Eulerian Prime Axes Approach
This simulator utilises 3D vector mathematics to model the true anatomical deviations of the bone, correcting the visual distortions inherently present in 2D measurements on a screen. A screen can be the one on which you are inspecting your digital radiographic image, the CT MPR reconstructions or the CAD software. It is a common misconception that we can perform 3D measurements on any of the above screens. Even if the software gives you the visual perception of a 3D body structure, the screen is in a 2D subspace of this 3D space. Any measurement on it is a 2D measurement.
In addition to that, a 3D visual representation of a body using, say, CAD software adds the element of perspective, which results in the distortion of reality. Imagine that you are looking at a right angle from above. Even if your eyes are straight on top of part of the lines forming the angle, there will always be parts of them that are far away from your eyes. Because this introduces a visual perspective, you can measure this angle as a non-right angle. The "camera" of the 3D software is your eyes in real life. Some software has the option for an orthographic visualisation which eliminates this distortion. The only way to perform 3D measurements is to utilise software that calculates distances and angles from the coordinates of bone landmarks in 3D space. Any other measurement is a "ghost" of our visual perception.
The purpose of this work is to guide bone deformity research in the right direction and disentangle it from 2D measurements that are currently seen as the holy grail of deformity correction. Also, it represents an effort to help the audience realise the difference between 3D and 2D measurements.
Enable the mathematical basis toggle in the controls to view these vectors:
In 3D mechanics, to describe the final orientation of a distal bone segment relative to a proximal segment, we cannot just apply one single rotation. We must apply three distinct, sequential rotations. These are known as Euler Angles.
Because these rotations happen sequentially, they create a strict parent-child hierarchy of coordinate frames. The notation uses "primes" (', '', ''') to track these frames:
A surgeon planning an osteotomy needs to know the absolute magnitude of the bend. If a bone has both Procurvatum and Valgus, it is bent in one single, diagonal 3D plane.
This highlights the exact convex wedge of bone that must be removed to straighten the anatomical axes. A pure closing wedge corrects angulation but leaves physical torsion unaffected.
This highlights a single plane and the exact rotation around its normal axis required to correct angulation AND torsion simultaneously in one movement.
By tracking the Eulerian hierarchy, we can mathematically break down the deformity into its true physical components and its optical projections:
When measuring this deformity on a 2D monitor, optical projections heavily distort the reality:
Abstract
Traditional quantification of long bone deformity has long been tethered to 2-dimensional (2D) orthogonal measurements, whether via plain film radiographs or Multi-Planar Reconstructions (MPR) in Computed Tomography (CT). While these methods provide essential clinical benchmarks, they carry an inherent mathematical limitation: they treat spatial planes as independent variables, ignoring the spatial coupling that occurs when an object is bent in 3D space. This report details a linear algebraic framework that characterises deformity as a single, compounded spatial rotation. By posing the distal bone segment as a rigid body relative to a proximal reference frame using intrinsic Eulerian sequences, we demonstrate how 3D hierarchical transformations expose the systemic projection distortions present in current clinical methodologies.
The core problem in classic 2D analysis is that it defines deformity relative to the observer's eye (the radiographic beam) rather than the bone's own geometry. To resolve this, we must first establish a canonical 3D coordinate frame anchored to the stable proximal segment.
We define a right-handed Cartesian system. To ensure mathematical validity, the medio-lateral (x), proximal-distal (y), and antero-posterior (z) axes must be mutually perpendicular. We verify this basis using the dot product, where the result must be zero for all pairs, and the cross product, which defines the third axis:
Fundamental to characterising a deformity is the definition of a "straight" baseline. This represents exactly where the distal anatomic axis would be if there was no deformity. Given that our proximal axis points upward, the reference distal axis (pointing downward) is defined as:
We must also establish a baseline for axial rotation. This vector represents the "torsion-free" orientation of the distal segment's width. In our 3D model, this serves as the static benchmark for measuring transverse plane projections.
To capture the geometric truth of the deformity, we treat it as a sequence of intrinsic rotations. This means each successive rotation occurs around the axes already moved by the previous step, establishing a strict parent-child hierarchy tracked via "primes" (X', Y'', Z''').
