Vertical Swing Plane (β_plane) and Path Shift Math
A rigorous mathematical and vector-geometric derivation of how the Vertical Swing Plane angle modulates the relationship between Attack Angle and horizontal Club Path shift.
geometry · 8 min · EXACT_GEOMETRY
Coordinate convention
This article uses the lab's right-handed frame: $+X$ points right of target, $+Y$ points down the target line, and $+Z$ points up. Angles are measured from $+Y$ toward $+X$. This matters: swapping the first two horizontal components changes the plane being described.
Let $\theta_{\text{dir}}$ be the horizontal swing-direction angle and $\beta$ the plane elevation above the ground. A unit normal for that plane in this frame is:
$\hat n_{\text{plane}} = [\cos\theta_{\text{dir}}\sin\beta, -\sin\theta_{\text{dir}}\sin\beta, \cos\beta]$
The clubhead direction at impact, using path angle $\theta_{\text{path}}$ and attack angle $\gamma$, is:
$\hat v = [\cos\gamma\sin\theta_{\text{path}}, \cos\gamma\cos\theta_{\text{path}}, \sin\gamma]$
Exact circle-plane relation
For an ideal planar swing, the velocity lies in the plane, so its dot product with the plane normal is zero:
$\hat v \cdot \hat n_{\text{plane}} = 0$
Substitution gives:
$\cos\gamma\sin\beta\sin(\theta_{\text{path}}-\theta_{\text{dir}}) + \sin\gamma\cos\beta = 0$
Therefore:
$\sin(\Delta\theta_{\text{path}}) = -\tan\gamma\cot\beta$
On the local physical branch around zero, the shift is the corresponding arcsine. For small angles, $\Delta\theta_{\text{path}} \approx -\gamma\cot\beta$ when angles are expressed in radians. This result follows from the stated frame and ideal planar-circle assumptions.
What the app computes
When path is coupled to AoA, the app solves the forward branch of the exact fixed-plane tangent relation:
$\theta_{\text{path}} = \theta_{\text{dir}} - h\arcsin(\tan\gamma/\tan\beta)$
Here $h=+1$ for RH and $h=-1$ for LH. A solution exists only when $|\tan\gamma/\tan\beta|\le1$. With independent path and AoA, both remain authored inputs; the circular arc preserves that delivery and reports its angular mismatch from the requested plane. The derived path-shift readout reports the actual path minus swing direction. Do not interpret a free delivery as if it satisfied the requested plane.