# A Visual Primer on Horizontal Approach Angle (HAA)

It’s been almost two years since I authored a visual primer on vertical approach angle (VAA) and almost three years since I first discussed VAA here at FanGraphs. The topic directly adjacent to VAA — horizontal approach angle (HAA) — has been marinating in my mind (and in my drafts) ever since.

In an auspicious turn of fate, while revisiting this draft, our beloved Eno Sarris reached out to ask if I had ever published anything about HAA. He had questions about Brandon Pfaadt — specifically, Pfaadt’s sweeper. Much was made about Pfaadt’s success this postseason, which can be attributed, in no small part, to his sweeper and how he weaponized it. (More on that later.) I was spurred into action. Thanks, Eno!

### HAA: Like VAA, but Not as Cool

Just as VAA captures the angle at which a pitch crosses home plate vertically, HAA captures the same horizontally. The pitcher’s release point (the beginning of the pitch’s trajectory), the pitch’s final location (its end), and the pitch’s velocity and acceleration vectors in all three dimensions ultimately determine the pitch’s movement through space. This movement through space includes the angles at which the pitch breaches the front plane of home plate.

vy_f = -sqrt(vy0² – (2 * ay * (y0yf)))
t = (vy_fvy0) / ay
vx_f = vx0 + (ax * t)
HAA = -arctan(vx_f/vy_f) * (180 / pi)

where, per Statcast’s documentation:

• vy0 = The velocity of the pitch, in feet per second, in y-dimension*, determined at y=50 feet. (*toward home plate)
• ay = The acceleration of the pitch, in feet per second per second, in y-dimension, determined at y=50 feet.
• y0 = 50 (“y=50 feet”).
• yf = 17/12 (home plate, converted to inches).
• vx0 = The velocity of the pitch, in feet per second, in x-dimension**, determined at y=50 feet. (**horizontally)
• ax = The acceleration of the pitch, in feet per second per second, in x-dimension, determined at y=50 feet.

Because HAA is a function of these other physical descriptors, it correlates with them to varying degrees — and, because of this, is also sort of obscured by them.

For VAA, I suggested adjusting it for vertical pitch location. VAA correlates so strongly with pitch height, it seemed almost meaningless to me to cite raw VAA numbers. A -4.0 degree VAA? Ok, but did the pitcher throw it at the top of the zone where VAAs of -4.0 degrees are commonplace, or did he throw it at the bottom of the zone where they almost always aren’t? Like all things baseball and sabermetric analysis, context is crucial.

Location is essentially the full extent of the adjustments needed for VAA (that, and pitch type, because approach angles vary so widely across pitch types). HAA is more complicated. Like VAA, we’ll adjust HAA for location, except it’ll be horizontal pitch location, not vertical. It’s also imperative to normalize for:

• Pitcher handedness: Horizontal movement patterns are unique to pitcher handedness, whereas vertical movement patterns are largely immune.
• Horizontal release point: A pitcher can vary his horizontal release point simply by shifting his starting position on the mound. If a pitcher throws two “identical” pitches to the exact same target from different ends of the rubber, they will naturally exhibit different HAAs. (Owen McGrattan discussed this when he researched HAA two years ago.) Unless a pitcher stands on a crate or digs a hole in the mound, he cannot alter his vertical release point in the same way without dramatically affecting his arm slot and mechanics (and all the pitch characteristics that result from them).

In Chris Gilligan’s deep dive on Pfaadt, Postseason Hero, he noted that after Pfaadt was optioned and recalled, pitching coach Brent Strom and pitching consultant Perry Husband worked with him to adjust his position on the rubber, moving from the third-base side to the first-base side. Husband (correctly) asserted that Pfaadt could change the appearance — and, thus, the performance — of his pitches without fundamentally changing their shapes. Husband preached tunneling and effective velocity. HAA does not address those topics, at least not directly, but it rests adjacent to them: by shifting toward the first-base side of the rubber, Pfaadt augmented his HAAs and his performance.