The total deformity matrix Rtot is the product of three elementary matrices. Because 3D rotations are non-commutative, the sequence of development matters. For an intrinsic X → Z' → Y'' sequence, the matrix is constructed via post-multiplication:
Before compounding the deformities, we define the fundamental 3D rotation matrices in their standard trigonometric form. Each matrix revolves the coordinate space around one of its principal axes by a given angle, utilising the cosine (cos) and sine (sin) functions:
By evaluating these trigonometric functions with patient-specific angular inputs, we translate clinical degrees into numerical linear algebraic arrays.
Consider a limb with a complex deformity sequence (X → Z' → Y''): 15° Procurvatum (φ), 15° Valgus (ψ), and 20° External Torsion (θ). Following mathematical conventions (Counter-Clockwise = Positive) and inserting these angles into the trigonometric matrices defined above, we generate the individual numerical rotational matrices:
Multiplying Rx · Rz · Ry yields:
Traditional radiographs cast shadows onto the flat walls and floor of the room. To find what a 2D radiograph "sees," we apply Rtot to the reference vectors to derive the actual spatial orientation.
We find the actual distal axis v by applying Rtot to u:
Using the atan2 function on the x and y components of v:
Apply Rtot to the reference medio-lateral axis xref, and project it onto the floor (the Global Transverse Plane). When a surgeon looks at the bone from top to bottom, this is what they see:
Notice how the inputted 20° of intrinsic true anatomical torsion appears as 16.4° on a flat CT slice due to the spatial coupling.
When a deformed bone has an internal, true 3D physical twist of θ degrees (in this example, 20°),in the presence of sagittal and frontal plane deformities, this twist projects onto the floor differently. We have named this projection the Projected twist, and it represents the real-life angle between the projection of the proximal and distal joint lines on the floor.
However, we usually correct the sagittal and frontal plane deformities with wedge osteoctomies before we correct torsion. So, from this deformity correction perspective, the calculated Projected twist may be a Ghost angle. What we are really interested in, is the projected angle after the sagittal and frontal plane deformities have been corrected. So, we need to check if this pre-operative Projected twist is indeed the same as the post-operative one. We can test that by first correcting the deformity using a wedge ostectomy and, then, measuring this angle again. It turns out that the projected twist changes after deformity correction. We can also notice that even if we set θ to 0°, φ to 15°, and ψ to 20°, after correcting the deformity with a wedge ostectomy, there is a Projected internal twist of 2.7°. This postoperative projected angle of twist must be the intrinsic bone twist induced in the bone by the sagittal and frontal plane deformities, which is called theWe calculate these in section 8.
The most robust clarification of our study is the reduction of these complex inductions into a single Axis-Angle (Single Cut) correction. Euler's Rotation Theorem proves that any complex rigid-body displacement with a fixed point can be expressed as a singular rotation around a unique axis.
Mathematically, every 3D rotation matrix possesses at least one eigenvector with an eigenvalue of +1. This means there is a specific vector (the rotational axis) that is completely unaffected by the transformation—it does not change its magnitude or direction. The spatial deformity can therefore be fully corrected by rotating the distal segment back around this exact axis.
To find the absolute magnitude of this single rotation, we rely on the Trace (the sum of the diagonal elements) of our total rotation matrix Rtot. A fundamental property of linear algebra is that the trace of a matrix is invariant under any change of basis.
If we were to theoretically align our coordinate system exactly with the true axis of rotation, the 3D rotation matrix simplifies to a standard 2D rotation in the remaining two dimensions, taking the form:
The trace of this simplified matrix is 1 + cos(Θ) + cos(Θ), which resolves to exactly 1 + 2cos(Θ). Because the trace is invariant regardless of the basis used, Euler's theorem establishes the following universal relationship for our calculated Rtot:
For our example matrix above, the trace is 0.908 + 0.933 + 0.931 = 2.772. This yields Θ ≈ 27.6°. This is the absolute spatial magnitude of the deformity.
To find the unique axis of rotation in 3D space, we must extract the anti-symmetric components of Rtot. By subtracting the transpose of the matrix from itself (R - RT), the symmetric diagonal components strictly cancel out. This isolates a skew-symmetric matrix that is directly proportional to the cross-product matrix of the rotational axis, scaled by 2sin(Θ).
Therefore, we can explicitly extract the individual x, y, and z components of the unit vector urot from the off-diagonal elements:
By rotating the bone by exactly -Θ around urot, the surgeon corrects procurvatum, valgus, and torsion in one mathematically precise gesture.
