Circular Current Loops

Axial field of a circular loop

Consider a circular loop of radius $a$ in the xy plane, carrying steady current $I$. The loop centre is the origin and the observation point lies on the z axis at coordinate $z$. Every current element is the same distance from that point,

Equal source-to-observer separation around the loop makes the axial Biot–Savart integral compact. The current element is tangent to the circle and the separation vector from the element to the observation point lies in a radial-axial plane. They are perpendicular, so the magnitude of the differential contribution is

The contribution is not wholly axial. Its component along the loop axis is obtained from the geometry of the separation triangle. The axial projection factor is $a/r$, giving

Every loop element has the same axial projection and direction at an axial point. Integrating $\mathbf{d}\ell$ around the circumference gives

Applying the right-hand rule to the complete current loop gives curl fingers with current and the thumb gives the positive loop-axis direction. A current reversal reverses the sign of $B_z$ but not its magnitude profile. The formula applies to a thin circular path with steady current; a coil of finite wire thickness or noncircular shape requires integration over its actual source geometry.

Carrying the axial integral through

Parameterise the loop by the azimuth $\varphi$. A source element at $\varphi$ has position $a\cos \varphi \hat{x}+a\sin \varphi \hat{y}$, and its directed length is $\mathbf{d}\vec{\ell }=a(-\sin \varphi \hat{x}+\cos \varphi \hat{y})\mathbf{d}\varphi$. An observation point at $z\hat{z}$ has source-to-observation vector

At every azimuth, $\mathbf{d}\vec{\ell }$ is perpendicular to $\vec{r}$. Its magnitude is $a\mathbf{d}\varphi r$, with no varying sine factor. The differential field magnitude is ${\mu }_{0}Ia\mathbf{d}\varphi /(4\pi r^2)$. Its direction is not parallel to the axis, however. Resolving that direction gives an axial factor $a/r$, so that

The scalar integrand no longer contains $\varphi$. Circular symmetry makes every axial contribution equal. Integration over one full turn yields

Keeping the vector step visible prevents a common mistake: integrating the full $\mathbf{d}B$ magnitude as though it were already axial. That would omit one factor of $a/r$ and produces the wrong distance dependence. A second check comes from the integral itself. The circumference contributes the factor $2\pi$; it cancels the $4\pi$ in Biot–Savart only after the axial projection has been included.

Field-map analysis and calibration

An axial map is more informative than one reading at the centre. Set the loop in a nonmagnetic holder and mark the plane through the conductor centreline. The probe coordinate $z=0$ belongs to that plane, not to a nearby support or to the outside surface of the wire. A translation stage with a readable scale establishes successive positions; at each position, turn the probe about two perpendicular axes until its response to the loop is maximal. This alignment procedure makes its sensitive axis parallel to the loop axis without relying on a printed arrow whose accuracy may be unknown.

Probe calibration should convert output voltage or digital count into tesla before the map is compared with the calculation. Record a zero-current reading at each scan position if the local background changes along the apparatus. With a stable background, a stronger method uses equal current magnitudes of both signs. If $S_{+}$ and $S_{-}$ are the two signed probe outputs, the current-dependent loop signal is

The average $[S_{+}(z)+S_{-}(z)]/2$ isolates the current-even background and belongs in the laboratory record. The difference method cancels a constant sensor offset and much of the static environmental field, but it cannot repair a probe whose gain drifts between the two readings. Alternate the current sign often enough that drift is small over one pair, and record current with a calibrated meter.

The plot to compare is $B_{\mathrm{odd}}$ against signed $z$, with uncertainty bars from repeated readings. The ideal curve is even: points at $+z$ and $-z$ have the same axial field. A pronounced odd part of the measured map is a diagnostic for an origin error, a tilted probe, or an asymmetric lead arrangement. Agreement only at the centre is weak evidence because several errors can leave one central value close to expectation while distorting the rest of the profile.

Radius, current, and limiting checks

For uncertainty work, retain the exact axial expression instead of using a centre approximation at every location. Its logarithmic sensitivity to current is one: $\partial \ln B_z/\partial \ln I=1$. The corresponding sensitivity to radius is

Thus radius uncertainty does not have one universal coefficient. At the centre, the coefficient is $-1$, consistent with $B_{\mathrm{center}}={\mu }_{0}I/(2a)$: a larger radius lowers the centre field. Well out on the axis it approaches $+2$, consistent with the far-axis dependence on $a^2$. The sign change occurs at $|z|=a/\sqrt{2}$, where the opposing effects of enlarging the loop—moving current away while increasing its enclosed area—cancel to first order for that particular axial point.

