Spiral bevel gears transmit rotation and torque between intersecting shafts. Their curved teeth engage gradually, so several tooth pairs can share load and motion is smoother than in straight bevel gears. The same property makes the contact pattern a sensitive indicator of mounting distance, shaft alignment, elastic deflection, and wear. In heavy-duty drives, a contact trace that migrates toward the tooth edge or becomes excessively narrow can precede pitting, scoring, or tooth fracture. Visual image tracking offers a non-contact way to follow the trace, but the trace is difficult to capture: at high rotational speed and high meshing density it is short-lived, low in contrast on a machined and oiled surface, and partly hidden by the approaching teeth.
What the Contact-Area Trace Represents
The contact-area trace is the locus of the instantaneous contact zone on the tooth flank. Under load, the theoretical point contact of a bevel gear pair becomes an elliptical contact patch; the trace is the centerline or footprint of that patch as it moves across the flank. Its position, length, and orientation are not arbitrary. They are set by the generating geometry, the mounting position, and the relative motion of the pinion and wheel. A trace near the center of the flank indicates a comparatively even load distribution. A trace biased toward the toe or heel indicates a mounting offset or a shaft-angle error. A trace that spreads to the edges concentrates stress and raises the risk of early failure.
Spiral bevel gears have concave and convex flanks. The two flanks do not produce the same visual signature. On the concave side, the contact region tends to be broader and the trace can occupy a larger area; on the convex side, the contact region contracts toward a smaller loaded zone. As a result, convex-flank traces are often shorter, more closely spaced, and harder to segment. A tracking method must therefore handle both a broad, scattered trace and a narrow, concentrated one without changing its decision criteria.
Geometric Model and Meshing Condition
Direct image search is inefficient because most of the tooth surface is not in contact at a given instant. A geometric model supplies a prior: it predicts where contact is possible and rejects regions that cannot contain a trace. The tooth flank can be parameterized by two surface coordinates \(u\) and \(\theta\), and the generating surface can be mapped into the gear coordinate system through a chain of homogeneous transformations:
$$
\mathbf{r}_c(u,\theta)=\mathbf{M}_{eb}\mathbf{M}_{ba}\mathbf{M}_{a1}\mathbf{r}_1(u,\theta,\phi_1)
$$
$$
\mathbf{r}_g(u,\theta)=\mathbf{M}_{gf}\mathbf{M}_{fe}\mathbf{M}_{ed}\mathbf{M}_{de}\mathbf{r}_c(u,\theta,\phi_1)
$$
Here \(\mathbf{r}_1\) is the vector equation of the contacting flank, \(\mathbf{r}_c\) is the corresponding surface in the cutter or generating coordinate system, and \(\mathbf{r}_g\) is the surface after transformation to the gear coordinate system. The matrices \(\mathbf{M}\) represent the successive translations and rotations between those systems. The unit normal follows from the partial derivatives:
$$
\mathbf{n}(u,\theta)=\frac{\mathbf{r}_u\times \mathbf{r}_\theta}{\|\mathbf{r}_u\times \mathbf{r}_\theta\|}
$$
Contact requires that the relative velocity of the two surfaces have no component along the common normal:
$$
f(u,\theta,\phi_1)=\mathbf{v}_{g1}\cdot \mathbf{n}=0
$$
where \(\mathbf{v}_{g1}\) is the relative velocity. The function \(f\) is the meshing condition. When \(f=0\), the surfaces are tangent and a contact point can exist; when \(f\neq 0\), the surfaces are separated or interpenetrating in the model. In the tracking pipeline, this condition acts as a gate. Only image regions that satisfy the predicted meshing condition are treated as trace candidates, which reduces false detections from scratches, oil marks, and illumination edges.
Segmenting the Trace from the Tooth Surface
Once the model has defined the search region, the image is segmented into background and trace classes. A fixed gray-level threshold is unreliable because the reflectance of a machined flank changes with lighting angle, oil film, and surface finish. The threshold is therefore selected from the image histogram. For a gray range \(0\) to \(Q_2\), a threshold \(T_2\) divides the pixels into a background class \(0\) to \(K_n\) and a target class \(K_n+1\) to \(Q_2\). The binary decision can be written as
$$
B(x,y)=
\begin{cases}
1, & \xi(x,y)\ge T_2,\\
0, & \xi(x,y)<t_2, $$=""
where \(\xi(x,y)\) is the gray level at pixel \((x,y)\) and \(B\) is the resulting mask. The threshold is chosen to maximize the between-class variance,
$$
\sigma_B^2(T)=\omega_0(T)\omega_1(T)\left[\mu_0(T)-\mu_1(T)\right]^2,
$$
where \(\omega_0\) and \(\omega_1\) are the proportions of pixels in the background and target classes, and \(\mu_0\) and \(\mu_1\) are their mean gray levels. This is the standard Otsu criterion: the threshold that best separates the two populations is the one that maximizes \(\sigma_B^2\). The resulting mask preserves the trace shape and local detail, which is important because the subsequent tracking step uses ridge orientation rather than a single centroid.