While urot untwists the entire 3D displacement, a closing wedge osteotomy only corrects the angulation between the proximal and distal anatomical axes.
The true spatial trajectory of the distal segment is represented by v = Rtot · u. The absolute magnitude of this angulation (the wedge angle) is found via the dot product:
The axis of rotation required to close this wedge is the vector perpendicular to the plane of maximum deformity (the plane containing both u and v). We find this using the cross product:
Rotating the distal segment about nwedge by -θw perfectly aligns v back to u. However, because this rotation ignores the transverse components of the matrix, it leaves behind the implicit torsion generated by the extrinsic sequence path.
To explicitly calculate the Codman-Induced twist which is the Projected Residual Twist (a') that remains after the wedge is physically closed, we mathematically simulate the osteotomy.
First, we construct the rotation matrix Rclose that rotates the distal segment around the nwedge axis by -θw. This is generated using Rodrigues' Rotation Formula, where I is the identity matrix and K is the skew-symmetric cross-product matrix of nwedge:
We apply this corrective rotation to our totally deformed bone matrix to find the final, post-wedge position of the bone:
In this new Rfinal state, the angulation is completely corrected, and the bone's longitudinal axis is perfectly re-aligned with the proximal baseline (u). We apply this final matrix to our reference lateral axis to find its post-wedge orientation:
Now that we know the orientation of the distal joint axis xfinal, we can calculate its angle between it and its original position xref. Since xfinal is the position of the joint axis after the wedge ostectomy, its orientation must be in the transverse plane. Similarly, xref is in the same plane. So, the angle between them is the remaining projected twist in this plane.
To find the angle between xref and xfinal in the transverse plane, why not just use a simple Dot Product?
While taking the inverse cosine (arccos) of a dot product gives the absolute magnitude of the angle between two vectors, it inherently loses the direction (sign) of the rotation. It will always return a positive value. Clinically, we must distinguish between internal and external torsion. To resolve this, we use a 3D trigonometric projection that calculates both the sine and cosine to establish the exact signed quadrant.
While the mathematics are robust, they cannot be performed in a standard DICOM viewer. 2D MPR slices are merely projections. To apply this logic, CT data must be imported into CAD software where bone segments are treated as independent rigid bodies.
By transitioning from trigonometric shadows to spatial transformations, we resolve the "cross-talk" errors that have historically complicated deformity analysis. Accurate quantification requires 3D rigid-body analysis, resolving multiple planes into a singular, mathematically precise axis of correction.
Clinicians rarely measure the 3D Total Intrinsic twist directly. Instead, they measure 2D projections on a screen. The true 2D footprint of the bone on the floor (Orthographic Projection) is entirely different from what a camera lens sees when looking at a tilted object (Perspective Distortion).
To illustrate this, we will follow a specific Eulerian case through the next sections: 15° Procurvatum, 15° Valgus, and 20° External Torsion.
This is the mathematical orthographic shadow on the transverse floor. It assumes parallel X-ray beams with no lens distortion. Using our atan2 algorithm on this 15°/15°/20° case yields the mathematical baseline:
Standard 3D software mimics the human eye. As a bone tilts away, the lens physically stretches and foreshortens the geometry to simulate depth, warping the visual angles on the monitor, which we must now calculate.
It is geometrically invalid to measure clinical angles visually on a 2D monitor using 3D rendering software unless the camera is perfectly orthogonal. Standard software mimics the human eye using a Perspective Matrix, which injects false torsion into visual measurements.
To mathematically map 3D space onto a 2D screen, we must perform translations (moving the camera). However, standard 3x3 matrices can only rotate and scale objects; they cannot translate them. To solve this, linear algebra employs Homogeneous Coordinates.
We append a 4th dimension, W = 1, to all our 3D vectors. This allows us to use 4x4 matrices, enabling complex translations while remaining mathematically reversible. No spatial data is destroyed during matrix multiplication.
The View Matrix mathematically translates and rotates the entire world so the camera lens acts as the absolute origin (0,0,0) looking straight down the -Z axis. If we position our virtual camera exactly overhead at (0, 30, 0) looking at the origin, the resulting 4x4 matrix is:
Notice how the Z-coordinates are now ~35 units away down the negative Z-axis of the lens.