For independent small standard uncertainties in current and radius, and with the probe coordinate treated as another measured input, first-order propagation gives

At $z=0$, the coordinate term vanishes to first order because the profile has a maximum there. This does not make centre placement unimportant: a substantial offset changes the reading at second order and can bias a claimed centre value. In the far-axis regime, the same expression predicts the familiar leading fractional budget ${\sigma }_B/B$ from one current term, two radius terms, and three distance terms. That trend also means that a distant measurement can carry less information than a modest off-centre one even when the probe has adequate resolution.

Two limiting substitutions provide a compact audit of a calculation. Setting $z=0$ must give ${\mu }_{0}I/(2a)$, with inverse radius dependence. Factoring $|z{|}^3$ from the exact denominator at $|z|\gg a$ must leave ${\mu }_{0}I{a}^2/(2|z{|}^3)$, with inverse-cube distance dependence. A result that fails either test has usually lost an axial projection, a circumference factor, or a power of separation.

Practical limits of the axial model

The formula is a field model for the loop itself, not automatically for every wire in the circuit. A laboratory loop needs two leads. Their fields can be small at the probe only when the lead routing is arranged deliberately: a close outgoing-and-return pair tends to cancel its external field, whereas widely separated leads can produce a detectable background. An axial scan with the loop current set to zero establishes the fixed background, but it does not remove a current-dependent field from poorly routed leads. That contribution must be reduced by geometry or included in a fuller source calculation.

The radius in the formula is the radius of the current path. With insulated wire, a ruler measurement to an outside edge is therefore not necessarily the appropriate value. Use the conductor centreline for a single turn; for several turns packed across an appreciable radial width, a single radius is an approximation. The central field of a tightly wound set of turns is often estimated by multiplying the single-turn answer by the turn count, provided the turns share nearly the same radius and axial location. When that is not true, add the individual loop contributions at the intended probe position instead of assigning the winding one nominal radius.

Instrument orientation matters as much as position. A one-axis Hall probe measures the component normal to its sensing face. If its sensitive axis makes an angle $\theta$ with the loop axis, the displayed loop contribution is $B_z\cos \theta$. Small alignment errors near the centre mostly change the scale; a probe displaced sideways also samples transverse field components and no longer tests the axial expression alone. A complete record therefore gives the current, centreline radius, signed axial coordinate, probe orientation, background procedure, and uncertainty in each. Such details trace a disagreement to geometry, calibration, or the source model.

Symmetry cancellation and limiting cases

The off-axis components of individual contributions cancel in pairs on the loop axis. For every source element at one azimuth, a diametrically opposite element has an equal magnitude contribution with transverse component opposite in direction. Their axial components have the same direction and add. This cancellation is specific to the observation point lying on the symmetry axis. Moving the observation point off axis breaks the pairwise cancellation and generally requires resolving vector components throughout the integral.

At the centre, $z=0$, the axial result becomes

For fixed current, a smaller loop produces a larger centre field because every source element is closer to the observation point. The result has units of tesla: ${\mu }_{0}I/a$ has units of $\mathrm{T}$. This centre limit is also an algebra check on any axial calculation; retaining an extra factor of radius or separation gives the wrong units.

Far from the loop, $|z|\gg a$, the denominator is approximately $|z{|}^3$, so

The inverse-cube decay differs from the inverse-distance result of an indefinitely long straight wire because a finite loop has no extended source length at large distance. The far-axis approximation is valid only after the observation distance is large compared with loop radius; applying it near the loop loses the measured peak and curvature of the exact axial profile.

Measurement along the loop axis

An axial scan tests the derivation directly. Mount the loop so its geometric axis is known, drive a measured steady current, and move a calibrated magnetic probe along that axis while keeping the probe's sensitive direction aligned with the axis. Record both signed current and signed probe output. Reversing current should reverse the measured axial signal, allowing a current-odd difference to suppress fixed sensor offset. The probe active area should be small compared with the distance scale over which the axial result changes; otherwise it reports a spatial average rather than the value at its marked location.

