players gear CB-OB relationships" secondary²⁷. The center-ball requirement is documented too: "A CTE shot with a center ball hit makes the alignment and pivot pointless" secondary²⁶.

What every system still demands, on Dr. Dave's summary page: adjusting for cut-induced throw, for squirt and swerve when english is used, and for speed, which the sources place at the top of what actually changes outcomes instructional⁴.

05 — Limited lines

The arithmetic that bounds every fixed menu of aims

A pool table allows infinitely many cut angles. A named system offers a fixed menu. The gap between those two numbers has been modeled, and the model is printed.

The model, from Dr. Dave's limited-lines analysis: for each object-ball direction there are "about 25 discrete cut angles per quarter ball of right and left cut", each slot about 3.6 degrees wide, worth about 1/16 inch of contact area measured³. The pocket is smaller than the menu needs: a corner pocket accepts about 4 degrees of object-ball angle on a typical approach, so a slot either lands inside the fan or it misses ³.

The coverage model, at five assumptions

Every row below is the same source's geometric modeling, printed with its assumptions {tg('measured')}{ref(3)}:

The same page's plain warning about the straight shot: with a straight-in shot at 11 degrees plus or minus, "The 1/4-ball and 1/2-ball lines of aim will result in a miss" {tg('instructional')}{ref(3)}.

06 — Efficacy

What the efficacy fight actually shows

There are no controlled studies of aiming-system efficacy. What survives is mechanism, and the mechanism side is unusually specific.

The honest baseline first: no controlled efficacy study of any aiming system exists in the corpus synthesis⁵⁸. Every claim on either side is mechanism, anecdote, or testimony. This section sorts those by what they can and cannot carry.

The mechanism argument for pivoting, from a contributor's analysis hosted on the benefits page: physically pivoting a correctly aligned cue moves the tip toward center ball, so the grip is brought back to the center in a repeatable, mechanical way anecdotal⁵. On the other side, the same page's boundary: the pivot argument "does not explain why the initial alignments are what they are or why they work as well as they do for certain shots but not others" ⁵.

The attention argument, also mechanism: a system "forces a person to focus on aim, alignment, and sighting consistently, which should result in more consistent aiming" instructional⁵. Dr. Dave: any system that enforces that focus "will be beneficial" for many players ⁵. Mike Page's version names the time: "maybe 2.5 seconds versus 1.5 seconds", and the pause is what works, because "once you are down in the stance and stroking you are no longer aiming" anecdotal⁵. JoeW's trust quote closes the loop: "My conscious mind couldn't be trusted with such an important task" anecdotal⁵.

The geometric critique of CTE is the sharpest published objection. Three shots inside one thick-cut category produce cut angles of about 10, 15, and 20 degrees depending on cue-ball and object-ball distance; the source's own conclusion: "this obviously cannot be the case" instructional⁸. The counter position, from the author himself: "Experience is our major teacher" secondary⁹. AtLarge's summary on the quotes page: after alignments and pivots, judgment fills the rest, "your visual intelligence, especially after many, many hours at the table" anecdotal⁹. The host's own verdict on the same page: "The alignments and pivots don't do the aiming for you" instructional⁹.

The practitioner record: one documented user, mohrt, reporting after nine months on a CTE forum: "my game is changing drastically, and in a very good way", with "a thousand shots or so" to get comfortable anecdotal²⁵. One user, self-reported, with no control arm. It is evidence of a learning curve, not of system superiority.

07 — How pros aim

How the documented pros aim: standing, and with experience

The best-documented characterization of pro aiming behavior fits on one page, and it starts before the player is down on the shot.

Dr. Dave's "How Pros Aim" (Billiards Digest, February 2024) is the corpus's direct treatment. The sequence: aim while standing, get down into your stance while holding the visual lock, drop straight down on the line of the shot, and make sure your vision center is aligned and your eyes are on the line of the aim. Then two optional checks: look back and forth from object ball to cue ball to verify alignment, or just focus on the object ball's contact point before the final stroke instructional¹².