This matrix prepares the geometry for the monitor. It uses the Field of View (e.g., 45°) to crush the 3D space into a 4-dimensional array called Clip Space. The brilliant mathematical trick of this matrix is that it copies the spatial depth (the Z-distance from the lens) and stores it into the W coordinate.
Up until now, the 4D matrices are completely reversible. The Perspective Divide is where the true flattening (and the illusion) occurs. The graphics engine divides the X, Y, and Z coordinates by the W depth coordinate to crush the geometry into 2D Normalised Device Coordinates (NDC).
The Illusion: Because the camera is at Y=30, the proximal cap is closer to the lens (W=24) than the distal cap (W ≈ 35.5). Dividing the distal coordinates by a larger depth value violently shrinks and warps the distal red line out of parallel alignment, simulating visual depth.
We now treat the red lines as flat 2D vectors directly on the glass of the monitor:
Applying the 2D atan2 function reveals the Apparent Screen Angle (the exact value a plastic protractor placed against the screen would measure):
Conclusion: In Section 10, we proved the True Orthographic Projected Twist on the floor is mathematically 16.4°. Yet, simply because we viewed the 3D model through a virtual camera lens overhead, the on-screen protractor reads 16.9°. The 0.5° discrepancy is the Camera Error.
The left viewport shows the Orthographic top-down mathematical reality. The right shows the Perspective camera view. Drag the right camera to see the Screen Angle falsely fluctuate while the true Projected Twist remains locked. Trying to make the perspective image look like the orthographic image will end up with a screen angle that does not match the actual projection angle.
A major flaw in reading raw 2D pixel rotation is that orbiting the camera can make an "Internal" rotation suddenly appear clockwise or counter-clockwise depending on the view. If we trust the 2D screen sign, the measurement will randomly flip.
To solve this, we use the Anatomical Sign Referee. We strip the mathematical sign from the 12.5° 2D Screen Angle and anchor it to the sign of the true 3D physical state (Total Intrinsic Twist). Because our example's Total Intrinsic Twist is 18.0° External, we force the Screen Angle to report 12.5° External.
Clinicians frequently measure intrinsic torsion in CAD software or an MPR (Multiplanar Reconstruction) viewer by establishing a proximal reference line on the screen, rotating the 3D volume to view the distal bone straight-on, and measuring the angle against a distal reference line. We must mathematically prove that this visual "Glass Screen" technique is identical to the surgical Closing Wedge Osteotomy.
In surgery, the proximal bone is clamped to the table. The distal segment (v) is physically hinged back to align with the proximal segment (u). The required rotational matrix is Rclose.
The total intrinsic twist is the angle measured between the original proximal reference (xref) and the newly aligned distal reference (Rclose · xfinal).
In MPR, the proximal reference xref is drawn and frozen on the monitor "glass." The user then rotates the entire 3D CT volume (matrix Rvol) until the distal segment looks straight into the camera (which is the u axis).
Because Rvol follows the exact same shortest-path arc to map v to u, Rvol is mathematically identical to Rclose.
Consider a deformity comprising a 40° Wedge Angulation and exactly 25.0° Total Intrinsic Twist. We apply the MPR methodology to extract the twist.
The Conclusion: Measuring the angle between the frozen line on the glass (xglass) and the rotated distal line (xnew) yields exactly 25.0°. The Kinematic Inversion proves that rotating a volume behind a fixed screen is mathematically indistinguishable from surgically bending a bone.
To demonstrate the scalability of the Eulerian framework, we evaluate a biapical deformity of the radius. The radiologist is viewing the bone from proximal to distal (looking down the shaft towards the carpus).