Reading residuals in an axial map

Once each reading has been reduced to a current-odd magnetic field, calculate a residual at every position,

Assess residuals as a pattern across the map. The largest absolute value alone does not identify a source of error. A nearly constant residual suggests an incomplete offset correction or a current-dependent background that the reversal procedure has not removed. A residual that changes sign between $+z$ and $-z$ points to a misplaced origin or to lead geometry that lacks the loop's mirror symmetry. A symmetric residual that is small at the centre and grows toward both ends is often caused by an error in the radius, the probe calibration scale, or the assigned probe coordinate. Each source has a different shape across the map, which is why a distributed scan is a more severe test than a centre reading.

Do not use the same map to alter every input until the curve agrees. The loop radius should first come from direct measurement of the conductor centreline, preferably at several diameters to expose an out-of-round winding. Current should come from a meter in series with the loop or from a calibrated current monitor, not from a nominal setting. Probe calibration and axial position should likewise have independent records. With those quantities fixed, compare the field with the single circular-path model within the stated uncertainties. Freely adjusting all parameters afterward can produce agreement from compensating errors.

Near the wire plane, preserve the sign convention carefully. The axial expression is positive on both sides of the centre for a selected current direction, because the axis direction is fixed while the observation coordinate changes sign. Confusing the even field profile with an odd one can make a valid scan appear wrong by an entire sign. The coordinate should therefore be recorded as signed distance from the loop plane, whereas the probe's positive sensing direction remains fixed in the laboratory.

The residual table should also state the probe's active-area dimension. The ideal formula is evaluated at a mathematical point. A probe with a finite sensitive area averages nearby values. Far from the centre that average is often negligible; a large sensor near a small loop can smooth the peak enough to mimic a radius or calibration error. Repeat central readings with a smaller probe or controlled displacement to test the spatial-averaging effect.

Off-axis fields and nonideal windings

The axial expression gives one component of the field on one special line. It does not predict the magnitude or direction at a point displaced sideways from that line. At an off-axis point, opposite source elements are no longer the same distance from the probe. Their transverse contributions therefore fail to cancel pair by pair, and the field generally has both an axial component and a radial component. Cylindrical symmetry still rules out an azimuthal component for a perfectly circular loop, but it does not reduce the remaining calculation to the single axial formula.

The distinction matters in a measurement. A probe moved a small distance sideways at fixed $z$ samples a changed vector field rather than a displaced copy of the on-axis value. A one-axis probe may report a lower value because the axial component has changed, because the total field has rotated away from the probe axis, or both. A scalar comparison against $B_z(z)$ cannot sort those possibilities out. Mapping such points requires the full Biot–Savart vector integral, a numerical summation over short current segments, or measurements of enough field components to reconstruct the local vector.

A near-axis check follows from the loop's reflection symmetry about its central plane and rotational symmetry about its axis, without claiming an off-axis formula. At $z=0$, the field at a small radial displacement remains directed along the axis; the radial component changes sign across the central plane and is zero in that plane. At $z\ne 0$, a radial component can occur and reverses when the radial displacement is reflected through the axis. These symmetry statements identify which measured components should vanish at special locations. They do not supply their nonzero values, so they must not be substituted for the axial result.

Mechanical alignment is consequently part of the model. A translation rail that is tilted relative to the loop axis causes an intended axial scan to acquire a sideways displacement that grows with distance. The resulting trace can lose its expected even shape even when the current and radius are measured accurately. Establish the axis with a sight line or alignment rod, check the rail at several positions, and retain a record of its transverse offset if high-accuracy mapping is required.

Conductor width, turn spacing, and the source path

The ideal derivation places all current on a circular line of radius $a$. Real wire occupies a cross-section, and adjacent turns may have different radii and different axial positions. These details matter most close to the winding, where the field varies noticeably over the dimensions of the conductor. Assigning a single radius to a broad winding can produce a central-field estimate that looks precise while hiding the dominant geometry uncertainty.

For one round wire carrying uniform current, the first approximation uses the radius of its centreline. The measured outside diameter should not be substituted directly: it shifts the current path outward by approximately half the wire diameter. For a rectangular conductor or a thick conducting band, current density may also vary across the cross-section, especially at high frequency. The steady-current treatment here assumes a known distribution; when the distribution is nonuniform, the source must be integrated over the conductor volume rather than represented by one centreline.

