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Deriving the Toleranced Feature
Spherical Features With Surface Data
Cylindrical Features with Surface Data under ASME Y14.5
Cylindrical Features with Surface Data under ISO 1101
Circular Features with Surface Data under ASME Y14.5
Circular Features with Surface Data under ISO 1101
Conical Features With Surface Data under ASME Y14.5
Conical Features with Surface Data under ISO 1101
Width Features with Surface Data under ASME Y14.5
For most specification types, the toleranced feature is the considered feature's surface data. However, with location and orientation geometric tolerances (position, concentricity, symmetry, perpendicularity, parallelism, and angularity) the toleranced feature is derived from the considered feature's surface data. This applies to spheres, cylinders, circles, cones, widths, slots, and notches. This also applies to the tangent plane modifier planar features. Each type of considered feature is handled differently. This topic covers features that have surface data, features without surface data (including mid-planes and mid-lines), and finally the tangent plane modifier. For information on the command types that have and do not have surface data, see "Feature Types With and Without Surface Data".
In several places below, we discuss the sample plane. Cylinder, circle, and cone auto features can have a sample plane:
When the feature has a sample feature, the sample feature is the sample plane.
When the feature has one sample hit, the sample plane passes through that hit and is perpendicular to the feature's nominal axis vector.
When the feature has three or more sample hits, the sample plane is the least squares plane for those sample hits.
When the feature has no sample hits nor sample feature selected, there is no sample plane.
When there is a sample plane, there is also a start plane that intersects the axis at the start point. The nominal sample plane is allowed to be offset from the nominal start plane, because the nominal sample plane might not intersect the nominal axis at the start point. The measured start plane is parallel to the measured sample plane, and is nominally offset from it.
As discussed in "Specification Versus Verification" in "Introduction to Geometric Tolerances and Feature Control Frames", we offer multiple math types for toleranced feature computation. PC-DMIS provides two such math types for features having measured surface data: DEFAULT and LSQ. What they do is detailed below. For the most part, DEFAULT is a good choice when the measurement uncertainty of the surface data is much less than the form error of the surface, because the mathematics is similar to the specification.
The LSQ feature math option does a plain least squares best fit to the surface data. This algorithm is mathematically rather different than the specification, but it is a better choice than DEFAULT when the measurement uncertainty of each point is much larger than the form error of the surface.
For more details, see "Specification Versus Verification".
The toleranced feature is a 3D point when the considered feature is a sphere. When the spherical feature has surface data, the toleranced feature is constructed in the following way:
A fit type is selected, based on the feature math type (DEFAULT or LSQ) and based on the material modifier. The LSQ math type always does a least-squares best fit. The DEFAULT math type does an inscribed or circumscribed best fit. When the material modifier is RFS (no material modifier) or MMC, the inscribed or circumscribed fit is chosen to be external to the material. When the material modifier is LMC, the inscribed or circumscribed fit is chosen to be internal to the material. Thus, the DEFAULT math type usually produces the unrelated actual mating envelope (UAME) unless the modifier is LMC. In that case, the math type produces the unrelated actual minimum material envelope (UAMME). Since traditional inscribed and circumscribed fits are notoriously unstable, PC-DMIS uses a constrained least squares algorithm to compute the inscribed or circumscribed fit in a stable way.
The center-point of the fitted sphere is the toleranced feature.
The toleranced feature is an axis when the considered feature is a cylinder. Under ASME Y14.5, when a cylindrical feature has surface data, the toleranced feature is constructed in the following way:
First, a fit type is selected, based on the feature math type (DEFAULT or LSQ) and on the material modifier. The LSQ math type always does a least-squares best fit. The DEFAULT math type does an inscribed or circumscribed best fit. When the material modifier is RFS (no material modifier) or MMC, the inscribed or circumscribed fit is chosen to be external to the material. When the material modifier is LMC, the inscribed or circumscribed fit is chosen to be internal to the material. Thus, the DEFAULT math type usually produces the unrelated actual mating envelope (UAME) unless the modifier is LMC. In that case, the math type produces the unrelated actual minimum material envelope (UAMME). Since traditional inscribed and circumscribed fits are notoriously unstable, PC-DMIS uses a constrained least squares algorithm to compute the inscribed or circumscribed fit in a stable way.
Second, an extrapolation is selected. It is based on whether a sample plane is available and whether a projected-zone modifier is present:
When there is no sample plane, and no projected zone modifier, the best-fit axis is extrapolated to the nominal end-faces of the cylinder. If there is a projected zone modifier, the extrapolation starts at the nominal start-face of the cylinder. It continues away from the end-face, until it hits the nominal projection plane parallel to the start plane and the projected distance away from it.
When there is a sample plane, the extrapolation starts at the measured start plane.
The extrapolated best-fitted axis is the toleranced feature.
The toleranced feature is an axis when the considered feature is a cylinder. Under ISO 1101, when a cylindrical feature has surface data, the toleranced feature is constructed in the following way:
First, PC-DMIS determines whether ISO 17450-3 : 2016 applies or not. To PC-DMIS, it applies when there is no material modifier, no projected-zone modifier, and the feature math type is DEFAULT.