Because the deformity has two distinct Centres of Rotation of Angulation (CORAs), it generates two independent Eulerian rotation matrices using the X → Z' → Y'' intrinsic sequence:
For Apex 1 (Rprox):
| 1 | 0 | 0 |
| 0 | cos(15°) | -sin(15°) |
| 0 | sin(15°) | cos(15°) |
| cos(15°) | -sin(15°) | 0 |
| sin(15°) | cos(15°) | 0 |
| 0 | 0 | 1 |
| cos(20°) | 0 | sin(20°) |
| 0 | 1 | 0 |
| -sin(20°) | 0 | cos(20°) |
Resolving the trigonometry:
| 1 | 0 | 0 |
| 0 | 0.966 | -0.259 |
| 0 | 0.259 | 0.966 |
| 0.966 | -0.259 | 0 |
| 0.259 | 0.966 | 0 |
| 0 | 0 | 1 |
| 0.940 | 0 | 0.342 |
| 0 | 1 | 0 |
| -0.342 | 0 | 0.940 |
| 0.908 | -0.259 | 0.330 |
| 0.323 | 0.933 | -0.158 |
| -0.267 | 0.250 | 0.931 |
For Apex 2 (Rdist):
Following the identical trigonometric expansion for 18°, 20°, and 16°:
| 0.903 | -0.342 | 0.259 |
| 0.398 | 0.894 | -0.207 |
| -0.161 | 0.290 | 0.943 |
To correct the radius, two separate osteotomies are required. The Eulerian formulas remain identical; they are simply applied independently. The Single Cut magnitude is derived purely from the trace of the respective local matrix:
It is a common clinical error to measure the projected twist of the proximal apex, measure the distal apex, and simply add them together. In 3D space, this is mathematically invalid.
Because matrix multiplication is non-commutative, the first deformity physically tilts the coordinate system for the second. Bending and twisting an already-tilted axis casts a completely different 2D shadow than bending a neutral axis.
To find the final carpal orientation relative to the proximal shaft, we multiply the matrices:
Rglobal = Rprox · Rdist
| 0.908 | -0.259 | 0.330 |
| 0.323 | 0.933 | -0.158 |
| -0.267 | 0.250 | 0.931 |
| 0.903 | -0.342 | 0.259 |
| 0.398 | 0.894 | -0.207 |
| -0.161 | 0.290 | 0.943 |
| 0.664 | -0.446 | 0.600 |
| 0.689 | 0.677 | -0.258 |
| -0.291 | 0.585 | 0.757 |
We isolate the geometric twist forced by the sweeps by zeroing the commanded torsion (θ = 0°) at both apices to create "Ghost" matrices:
| 0.966 | -0.259 | 0 |
| 0.250 | 0.933 | -0.259 |
| 0.067 | 0.250 | 0.966 |
| 0.940 | -0.342 | 0 |
| 0.325 | 0.894 | -0.309 |
| 0.106 | 0.290 | 0.951 |
| 0.823 | -0.562 | 0.080 |
| 0.511 | 0.673 | -0.534 |
| 0.246 | 0.481 | 0.841 |
Extracting the lateral X axis (xghost) from column 1 and projecting onto the floor (zeroing Y):
We find the true physical twist by comparing Rglobal to a shortest-path pure bend (Qalign). The final distal axis (Y) from Rglobal is yfinal = [-0.446, 0.677, 0.585]T. Creating the alignment quaternion to this vector and extracting the residual torsion yields the physical truth:
We extract the true final lateral X axis from Rglobal and flatten it onto the global transverse plane:
Assuming a camera position C = [5, 10, 5], we map xfinal to the 2D monitor using the View-Projection matrix (VP). If the raw 2D pixel maths returns a magnitude of 31.5°, the Sign Referee checks the Total Intrinsic Twist (32.1° External) to force the sign:
The cumulative optical distortion injected by the perspective lens:
In Section 13, we proved that Multi-Planar Reconstruction (MPR) neutralizes optical distortion by rotating the viewing plane to be exactly orthogonal to a deformity's True Oblique Plane. A critical clinical question arises: can a single global MPR plane (aligning the proximal Segment 1 and distal Segment 3) be used to correct a biapical deformity?
Mathematically, a 3D viewport can always be rotated to find a single plane where Segments 1 and 3 appear parallel. However, attempting to correct a biapical deformity based on this single global view results in two failures:
If a surgeon executes a single wedge osteotomy to align Segment 1 and Segment 3, they are treating the bone as uniapical. Because the intermediate Segment 2 is physically angulated off the mechanical axis, forcing Segment 3 to align with Segment 1 via a single cut will translate Segment 2 entirely outside the leg's weight-bearing axis. The angulation is cured, but a massive translation deformity is created.
It is mathematically impossible to divide the global torsion found in a 1-to-3 MPR into the two individual apices. As proved in Section 8, when Segment 2 is angulated, it acts as a mechanical crank handle. As it swings through a diagonal 3D arc, it physically generates geometric Codman torsion. Therefore, the global intrinsic torsion (Tglobal) is non-linear:
Conclusion: Because every apex possesses its own independent True Oblique Plane, and the intermediate segment generates hidden Codman torsion, a single global MPR is clinically invalid. To accurately restore the mechanical axis without inducing translation, a clinician must perform two independent local MPR alignments, measuring and correcting the torsion at each apex separately.