A compact group of $N$ turns has the simple centre estimate $B_{\mathrm{center}}\approx N{\mu }_{0}I/(2a)$ only if all turns have nearly equal radii and lie close to the same plane. A more faithful representation retains a radius $a_j$ and axial displacement $z_j$ for each turn. At an observation coordinate $z$, their axial contributions add as

The expression is a superposition of circular-loop results. Turn spacing broadens the axial peak, and radial layering changes its scale. When the total winding width is small compared with both its mean radius and the probe distance of interest, replacing the set by $N$ coincident turns is normally adequate. The approximation should be checked against the actual winding dimensions, not inferred from a large turn count alone.

Outcome of the calibration comparison

The numerical agreement establishes a limited claim: at the stated current, geometry, and probe location, the observed axial component is compatible with the circular-turn calculation. It does not establish the model at every point in space or at every current. A calibration record therefore includes the coordinate, current sign, probe sensitivity, turn count, mean radius, winding width, and the treatment of background readings. Another group can then repeat the comparison without guessing which nominal dimensions were used.

Several follow-up checks separate a robust calibration from an accidental match. Repeat the map at a second current, for example $0.150\mathrm{A}$, and divide each current-odd result by the measured current. The normalized traces should agree within their uncertainties if the probe and winding remain in the linear steady-current regime. A change of scale that tracks neither current nor calibration indicates an instrument or lead-geometry problem. Repeating the scan after reversing the physical orientation of the loop, while retaining the laboratory probe convention, gives a separate sign check.

The selection of axial positions should also serve the uncertainty budget. Near the centre, the field is large and coordinate error affects the result only at second order, making those points suitable for checking the current-to-field scale. Around a substantial fraction of a radius from the centre, the curve has more shape and is more sensitive to an incorrectly assigned radius or origin. Farther out, the signal drops rapidly and distance uncertainty becomes costly. A calibration run that includes all three regions can expose errors that a collection of closely spaced central readings would leave hidden.

Finally, distinguish uncertainty from correction. A known probe sensitivity offset is a correction to apply before comparison; the remaining uncertainty in that correction belongs in the budget. A nearby steel object whose effect has not been characterized is not a small statistical uncertainty to be appended automatically. It is an uncontrolled source contribution, best removed by changing the apparatus or measured independently with the loop current set to zero and with appropriate current-reversal tests. This distinction keeps the reported calibration tied to quantities that were actually measured.

Winding superposition and numerical prediction

Each circular path contributes an axial field whose sign is set by its current orientation. If a common positive axis has been chosen, introduce $s_j=+1$ for a turn whose right-hand-rule field points in that direction and $s_j=-1$ for a turn whose field points oppositely. For paths with radii $a_j$, axial locations $z_j$, and currents $I_j$, the axial model becomes

The sign belongs to the source orientation, not to the observation coordinate. A turn that is reversed changes its contribution at every axial point; moving the probe from positive to negative $z$ does not reverse the axial component of a single turn. Keeping those two facts separate avoids a frequent sign error in multi-turn calculations. The signed form also accommodates a return path that makes one or more nearly circular loops in the opposite sense. Treating all turns as positive because the supply current has one stated polarity can then overpredict the measured field substantially.

For nearly coincident turns carrying the same current and orientation, the sum reduces to $N$ times the one-turn result. That reduction is a consequence, not an added rule: set every $a_j=a$, every $z_j=0$, every $I_j=I$, and every $s_j=+1$ in the sum. If one subgroup is wound in the opposite sense, its contribution subtracts. Two equal, oppositely oriented groups in the same plane cancel on the axis, including at the centre. If the two groups are separated along the axis, their centre-plane contributions can still cancel while the field changes rapidly away from that plane. Such a source cannot be represented by a single scaled loop curve.

Sign checks are simple to build into the laboratory record. Mark the current direction on each winding layer, define the positive axial direction on the apparatus, and make a predicted-sign column before taking probe readings. Reverse the supply current once the map has been recorded. Every source contribution that follows that supply branch should reverse; a static background should not. A disagreement in sign is often more diagnostic than a modest scale disagreement because it exposes an orientation or probe polarity mistake immediately.