Then the limits of the question. Asked how they aim, "most pros are not able to express a clear, accurate, and satisfying answer" instructional¹². On forums and in TAR podcasts, John Schmidt put it two ways. First: "The pros scoff at that stuff" secondary⁹. Second, on what actually carries the aim: "We've hit a million pool balls — that's our system" secondary⁹. He calls the systems "aiming's adorable — but you still have to deliver — so you could aim perfect" secondary⁹.

The host's synthesis of the pages: top players rely on instinct and experience built from intense practice; they aim while standing and keep the eyes still at the end of the stroke; they are excellent judges of distance, speed, throw, and cue-ball control; and no aiming system substitutes for any of it instructional⁴¹². Patrick Johnson's one-line version: "You'll always aim by feel" anecdotal⁵.

08 — Why you miss

Why players miss: the source's ranked list

The ranked list of miss causes reads like a practice plan, and the first item is not the stroke.

Dr. Dave's ranked list, in order: 1, failure to visualize and aim the shot; 2, failure to physically execute the shot as visualized, including speed control; 3, shot difficulty and table difficulty; 4, "SNOOKERED" or tricky layouts; 5, poor stroke instructional¹¹. He ranks item 1 as the most common cause, and the list comes with a telling observation from Cliff Thorburn: "I never miss a shot I know the speed of" secondary¹¹.

The english traps get their own entries. Unintended inside english on outside-spin shots, and unintended outside on inside-spin shots, are named failure modes from an off-line or crooked stroke instructional¹¹. Dr. Dave's stance: an off-line or crooked stroke is the most common cause of misses for most players, provided the aim is good, and it is "just as important as good aiming" instructional²². The safe harbor he offers: a purely vertical cue motion, if the stroke is straight, affects the shot minimally; small deviations are acceptable as long as the upper arm and shoulder joint stay still instructional²².

The practice method named for closing the gap: the "Millions of Full-length Understanding Drills And Techniques", HAMB instructional¹². Schmidt's million balls, again ⁹. The aiming prerequisites list covers the rest: consistent stance, bridge, stroke, and alignment, all named before any system can help instructional⁴.

09 — Throw and cling

After the aim: throw is real, big, and manageable

A correctly aimed shot can still miss, because friction bends the object ball off the impact line. The size of the bend is published.

The 90-degree rule sets the frame: after a stun impact, cue ball and object ball separate at 90 degrees, with the cue ball persisting along the tangent line instructional¹⁸. John Matranga's trick for seeing it: with the cue ball a diamond from the corner pocket and a half-ball hit on an object ball a half diamond from the corner, the cue ball's tangent path enters "within three fingers of the opposite pocket", and each of his finger spans is about 2.25 inches, one ball width anecdotal¹⁸.

Then friction edits the picture. Maximum throw: "about 1 inch per foot of OB travel, or 1/2 a ball per diamond on a 9' table, which is about 5 degrees" instructional¹³. The conditions are named: slow speed, stun, half-ball hit ¹³. The drawn peak case in the lab below is true scale: over about 3.5 feet of object-ball travel, about 3.5 inches of deflection, wider than most pocket mouths.

Speed, in the range that matters, mostly does not matter: below about 20 degrees of cut, cut-induced throw is independent of speed instructional¹³. Spin type does matter: topspin or bottom spin reduces throw by up to 75 percent for cut angles up to about 30 degrees ¹⁴. And gearing outside english removes it: "at a half-ball hit, about 50% sidespin will result in 0 degrees of throw" instructional¹⁴. The term is a definition, not a style point: gearing means the ball surfaces are rolling with no relative slipping at the contact point ¹⁴.

Then there is cling: "Clng occurs when there is significantly more friction between the cue ball and object ball than normal... usually from a chalk deposit" instructional¹⁵. The source is blunt about responsibility: "a bad hit has nothing to do with the shooter or the stroke" ¹⁵. Its hygiene rule: "Definitely wipe off the cue ball before each break shot or any time you have ball in hand" ¹⁵. Draw and follow reduce cling effects more than speed does ¹⁵, and the static electricity hypothesis is called "highly questionable" in the source's own experiment ¹⁵.