### Horizontal Approach Angle Above Average (HAAAA)

Just as I adjusted VAA for location to create VAAAA, I have adjusted HAA for location, handedness, and release point within each Statcast pitch type classification to create Horizontal Approach Angle Above Average, or “HAAAA.” Would “Adjusted HAA” have been better? Maybe. Is HAAAA spectacularly uglier? Sure. Is it fun to scream at the top of your lungs? Absolutely.

Every pitch type will have a different league-average HAA value (sliders and cutters have much sharper angles on average than sinkers or changeups), but they’ll all have HAAAAs that center on zero, with “plus” or “minus” values to indicate above or below average for that pitch type. To wit: listed below are five pitchers whose average HAAs are nearly identical, both with one another and to the 2023 league average (1.3 degrees, per my Pitch Leaderboard). Looking at HAA alone, one might misconstrue their four-seamers’ HAAs as identical to one another and/or as “average” on merit.

2023 Four-Seamer HAA(AA)
Name HAA HAAAA
Charlie Morton 1.3° -0.49°
Kevin Gausman 1.3° -0.21°
Gerrit Cole 1.3° -0.09°
Cristian Javier 1.3° +0.15°
Dylan Cease 1.3° +0.31°

As you can see, in adjusting for the unique combinations of release points and locations (not on average but for all of the four-seamers they threw in 2023), HAAAA depicts these five offerings as having significantly different approach angle profiles.

### Let’s Look at Four-Seamers First

Before we proceed, please acquaint yourself with these legends. These illustrations are presented from the catcher’s perspective.

Because we are working with two laterally-oriented metrics — horizontal location and HAA — we are burdened with forcing one of them to conform to the vertical Y-axis. HAA drew the short straw. The farther up the Y-axis, the sharper the HAA toward left-handed hitters; the farther down, the sharper toward righties.

The third legend below synthesizes the first two. Honestly, I can’t tell you how many times I’ve had to refer back to this. It’s turning my brain to mush. I should have it tattooed on my forearm, or maybe on the insides of my eyelids.

As far as color-coding is concerned, red means hot and blue means cold, if you can believe it. The metrics that underpin these heat maps account for ball-strike count. Swing rates on 0-0 are vastly different from swing rates with two strikes, for example. Pitchers inadvertently create selection bias by using different pitches (with varying degrees of quality) in different situations. It’s important to ensure we compare apples to apples. The intensity of the heat represents the magnitude of each metric above or below average, neutral of count.

OK!

For pitchers throwing four-seamers against hitters of the same handedness (i.e., righties versus righties), sharper approach angles toward the hitter correspond with higher swing rates on the inside edge and lower swing rates on the outside edge. As the approach angle shifts away from the hitter, swing rates drop inside and increase outside.

I see two trends:

1. Specifically at the edges of the zone, sharper HAAAAs beget more exaggerated plate discipline outcomes; and
2. Sharper HAAAAs afford themselves wider locational targets. For example, a four-seamer with -0.6 degree of HAAAA that is located an entire ball’s width off the inside edge of the plate will induce swings at a rate more or less commensurate with an average (+0.0 degrees) four-seamer directly on the black. That’s more swings on would-be balls (and fewer swings on would-be strikes) for pitchers with sharper approach angles.

Per MLB’s Mike Petriello, Strom said this of Pfaadt during his times of turbulence:

“Before, he was throwing balls out of his hand which became strikes,” Strom said in July. “Now we have pitches that are strikes that can become balls, which is what I was trying to achieve.”

This is good coaching! Irrespective of tunneling, effective velocity, and Strom’s/Husband’s other points of focus, Strom leverages HAA in a meaningful way. He understands the value of location and (again, correctly) implies that, by simply altering his release point vis-à-vis his position on the rubber, Pfaadt can augment his sweeper’s performance without changing his mechanics or the shape of his pitch(es).