Numerical quadrature for a nonideal winding

The separate-loop sum is efficient when the paths are visibly circular. A distorted winding, a lead that approaches the probe, or a deliberately shaped conductor calls for a direct numerical Biot–Savart evaluation. Represent the current path by ordered points ${\vec{r}}_k$. The short directed segment from one point to the next is $\Delta {\vec{\ell }}_k={\vec{r}}_{k+1}-{\vec{r}}_k$, and the segment midpoint is used as its source location. At a probe position $\vec{P}$, the midpoint approximation is

where ${\vec{r}}_k^{\ast }$ denotes the midpoint of segment $k$. The cross product preserves both direction and sign, so this calculation can predict off-axis components as well as the axial one. Closed circular paths submitted to the same routine provide a convergence check: as the segment count is increased, the numerical axial result should approach the analytic circular-loop expression.

Segment length is a model-control parameter. Halve the largest segment length and repeat the calculation. If the predicted probe field changes by more than the desired numerical tolerance, the original segmentation was too coarse. For smooth paths with midpoint segments, the leading integration error commonly falls roughly with the square of segment length, but a sharp bend or a nearby probe can spoil that behaviour. Refine those regions locally. Long, smooth portions of the path need fewer segments.

The centreline model assumes that the probe is not so close to the conductor that its cross-section is resolved. When the probe distance is comparable with wire radius, replace one centreline with several nearby paths that share the measured current in proportion to area, or integrate the current density over the conductor cross-section. The result should then be tested for convergence both in the number of path segments and in the number of cross-sectional sample paths. A numerical answer with many digits but no convergence check is only an untested approximation.

Measured winding coordinates can be obtained from a photographed grid, a mechanical scan, or a design drawing, provided the coordinate system is registered to the probe position. The path order must follow the actual current direction. Accidentally joining two points across a gap creates a fictitious straight segment and can dominate the calculated field near the gap. Plotting the reconstructed path before evaluating the field is a practical way to catch that input error.

Residual structure over a full map identifies the next measurement. A nearly uniform scale error at all positions suggests current-meter or probe-sensitivity calibration. A discrepancy that grows near one side of the apparatus points toward a lead or a local magnetic object. A residual that changes sign when the probe is displaced sideways is consistent with axis misalignment or with a source path that is not circular. None of these patterns is repaired honestly by changing the nominal turn count after the data are known. Measure the suspected source geometry, update the signed path model, and state the revision in the comparison record.

Boundaries of a numerical comparison

A numerical path calculation should be compared with data at locations selected before the result is viewed. Choosing only positions where the curve happens to agree changes the task from a model test into a display exercise. Include a centre-near point, one or more points displaced by a substantial fraction of the winding radius, and a point far enough away to test the predicted decline. At each location, retain the raw positive- and negative-current probe outputs as well as their reduced current-odd value. That record allows a later check for drift, asymmetric current magnitude, or background changes that were concealed by the reduction.

Geometry uncertainty and numerical error should not be merged without explanation. The former reflects incomplete knowledge of the physical source: turn centres, lead paths, conductor dimensions, and probe position. The latter reflects the accuracy of the chosen calculation after a particular geometry has been supplied. Report the segment-refinement change separately from the uncertainty caused by moving an input coordinate within its measurement tolerance. If refinement changes the result by $0.05\mu \mathrm{T}$ while geometric uncertainty is $0.50\mu \mathrm{T}$, further subdivision is unlikely to improve the physical prediction. If the refinement change is comparable with the measured residual, the numerical model has not yet earned a comparison with the experiment.

An observed discrepancy is most informative when it can be connected to a proposed independent test. A suspected nearby lead can be rerouted while every other source is held fixed. A suspected probe-angle error can be checked by rotating the probe through a known small angle and observing the expected component change. A suspected path coordinate error can be tested against a new mechanical survey. These interventions turn residuals into information about the apparatus rather than reasons to add an unspecified correction factor. The final field prediction remains credible only to the extent that its source geometry, numerical convergence, and measurement protocol are each documented at the same level of care.

Winding coordinates and sign conventions

A winding model begins with a coordinate convention that can be reconstructed from the apparatus. Put the origin at the chosen reference plane of the winding, direct the positive $z$ axis along the mechanical axis, and use a right-handed transverse pair $(x,y)$. A layered winding requires each path to be recorded by mean centreline radius $a_j$, centre-plane coordinate $z_j$, and orientation sign $s_j$. The sign should be tied to a visible current-direction mark: with the observer looking from positive $z$, counterclockwise current has the right-hand-rule axial direction used as positive in the preceding sums.