Between the two limits sits the everyday call: outside english reduced or removed throw, and inside english increased it, with the fast-inside-english exception on thin cuts instructional¹⁴. The league-tournament version of the same list lives in section 14.

10 — Squirt and squerve

Squirt, swerve, and the net: squerve

Sidespin does not aim where the stick points. The deflection has two parts with different physics, and the published numbers are robot measurements.

Definitions first, because the vocabulary is where most confusion lives instructional¹⁶: squirt, also called deflection, is the cue ball's lateral deviation from the stick line as it departs the tip, from the tip's off-center impact. Swerve is the cue ball's curve back toward the stick line while the ball slides, from a tilted stick with sidespin. Net deflection, squerve, is the sum at impact, and it is what your aim must compensate.

The measured numbers: Platinum Billiards' robot tests, "Iron Willie", shot at a controlled 15 mph with 6 and 12 mm tip offsets, aimed at a target "exactly the distance between the foot string and the head spot on a 4 1/2 x 9 pool table", 50 inches measured¹⁷. Deflection across 46 tested shafts spans 1.3 to 2.3 inches over that distance, about 1.3 to 2.3 degrees, with the 314-2 near the low end at about 1.3 inches ¹⁷. Across the earlier literature, deflection ran 0.5 to 2.3 degrees and effective pivot length 50 to 10 inches ¹⁷. Speed does not change it, effect 2 in the source's numbered list instructional¹⁴.

Swerve behaves differently: it occurs "with practically all sidespin shots", though it is small at low cue elevations, effects 5 and 6 instructional¹⁴. It grows with cue elevation, with more speed for the same angle, and with a soft, non-slick cloth ¹⁴. With the cue elevated to clear a rail, the curve is visible: the cue ball departs the stick line with squirt and bends back. At certain speed and elevation combinations the net of the two is zero, effect 13, "squerve" instructional¹⁴¹⁶.

The compensation systems are named and old: back-hand english sets the tip's side position and pivots about the bridge hand; front-hand english does the same from the front hand. Both, "if the pivot length is equal to the effective pivot length of the cue, ... result in perfect cancellation of squirt", effect 15 instructional¹⁴. The pivot length is a shaft number, and the source gives the definition: the point at which the cue pivots for BHE, "where the front hand is placed" anecdotal¹⁷. The caveat is printed in bold in the robot study: these numbers are for each shaft's effective pivot length; "most cues will have a much longer effective pivot length", which is why the word effective matters measured¹⁷.

11 — Center or english

Center ball unless you have a reason

The english decision is two separate questions: aiming compensation for squirt, and position play. Collapsing them is the standard error.

The sources' guidance is unambiguous about the baseline: "In general, center ball aiming is more accurate" instructional¹⁴. English multiplies the error budget: squirt shifts the line, swerve bends it, and cling waits for a chalk mark ¹⁴¹⁵¹⁶.

Outside english has one documented consistency bonus: it "reduces the chance of a late collision or cling-related throw error", and inside english "increases the effective pockets if you accidentally hit a wall before the pocket", effects 38 and 39 instructional¹⁴. That is an argument about insurance, not about accuracy.

And the alignment rule from section 04 repeats: any english adjustment requires a compensated aim or a pivot from the correct point. A CTE alignment with a center-ball intent, then an english hit with no compensation, is the documented failure mode synthesis⁴¹⁴.

12 — Stroke and control

The stroke question, and the cue-ball rules aiming presumes

Aiming hands you a line. Two more layers decide whether the ball obeys it.

On the vertical-cue debate, the source's position has a caveat in it: a purely vertical cue motion is recommended because it is repeatable, but the truth is "As long as you keep your upper arm and shoulder joint still during the stroke, ... little effect on the straight motion of the cue" instructional²². The mass-dense-to-under-delayed-technique, MOFUDAT, is named for the stroke itself ²².

The rules themselves are indexed, not re-derived: leave an angle on the next ball, come into the line of a shot rather than straightening out, cheat the pocket from long range to protect shape, break clusters early, and know the three-times-the-angle draw rule and the 45-degree-rule two-rail escape "fairly close to the center of the table along a scratch-free path" instructional²⁰²¹. The 30-degree rule covers rolling-ball position: "between a 1/4-ball hit (49 degree cut) and 3/4-ball hit (14 degree cut)... about 34° closer to a 1/2-ball hit and... about 27°" near the ends instructional¹⁹. Its technique is the peace sign ¹⁹.