Ridge-Line Orientation and Trace Tracking
The trace itself can be faint, but its edge often lies along a contact ridge line on the flank, which remains more visible than the interior of the contact patch. The tracking method uses this ridge as a geometric reference. Directional convolution is applied along two orthogonal directions \(X\) and \(Y\), with responses \(S_X\) and \(S_Y\). The local orientation of the ridge is
$$
\theta_1=\arctan\left(\frac{S_Y}{S_X}\right).
$$
After edge extraction, the ridge points are grouped into line segments. For a contact point \(p(a_1,b_1)\), the model identifies a nearby ridge intersection and computes the ridge slope \(\zeta\). The slope is then compared with the predicted trace direction. A correlation index \(F(X)\) summarizes the comparison: \(F(X)=1\) means that the ridge and the trace coincide, so the intersection is accepted as the current trace location; \(0<f(x)<1\) \(f(x)="0\)" an="" and="" at="" correlation="" follows="" has="" is="" means="" no="" offset="" offset,="" p="" point="" point,="" position;="" positive="" rejected.
This slope-based comparison has two practical advantages. First, it converts a difficult appearance-matching problem into a local geometric test, which is less sensitive to brightness changes. Second, it provides a continuous sequence: as the gear rotates, the accepted ridge intersections form a track, and the difference between the measured position and the model prediction is used as an offset correction. Within a short rotation interval, the trace moves approximately linearly, so the sequence of accepted ridge intersections can be extrapolated from one frame to the next. The correction keeps the track stable when the trace is temporarily obscured or when the segmentation misses a low-contrast segment. The same rule is applied to concave and convex flanks, which avoids separate thresholds for the two sides.
Simulation Evaluation
The evaluation was simulation-based. A pinion and wheel pair was represented with the parameters in the table below, and images of the concave and convex flanks were generated for tracking tests. The gear pair had a 90° shaft angle, a 38° spiral angle, and a 3.5 module. The pinion had 14 teeth and a left-hand spiral; the wheel had 30 teeth and a right-hand spiral. The face width was 4 mm on the pinion and 16 mm on the wheel.
| Parameter | Pinion | Wheel |
|---|---|---|
| Number of teeth | 14 | 30 |
| Module | 3.5 | 3.5 |
| Spiral angle (°) | 38 | 38 |
| Face width (mm) | 4 | 16 |
| Shaft angle (°) | 90 | 90 |
| Hand | Left | Right |
On the concave flanks, the tracked traces were irregular and varied in length. This is consistent with the contact mechanics: at high rotational speed, the initial engagement can produce a relatively large meshing area, and the centerline of that area forms the trace. As rotation continues, the flank separates from the mating surface and the visible trace shortens. Because contact is re-established at successive positions, the concave trace appears as a scattered set of segments rather than a single continuous line. The trace range on the wheel was larger than on the pinion, which follows from the larger contact area of the wheel.
On the convex flanks, the traces were more regular and more closely spaced. The convex contact region contracts toward a small loaded zone, so the trace is shorter and more concentrated. The tracker followed this concentrated pattern, and the close spacing is the main source of difficulty in convex-flank images. The difference between the concave and convex results is not a defect of the method; it reflects the different contact morphology of the two flanks. Both patterns were tracked within the same ridge-comparison framework.
The feature-extraction stage was also compared with two alternative approaches: one based on installation adjustment of the gear set and one based on contact analysis of a straight bevel gear pair. Over a contact-point count range of approximately 20 to 100, the ridge-based method gave the highest and most stable extraction accuracy. The alternative methods were more sensitive to the number of contact points and showed a decline in accuracy as the data volume increased. The likely reason is that neither alternative used a detailed motion model of the spiral bevel gear pair, so their trace candidates were less constrained when the contact pattern changed.
Practical Interpretation and Remaining Work
The value of the approach is that it combines a physical contact model with a lightweight image pipeline. The model does not replace the camera; it narrows the search and supplies the expected trace orientation. The segmentation does not need to recover a full three-dimensional surface; it only needs to separate trace pixels from the surrounding flank. The ridge comparison then provides a local correction that keeps the track continuous. This division of labor keeps computation low enough for repeated inspection and for use as a pre-processing step in a condition-monitoring system.
The limitations are equally clear. The evaluation is simulation-based, so it does not capture all field effects: oil streaks, metallic reflections, vibration, and motion blur at production speed can all reduce segmentation quality. The method also depends on a visible ridge line; if the contact zone is completely hidden by the mating tooth, the model can predict where the trace should be but cannot confirm it from the image alone. A practical installation would need controlled illumination and a trigger synchronized with shaft rotation. Finally, the present pipeline tracks the trace but does not yet invert the track for kinematic parameters such as mounting error or transmission error. Linking trace motion to those parameters is the next technical step, and it would turn contact-pattern tracking from a visual inspection aid into a quantitative diagnostic tool for spiral bevel gears.
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