The reference plane needs an operational definition. A convenient choice is the midplane between the outermost turn planes, measured from conductor centrelines. A stage reading $q$ then becomes the model coordinate $z=q-q_0$, where $q_0$ is the stage reading at that reference plane. The uncertainty in $q_0$ is correlated among every point in an axial scan: moving the whole coordinate origin changes all listed $z$ values together. Treating it as unrelated random error at each point can understate a systematic displacement of the map.

Radius and axial position must refer to the same current path. A radius measured to the outside insulation and a plane measured to the wire centre do not define one consistent loop. When the winding is surveyed from a photograph or drawing, place fiducial marks in the same physical plane as the probe travel and state the scale used to convert pixels or divisions to length. The convention becomes especially important when two layers are close enough that their separate axial coordinates affect the predicted field by more than the probe uncertainty.

Probe alignment and measurement protocol

A Hall probe reports the component of magnetic field along its sensing normal. Over its calibrated range, a simple representation is $V=V_0+S\vec{B}\cdot \hat{n}$, where $V_0$ is the zero-field output and $S$ is sensitivity. At an axial point, a tilt angle $\theta$ between $\hat{n}$ and the winding axis gives a loop signal proportional to $\cos \theta$. Align the probe by rotating it at a known axial point until the current-odd response is maximal, then lock that orientation before the translation scan. The maximum provides an alignment datum without relying on the orientation of a probe housing or support rail.

Calibration requires both a scale and a zero procedure. Determine $S$ from a known reference field or from a traceable calibration, then record a zero-current value with the probe in the same position and orientation used for the scan. At each map point, the alternating-current estimate $[V(+I)-V(-I)]/(2S)$ removes the constant part of $V_0$ and most static background. A current reversal does not correct gain drift, probe saturation, or a field from leads that reverses with the loop current; those effects must be checked by repeated calibration, operating-range limits, and lead geometry.

Spatial resolution is set by both the scan increment and the active sensor size. The ideal axial curve falls to one half of its centre value at $|z|=0.766a$. A step no larger than roughly $a/10$ gives more than fifteen intervals across the corresponding full width and usually resolves the peak shape of a single compact loop. Coarser steps can still verify a scale, but they can miss a shifted maximum or obscure broadening caused by turn spacing. If the active area or its axial extent is not small relative to $a$, the probe measures an average over that area; reducing the step alone cannot restore the lost spatial detail.

A compact field-map protocol

A 24-turn winding with mean radius $a=46.0\mathrm{mm}$, current $I=0.4000\mathrm{A}$, and a probe sensitivity of $2.00mV/\mu \mathrm{T}$, begin by surveying the two winding layers and entering their separate $a_j$ and $z_j$ values in the signed source list. Use the mean radius only to set an initial scan range. The single-radius centre estimate is $N{\mu }_{0}I/(2a)=131\mu \mathrm{T}$; it provides a scale check before the layered model is evaluated.

Set the stage reference at the surveyed winding midplane. Record points at $z=0$, $\pm 23\mathrm{mm}$, and $\pm 46\mathrm{mm}$, then add intermediate points at 4 or 5 mm spacing if the profile itself is to be tested. At every location, take a short sequence $+I,-I,+I,-I$ after the probe has settled. Convert the mean pair difference to tesla with the calibration sensitivity and retain the individual voltages. At the centre, the preliminary scale predicts about $0.262\mathrm{V}$ for a positive-current signal; at $|z|=a/2$ and $|z|=a$, it predicts about $0.188\mathrm{V}$ and $0.0928\mathrm{V}$, respectively. These are planning values, not replacements for the source model using the surveyed layers.

Calculate the field at each recorded coordinate from the signed turn sum, or from the converged segment model when lead paths must be retained. List the current-odd probe field, its standard uncertainty, the model value, and the residual in the same order as the scan coordinate. Examine symmetry first: paired points should agree within uncertainty for a winding and lead arrangement that is symmetric about the reference plane. Then inspect the residual sequence for a constant scale bias, a coordinate shift, or a one-sided disturbance. Only after those checks should a geometry parameter be revised, and the revised value must be backed by a new survey rather than chosen to force the plotted curves together.