None of that is aiming. All of it is what the aiming was for synthesis²¹.

13 — The numbers

The pro metric: TPA, and what it will not tell you

The only published per-match shooting metric in the corpus is Accu-Stats TPA. It measures match performance, not aiming method, and it must not be converted into a make percentage.

The definition, from Accu-Stats' own scoresheet instructions: "TPA = Balls Pocketed / (Balls Pocketed + Errors)" measured²³. An error is one of five kinds: a miss, a safety, a jump, a push, or a foul, and the definition's teeth are procedural: "A MISS THAT IS 'EASIER THAN A SPOT SHOT' COUNTS AS TWO ERRORS" measured²³.

The 2024 Derby City Classic numbers, as measured and published by Accu-Stats measured²⁴:

Two readings the data does support. First, the elite tier lives between .90 and .975 in these matches, with .975 as the observed ceiling, Filler over Morra measured²⁴. Second, TPA is not a potting percentage: Shaw outplayed Thorpe while scoring lower, .843 to .890, because the metric prices error-free play, not shot difficulty measured²⁴. A community explainer of the stat reads .950 as missing "19 of every 20 shots. For every 20 shots, you'll miss less than one" secondary²⁸, a useful intuition that no official documentation confirms; take it as a reading aid, not a definition.

The scoreboard column closes the argument this episode has been building: in seven of these eight matches the higher TPA won; in the eighth, the loser outshot the winner on the composite and lost 9-8 anyway measured²⁴. Aim well and the match still needs the rest of the game.

14 — League vs tournament

From league night to the tournament table: what actually changes

Your aiming homework does not change with the room. The conditions around it do, and the documented changes are specific.

The tables are the headline. The research episode on table variance is its own document; the aiming-relevant summary: 7-foot bar boxes shrink the long cut shots where pocket tolerance bites hardest and reduce swerve accumulation, since swerve grows with distance and elevation synthesis³¹⁴²⁹. The 50-inch squirt test distance is itself the 9-foot-table standard, foot string to head spot measured¹⁷.

Cloth and chalk care carry over verbatim: wipe the cue ball before the break and at every ball in hand instructional¹⁵. The break matters downstream of the aiming decisions: peer-reviewed work on break performance found the cue ball finishing near table center after the break correlated with more balls potted in the following visit secondary²⁹³². If the table plays dirty, the cling rules of section 09 apply with interest ¹⁵.

Equipment: the deflection table of section 10 is shaft-specific and dated measured¹⁷. Your own pivot length is yours to learn; the sources' shaft list is a map of the spread, not a promise about your cue synthesis¹⁷³¹. And the honest note stands: no published deflection number for current carbon-fiber shafts survived verification this session, so the recommendation is the method, not the number synthesis¹⁷.

Rights and rules differ by ruleset, and the break-rules episode documents them secondary³⁰. For aiming, the differences that matter are none: the geometry of sections 02 and 05, the throw of section 09, and the squerve of section 10 travel with you to any table, any league, any tour synthesis¹³¹³¹⁴.

15 — Conclusions

What survives the sources

Every number in this episode carries its source and tier. These are the claims strong enough to act on, and the honest gaps beside them.

1. 1. The pocket, not the system, is the constraint

   A corner pocket accepts about 4 degrees of object-ball angle on a typical approach, and the table offers about 25 pocketable cut-angle slots per direction for a spot shot. Coverage math follows: three fractional lines cover everything under a foot and under half the shots beyond two feet; six angles cover about a quarter of the table's area; full coverage needs about 19 lines. No fixed menu of aims covers the table. That is geometry, not opinion. measured ³

2. 2. Ghost ball is the reference; every named system is a different door into it