In the absence of swings, we have non-swings that, depending on location, result in called striks. The combinations of approach angle and location that see fewer swings — sharper HAAAAs toward the hitter on the outer edge — naturally produce more called strikes. Presumably these sharper approach angles in these locations create the illusion of a pitch that’s on a trajectory outside the zone but cuts back in at the last second. Indeed, sometimes they actually are, and it’s the umpire who has been fooled:

(I have long thought this phenomenon, in addition to extreme VAAAAs at the top and bottom of the zone, has implications for catcher framing — a discussion for another day or a smarter mind than mine.)

In terms of contact on swings, the farther inside, the worse the hitter’s contact rate (better for the pitcher, in red), exhibiting a gentle trend in favor of sharp angles toward, inside over sharp angles away, outside:

Flattening swings and contact into one figure, we see general favoritism for sharp approach angles away from the hitter, even in vulnerable locations like the center of the zone:

Further flattening everything into called strikes plus whiffs (CSW), ultimately the best approach is to capture called strikes on the outer edge with pitches breaking sharply toward from the hitter. This is a convenient and somewhat intuitive outcome: VAA(AA) is a swinging-strike weapon, HAA(AA) is a called-strike weapon.

This, of course, focuses exclusively on plate discipline and neglects hitter contact quality, which is a vital component. When incorporating hitter contact quality, we see a few interesting trends: the locational margin of error on the outer edge is dramatically smaller, whereas it’s much wider inside. In fact, sharp approach angles toward the hitter that are located healthily inside are especially effective:

In other words, there’s a time and place for chasing strikes, especially called strikes, which come much more easily than swinging strikes — but hitter batted ball outcomes are markedly inferior away from the locations where those called strikes come easiest. At least for four-seamers, mostly we see that more intense HAAAAs do not exactly translate to superior performance. There are apparent plate discipline gains with sharper HAAAAs, but they are sort of rendered inert by the volatility of batted ball events.

Rather than looking at other pitch types individually, I merged them all into one heat map. We lose the granularity of each pitch type’s quirks and nuances but achieve a broad understanding of approach angles locationally — which, ultimately, are not distinctly different from four-seamers anyway. Here’s righties versus righties for all pitches:

As with four-seamers, the inner edge is king, especially for pitches with ample arm-side. There are exaggerated effects for ample glove-side cut off the outer edge, too, more so from this all-pitch perspective than for four-seamers alone. Being able to generate above-average outcomes on pitches outside the zone? That’s beneficial! That margin of error for sweepy pitches off the outer edge, however, is visibly smaller.

The heat map against left-handed hitters is effectively the same, just rotated 180 degrees:

As a bonus, to bring it back to Pfaadt one last time, here’s performance from righties against all hitters for sweepers and sliders. There may be some gentle bolstering effects from sharper HAAAAs, but location prevails first and foremost.

### TL;DR

The most important HAAAA effects are shown for approach angles breaking sharply toward the hitter on the outside edge (in terms of CSW and called strikes) and also on the inside edge (in terms of weak contact quality). Those gains cannot be realized without fairly well-executed location, although, like VAAAA, extreme HAAAAs make command less important. That, I think, would support the argument for capital-‘s’ Stuff over command. But location is crucial for HAAAA, arguably much more so than for VAAAA.

### Reminder

By design, this is an overly simplistic exercise. Binning and averaging, while plenty illuminating in its own right, is wildly primitive compared to regression analysis or machine learning. There are effects shown here that in a broader, more rigorous analysis might be shown to be over- or understated.

I must also reiterate that this analysis critically lacks the contextualizing benefits of many other pitch characteristics (Although HAA/AA does inherently account for velocity, acceleration, release point, and location. In its defense, that’s actually a lot of things.) Like any evaluation effort in baseball, an individual variable — in this case, HAAAA — should probably never serve as a sole metric for evaluating a pitch. Let it supplement or be supplemented.