To fully understand the clinical impact of optical distortions, we must model how the Projected Twist (b) and Screen Angle (S) behave as continuous mathematical functions of the true physical state of the bone, the Total Intrinsic Angle (T).
We established in Section 8 that the true physical twist embedded in the bone ($T$) is the sum of the user-commanded torsion (θ) and the geometric Codman-induced torsion (a'). Therefore:
To plot our optical projections strictly as a function of the true physical twist, we replace all instances of θ in our Eulerian expansions with (T - a').
As derived in Section 5.2, the orthographic projection of the lateral axis on the X-Z floor is found by applying Rtot to xref = [1, 0, 0]T. Expanding the trigonometric sequence yields the 3D coordinates of the new lateral axis:
To find the clinical external torsion angle, we use the 2D atan2 function. Since Anterior (+Z) correlates to Internal Rotation and Posterior (-Z) to External Rotation, the mathematical projection evaluates as atan2(-Znew, Xnew). Substituting θ = T - a' yields the final analytical orthographic function:
The Screen Angle ($S$) passes the 3D coordinates through the camera matrices to get Normalised Device Coordinates (NDC). However, to calculate the true angle, we must cancel out the aspect ratio of the monitor by multiplying the NDC coordinates by the physical Width and Height of the pixel grid:
Because perspective depth (Wclip) scales non-linearly with physical translation, S(T) injects a continuous sinusoidal harmonic error (Camera Error) directly on top of the orthographic projection.
Adjust the angulation below to see how the camera error (the gap between the green and white lines) violently fluctuates across the entire spectrum of intrinsic torsion.
To fully understand the clinical impact of optical distortions in complex trauma, we must model the Total Projected Twist (btot) and Total Screen Angle (Stot) as continuous mathematical functions of the physical Total Intrinsic Angle (Ttot) across a Biapical deformity.
In a biapical deformity, the Total Intrinsic Torsion is the geometric sum of the torsions embedded at both apices, combined with their respective Codman-induced twists. We define a continuous total commanded torsion ($\theta_{tot}$) distributed across Apex 1 and Apex 2 based on a mechanical ratio ($k$).
By sweeping $\theta_{tot}$ computationally, we isolate the true physical $T_{tot}$ and plot the resulting optical illusions strictly as a function of this final physical twist.
The Screen Angle ($S_{tot}$) is uniquely volatile in biapical deformities. Because the angulation at Apex 1 physically translates the intermediate bone segment away from the mechanical axis, Apex 2 is pushed further toward the margins of the camera's perspective lens. We must multiply the local coordinate vector by both rotational matrices and the intermediate translation matrix ($M_{trans}$) before applying the View ($V$) and Projection ($P$) matrices:
Adjust the angulations for both apices below. Notice how the Camera Error (the gap between the green and white lines) becomes highly asymmetric due to the spatial translation of the intermediate bone segment under the perspective lens. The domain is restricted to ±45° to focus on clinically survivable torsion limits.
Throughout this report, we have visualised deformities as 2D intersections. However, the Codman-induced twist (a') is fundamentally a continuous, multivariable geometric phenomenon. It is generated by the simultaneous interplay of both the Sagittal (φ) and Frontal (ψ) planes.
In previous sections, the physical deformity was represented by the matrix Rtot, which includes the surgeon's commanded intrinsic torsion (θ). To map the pure geometric Codman effect without contamination, we must create a "Ghost Matrix" (Rghost).
Rghost is exactly Rtot, but with the θ parameter forcefully muted (set to 0°). This represents a bone that has been bent in two planes but expressly forbidden from twisting on its own axis. Because we use column vectors, the rotation matrices are applied from right to left (Sagittal X, then Frontal Z, then Torsion Y):
To find the hidden twist, we mathematically "straighten" this ghost deformity using a single wedge osteotomy and measure what happens to the bone's lateral axis. The derivation requires 5 steps:
Plotting this continuous function, f(φ, ψ), across all clinical angulations generates the interactive Codman Surface below.
Use the sliders below to dial in a specific deformity, or left-click and drag to orbit the 3D surface. Hovering your mouse over the landscape will dynamically update the sliders and reveal the exact geometric torsion embedded at that coordinate.
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