   The documented family is small: fractional-ball with its standard fractions, CTE in four versions, 90/90, SAM. The host's verdict is that cut-shot systems can initially help strugglers while ghost-ball aiming remains the better teaching approach, "training wheels" versus "crutches". What every system still requires is the same list: throw, squirt, swerve, and speed adjustments. No source shows a system removing them. instructional ^{4 7 10 6}

3. 3. The efficacy literature is absent; the mechanism and attention arguments are real

   No controlled study of any aiming system exists in this corpus. What the sources support: a system that forces consistent focus on alignment and sighting benefits many players; pivoting has a real mechanical effect on tip return; the pause it enforces buys aiming time the stance cannot. Against that: the geometric critique of fixed-thickness cuts is printed and unanswered, and the system author's own appeal is to experience. One documented user's nine-month self-report is a learning curve, not a verdict. synthesis ^{5 8 9 25}

4. 4. The first place you miss is the aim, not the stroke

   The source's ranked list of miss causes puts failure to visualize and aim first and poor stroke fifth. Colenso's fixed-bridge fixture pocketed 20 of 20 without a stroke. The lesson for a league player is uncomfortable and cheap: before rebuilding the stroke, buy practice balls, line up striped balls on both lines, and check where the aim actually points. instructional ¹¹

5. 5. Throw is real, and it is bigger than most pocket tolerances

   Peak throw, slow stun half-ball, is about 5 degrees: about an inch per foot of object-ball travel, over 3.5 inches on a 3.5-foot shot, against a pocket that accepts about 4 degrees of angle. Cut-induced throw does not care about speed below about 20 degrees of cut. It shrinks by up to 75 percent with draw or follow, and gearing outside english, about 50 percent side at a half-ball hit, makes it zero. Wipe the ball; cling is a chalk event, not a stroke event. instructional ^{13 14 15}

6. 6. With sidespin, aim for the net

   Squirt is speed-independent and shaft-specific: 1.3 to 2.3 inches of deflection over the 50-inch robot distance across 46 shafts. Swerve grows with elevation and soft cloth. The only number that aims is the net, squerve, and the pivot systems exist to zero it, provided the pivot length matches the shaft's effective pivot length. Center ball remains the accuracy default; english is a purchase, and the price is always paid at the object ball. measured ^{17 14 16}

16 — Sources

Sources

Each source was opened, read, and used for exactly the claims tagged to it. Tier and confidence follow the Series Bible's seven-tier system.

Flagged: not verified

These claims appear in the research corpus or circulate in the wild but did not survive verification against a primary source. They are printed where relevant with the unverified tag; they are collected here so the gap is impossible to miss.

• ⚑"90/360" as a named aiming system. Appears in no source located this session. The documented pivot-based family is Houle's 3-angle system, early CTE versions, Shuffett's Pro One, and Vitello's 90/90. If you meet the term, treat it as noise.
• ⚑"Aiming systems outperform good fundamentals," and its reverse. No controlled study exists on either side. Both strong claims circulate on testimonials. Section 06 is built on that absence.
• ⚑Carbon-fiber shaft deflection numbers. The verified robot table is Platinum Billiards data from circa 2008. No comparable published figure for current carbon shafts survived verification. None is printed anywhere in this episode.
• ⚑Poolology and other named commercial systems. Not verified from a primary source this session, so not described here.
• ⚑"Pros make 95% of their shots" as a universal statistic. No authoritative per-shot make-percentage table exists in the corpus. TPA is a composite and must not be converted into one.
• ⚑Quiet-eye numbers for pool specifically. The 2.5- versus 1.5-second gaze figures are one contributor's summary of research on other aiming tasks. Directional, secondary.
• ⚑Mike Page's five-element aiming model. A contributor's framework on the benefits page, not a tested model.
• ⚑"A thousand shots to learn CTE." One practitioner's 2011 estimate. No verified learning-time claim exists.
• ⚑Static electricity as a direct cause of cling. Called "highly questionable" in the primary source's own experiment. Not presented here as a mechanism.
• ⚑Exact fractional cut angles beyond the half-ball 30. Only ranges verify: 3/4-ball about 14 to 15 degrees, 1/4-ball about 48 to 49, depending on the source's rounding. This episode prints "about" everywhere except the exact 30.