The Reference-Point Problem in Club Delivery Data
The Short Version
A rotating clubhead has a velocity field. Two points can have different velocities and travel directions, so a reported club path needs a reference point. They need not differ: points separated along the angular-velocity axis have the same velocity, and some velocity differences change speed without changing direction.
The correction is exact when the geometry and full angular velocity are known. It is not a universal number of degrees, and correcting a measurement convention does not itself change the ball’s flight.
The practical question is: which point, in which frame, at which instant? Those three details must agree before two delivery measurements can be compared.
One Head, Several Reference Points
Use a right-handed world frame: \(x\) along the target line, \(y\) up, and \(z\) right when looking along the target line. Define horizontal path as \(\psi=\operatorname{atan2}(v_z,v_x)\) and attack angle as
\[ a=\operatorname{atan2}\left(v_y,\sqrt{v_x^2+v_z^2}\right). \]
Positive path is toward the right in this frame; its usual in-to-out interpretation assumes a right-handed golfer delivering generally along \(+x\). Positive attack angle is upward.
Let \(Q\) be a named reference point and \(P\) the point whose velocity is required. For a rigid head,
\[ \mathbf v_P=\mathbf v_Q+\boldsymbol\omega\times\mathbf r, \qquad \mathbf r=\mathbf p_P-\mathbf p_Q. \]
All quantities must be expressed in the same frame at the same time. This relation is independent of whether a device uses radar, cameras, or another measurement principle. The choice of point and the accuracy of its estimated motion are separate questions.
A geometric centre, centre of mass, face centre, tracked marker, and actual impact point are different definitions. Their separations depend on the head and strike. Do not substitute an assumed “driver depth” for measured geometry, or assume that every device reports the same point.
The rigid relation applies within the head approximation. Applying it from the handle to the head additionally assumes a rigid shaft; shaft bending and torsion require their own relative motion.
Impact Timing Is Part of the Definition
First contact, maximum compression, and separation are distinct instants. A measurement reconstructed from samples before contact is also different from a directly measured quantity during contact.
If the head rotates while the ball deforms, its face normal changes. The impulse can alter head speed and angular velocity. A reference-point correction at one instant cannot also correct an undocumented time offset.
For each device or dataset, use its stated point, frame, filtering, and timing definition. The vendor reference and technology review are starting points for locating those definitions; their values should not be treated as universal conventions.
Direction Change and Speed Change Are Different Projections
Write \(\delta\mathbf v=\boldsymbol\omega\times\mathbf r\) and \(v=\|\mathbf v_Q\|>0\). For a small velocity change,
\[ \delta v_{\rm speed} =\widehat{\mathbf v}_Q^{\mathsf T}\delta\mathbf v +O(\|\delta\mathbf v\|^2/v), \]
whereas the unit direction changes by
\[ \delta\widehat{\mathbf v} =\frac{(I-\widehat{\mathbf v}_Q\widehat{\mathbf v}_Q^{\mathsf T}) \delta\mathbf v}{v} +O(\|\delta\mathbf v\|^2/v^2). \]
Thus the approximate three-dimensional angular separation is
\[ \Delta\theta\approx \frac{\|(I-\widehat{\mathbf v}_Q\widehat{\mathbf v}_Q^{\mathsf T}) (\boldsymbol\omega\times\mathbf r)\|}{v}. \]
The often quoted \(\omega d/v\) additionally requires the lever to be perpendicular to angular velocity and the resulting velocity correction to be perpendicular to travel. Horizontal path needs a further projection:
\[ \delta\psi\approx \frac{v_x\delta v_z-v_z\delta v_x}{v_x^2+v_z^2}. \]
The formula becomes ill-conditioned when horizontal velocity approaches zero because horizontal path itself becomes undefined.
A Reproducible Perpendicular Example
Assume \(\mathbf v_Q=53.6448\mathbf e_x\) m/s, corresponding to 120 mph, \(\mathbf r=0.035\mathbf e_x\) m, and \(\boldsymbol\omega=(2000\pi/180)\mathbf e_y\) rad/s. Then
\[ \delta\mathbf v=-1.22173\mathbf e_z\ {\rm m/s}. \]
The exact path change is
\[ \Delta\psi=-\arctan(1.22173/53.6448)\approx-1.305^\circ. \]
The speed increase is
\[ \sqrt{53.6448^2+1.22173^2}-53.6448 \approx0.0139\ {\rm m/s} \approx0.0311\ {\rm mph}. \]
This example has a first-order direction change and a second-order speed change. A different cross-product direction can produce a first-order speed change as well. Rotation is not intrinsically “direction only.”
These are assumed inputs, not population averages or a validation of any instrument. If a parameter is adjusted until the calculation matches a published offset, that is calibration to that offset. Independent validation requires a separate measurement.
Face-Angle Rate Is Not Total Angular Velocity
Let \(\mathbf n\) be the unit face normal, with horizontal face angle
\[ \phi=\operatorname{atan2}(n_z,n_x). \]
Rigid-body kinematics gives \(\dot{\mathbf n}=\boldsymbol\omega\times\mathbf n\). Hence
\[ \dot\phi= \frac{n_x(\boldsymbol\omega\times\mathbf n)_z -n_z(\boldsymbol\omega\times\mathbf n)_x} {n_x^2+n_z^2}. \]
This is an angle derivative, not generally \(\|\boldsymbol\omega\|\) or one fixed component of it. Near \(\mathbf n=\mathbf e_x\), the stated convention gives \(\dot\phi=-\omega_y\): positive vertical-axis rotation closes a right-facing normal toward the left.