### Making the Adjustments to HAA

HAAAA exhibits much smaller location-adjusted differences than VAAAA does. Whereas the the flattest (steepest) pitches can be up to 1.5 degrees or even 2.0 degrees flatter (steeper) than the average pitch in their class, the runniest/sweepiest pitches rarely surpass 0.5 degrees of excess HAA. It’s possible that if we adjust HAA by something more broadly defined, like pitch class (“fastballs”) instead of pitch type (“four-seamers,” “sinkers”), then our magnitudes of HAAAA would like increase. I’m not sure how much more clarity it would provide us.

VAAAA is largely a weapon for fastballs but has use cases for other pitch types. HAAAA may have more utility for breaking (and off-speed) pitches given their movement profiles. When I adjust HAAAA for horizontal location (in addition to pitcher handedness and horizontal release point), I do so by pitch type. I want to know how sharp a fastball’s approach angle is compared to other fastballs. That’s potentially not as helpful for breaking pitches, not just because breaking pitches vary so widely in their movement profiles but also because breaking pitches now adopt so many different names and terminologies. And frankly, that’s great. I support that effort completely. Call a pitch whatever you want to call it to best describe its movement traits. But that makes it difficult to compare, for example, sliders to sliders. Some of those sliders are called sweepers now. Some of those sliders are called sliders but are actually sweepers. Some of those sweepers are actually sliders. Some are slurves! And some are tiny hamburgers. Who even knows anymore. Taxonomy is a real issue, though, and it may constrain this analysis (in addition to affecting the “adjustments” made on the basis of the aforementioned pitch type classifications).

### The Stuff-Versus-Command Wars

Accentuated HAAAAs can often fail in bad locations where VAAAAs might thrive. Why?

To me, the explanation is fairly intuitive. A hitter’s swing has a much wider margin of error laterally than it does vertically. Horizontally, an inch off the barrel probably still results in contact, just inferior contact. In fact, with a miss of several inches, a hitter can still foul one off or even luck into a base hit. On the other hand, a one-inch miss vertically is, at best, a foul tip; at worse, it’s a straight-up whiff. And a whiff, on average, is unequivocally worse than inferior contact. This helps explain not only why elite VAAAAs (more than elite HAAAAs) play up in bad locations, but why elite VAAAAs play up better than elite HAAAAs generally.

Indeed, HAAAA remains dependent on location. There’s a chicken-and-egg situation here. Stuff is a better predictor of success, much to the chagrin of old-school types and innumerates. For HAAAA, though, stuff can’t exist without command to some extent. It’s only with some semblance of good location — the strike zone’s edges, specifically — that HAAAA can play up. Even then, location shoulders most of that load. But good HAAAA-related stuff broadens the pitch target, reduces the locational dependency, buys more wiggle room. That is, good HAAAA-related stuff elevates a command-oriented profile.

### Pitch Modeling

Before being hired to the Rockies’ R&D department, Ethan Moore wrote a seminal article on approach angles. He found in modeling pitch characteristics that HAA demonstrated exorbitant run prevention effects, although he admits “we have to take this with a grain of salt.” It’s possible that applying adjustments to HAA (beyond including location, release point, etc. as independent variables in the model) might yield a more reasonable (i.e., smaller) coefficient for HAA, especially as HAAAA yields a much narrower distribution of possible values.

### Lastly…

I apologize if I missed (and subsequently did not cite) your research on HAA. A cursory Google search of “horizontal approach angle” turns up maybe half a page worth of posts before devolving back into posts about vertical approach angle. I’ve seen folks from Driveline and Tread evoke VAA and HAA on Twitter, but no research came up. If I’ve missed anything, please feel free to drop it in the comments.

Editor’s note: Subsequent to publication, the graphics in this piece were updated to correct an earlier error.

Two-time FSWA award winner, including 2018 Baseball Writer of the Year, and 8-time award finalist. Featured in Lindy's magazine (2018, 2019), Rotowire magazine (2021), and Baseball Prospectus (2022, 2023). Biased toward a nicely rolled baseball pant.