Other quantities called closure rate answer different questions:
| Quantity | Required Definition |
|---|---|
| Handle twist velocity | Component of handle angular velocity along the shaft axis |
| Head angular velocity | Full vector describing head attitude change |
| Horizontal face-angle rate | Derivative \(\dot\phi\) of a projected face direction |
| Face-to-path rate | Derivative \(\dot\phi-\dot\psi\) in the same convention |
| Closure per distance | \(\dot\phi/v_Q\) along the travel of a specified point |
A relation such as \(\mathrm{HTV}\sin\lambda+\mathrm{SPV}\cos\lambda\) is a projection within a chosen two-component geometry. It is not a general identity converting measured handle twist into the derivative of face azimuth. The assumed axes, lie convention, omitted angular component, shaft flexibility, and meaning of “closure” must be stated.
What the Instantaneous Screw Axis Actually Gives
For nonzero \(\boldsymbol\omega\), rigid-body motion admits an instantaneous screw-axis description. The velocity of a point has a component parallel to that axis and a rotational component perpendicular to it.
Let \(v_\parallel=\mathbf v_Q\cdot\widehat{\boldsymbol\omega}\) be the axial velocity and \(R_{\rm ISA}\) the distance from \(Q\) to the instantaneous axis. Then
\[ \|\mathbf v_Q\|^2 =v_\parallel^2+\|\boldsymbol\omega\|^2R_{\rm ISA}^2, \]
so
\[ R_{\rm ISA}= \frac{\sqrt{\|\mathbf v_Q\|^2-v_\parallel^2}} {\|\boldsymbol\omega\|}. \]
Only when axial velocity is zero does \(R_{\rm ISA}=v_Q/\|\boldsymbol\omega\|\). Even then, replacing a full-vector angular speed with scalar closure rate is another assumption.
For example, an angular speed of 20 rad/s, radius 0.7 m, and axial speed 8 m/s produce total speed \(\sqrt{14^2+8^2}\approx16.1\) m/s. Dividing total speed by angular speed gives about 0.806 m, not 0.7 m.
Likewise, a path offset and reference-point separation alone cannot generally identify a screw axis. They give insufficient information about the vector geometry and axial motion. The underlying rigid-body theory is developed in Murray, Li, and Sastry, A Mathematical Introduction to Robotic Manipulation.
When Speed Normalisation Helps
If a motion is uniformly sped up while its geometry stays the same, both \(\mathbf v_Q\) and \(\boldsymbol\omega\) scale by the same positive factor. The two-point path difference then stays unchanged.
This is a conditional invariance, not a claim that all faster swings have the same offset. Changing tempo can also change posture, release, shaft deformation, and the angular-to-linear velocity ratio.
Closure per distance, \(\dot\phi/v_Q\), is useful when comparing the evolution of face angle along a specified path. It does not generally equal \(1/R_{\rm ISA}\).
Bias, Variability, and Adaptation
For a specified delivery, the two-point offset is deterministic. Across repeated swings it may have a nonzero mean, vary from shot to shot, or both. There is no mechanical rule making it purely dispersion rather than bias.
If the offset is a function \(g(\mathbf v_Q,\boldsymbol\omega,\mathbf r)\), its local variance is approximately
\[ \operatorname{Var}(g)\approx \nabla g^{\mathsf T}\Sigma\nabla g, \]
where \(\Sigma\) includes the covariances of the measured inputs. Correlated changes in velocity, angular velocity, and strike location matter. Treating all variation as independent “timing noise” can misrepresent the delivery.
A small time shift along a prescribed face-angle trace gives
\[ \delta\phi\approx\dot\phi\,\delta t. \]
At an assumed 2100 degrees/s, a 1 ms shift corresponds to 2.1 degrees. This is a local sensitivity calculation. It does not establish the size of a golfer’s timing variability, independence from other errors, or a resulting fairway percentage.
Cheetham’s 2014 study paired a captured driver swing with same-year tour statistics. It reported an HTV mean of 1307 degrees/s for 94 professionals; the accuracy subgroup had 70 players and correlation \(r=-0.14\). Accuracy was seasonal fairways hit, not dispersion measured for the captured swing. Cheetham, 2014, Study 1.
That observational result does not identify the effect of changing one golfer’s closure rate. Nor does failure to detect a group difference prove equivalence. Players may adapt their delivery, but adaptation and cancellation between mechanisms need their own evidence.
What a Coaching Target Must Specify
If a coach asks for zero path at \(Q\), the ball may encounter a different path at \(P\). That is a reason to define the target carefully and observe ball output. It does not by itself prove that the target causes a fade: face normal, strike, tangential contact, and spin also enter the collision.
When comparing two hypothetical deliveries, state what is held fixed. Holding head-centre velocity fixed while changing rotation changes contact velocity. Holding contact velocity fixed requires a compensating centre-velocity change. The rotation-induced spin companion derives why those experiments can produce different conclusions.
Using the Interactive Explorer
The Rate of Closure Impact Explorer is useful for exploring its chosen geometry. Treat its assumed depths, angular-rate decomposition, and example deliveries as inputs to a model.
Agreement between two implementations checks consistency of the code. It does not establish that the geometry represents a particular golfer, that an assumed rate is a measured head rate, or that the model predicts accuracy. For an empirical comparison, supply independently measured point locations and the full head motion, then compare corrected quantities at matched instants.
The useful outcome is a consistent chain: body and shaft motion determine head motion; head motion determines contact velocity; the collision determines ball launch. A reference convention should make that chain easier to test.
Citation
@online{olson2026,
author = {Olson, Dieter},
title = {The {Reference-Point} {Problem} in {Club} {Delivery} {Data}},
date = {2026-08-03},
url = {https://affinedrift.com/articles/reference-point-problem.html},
langid = {en}
}