When ISO 17450-3 applies, and the surface data were measured in cross sections, the toleranced feature is an imperfect axis. Each cross section has a least-squares circle fitted to it. The vector of each circle is the vector of the least-squares axis of the entire cylinder. The center-points of the circles form the toleranced feature. This process matches closely with the specification in ISO 17450-3. When the surface data were not measured in cross sections, it is not possible to adhere so closely to the description in ISO 17450-3, and so an approximation is employed. Specifically, the toleranced feature is the axis of the least-squares cylinder, extrapolated to the end-points of the measured surface data.
When ISO 17450-3 does not apply, a fit type is selected, based on the feature math type (DEFAULT or LSQ) and based on the material modifier. The LSQ math type always does a least-squares best fit. The DEFAULT math type does an inscribed or circumscribed best fit. When the material modifier is RFS (no material modifier) or MMC, the inscribed or circumscribed fit is chosen to be external to the material. When the material modifier is LMC, the inscribed or circumscribed fit is chosen to be internal to the material. Since traditional inscribed and circumscribed fits are notoriously unstable, PC-DMIS uses a constrained least squares algorithm to compute the inscribed or circumscribed fit in a stable way.
When ISO 17450-3 does not apply, and after a fit has been computed, an extrapolation is selected. The extrapolation is based on whether a sample plane is available, and based on whether a projected-zone modifier is present:
When there is no sample plane, and there is no projected zone modifier, the best-fit axis is extrapolated to the nominal end-faces of the cylinder. If there is a projected zone modifier, the extrapolation starts at the nominal start-face of the cylinder. It continues away from the end-face, until it hits the nominal projection plane parallel to the start plane and the projected distance away from it.
When there is a sample plane, the extrapolation starts at the measured start plane.
The extrapolated best-fitted axis is the toleranced feature.
The toleranced feature is a 2D point when the considered feature is a circle. Under ASME Y14.5, when a circular feature has surface data, the toleranced feature is constructed in the following way:
First, a fit type is selected, based on the feature math type (DEFAULT or LSQ) and on the material modifier. The LSQ math type always does a least-squares best fit. The DEFAULT math type does an inscribed or circumscribed best fit. When the material modifier is RFS (no material modifier) or MMC, the inscribed or circumscribed fit is chosen to be external to the material. When the material modifier is LMC, the inscribed or circumscribed fit is chosen to be internal to the material. Thus, the DEFAULT math type usually produces the unrelated actual mating envelope (UAME) unless the modifier is LMC. In that case, the math type produces the unrelated actual minimum material envelope (UAMME). Since traditional inscribed and circumscribed fits are notoriously unstable, PC-DMIS uses a constrained least squares algorithm to compute the inscribed or circumscribed fit in a stable way.
The axis vector of the best-fit circle is the nominal axis vector unless there is a sample plane. If there is a sample plane, the best-fit circle axis vector is the surface normal of the sample plane.
Second, a projection is selected, based on whether there is a sample plane:
When there is no sample plane, the best-fit center-point is projected to the nominal start-plane of the circle.
When there is a sample plane, the best-fit center-point is projected to the measured start plane.
The projected point is the toleranced feature.
The toleranced feature is a 2D point when the considered feature is a circle. Under ISO 1101, when a circular feature has surface data, the toleranced feature is constructed in the following way:
All of the possibilities involve fitting a circle. The axis vector of the best-fit circle is the nominal axis vector unless you took at least three sample hits. In that case, the best-fit circle axis vector is the surface normal of the least-squares plane of the sample hits.
First, PC-DMIS determines whether ISO 17450-3 : 2016 applies or not. To PC-DMIS, it applies when there is no material modifier and the feature math type is DEFAULT.
When ISO 17450-3 applies, the toleranced feature is the center-point of the least-squares circle.
When ISO 17450-3 does not apply, a fit type is selected, based on the feature math type (DEFAULT or LSQ) and on the material modifier. The LSQ math type always does a least-squares best fit. The DEFAULT math type does an inscribed or circumscribed best fit. When the material modifier is MMC, the inscribed or circumscribed fit is chosen to be external to the material. When the material modifier is LMC, the inscribed or circumscribed fit is chosen to be internal to the material. Since traditional inscribed and circumscribed fits are notoriously unstable, PC-DMIS uses a constrained least squares algorithm to compute the inscribed or circumscribed fit in a stable way. The axis vector of the best-fit circle is the nominal axis vector unless there is a sample plane. In that case, the best-fit circle axis vector is the surface normal of the sample plane. Next, a projection is selected, based on whether there is a sample plane:
When there is no sample plane, the best-fit center-point is projected to the nominal start-plane of the circle.
When there is a sample plane, the center-point is projected to the measured start plane.
The projected point is the toleranced feature.
The toleranced feature is an axis when the considered feature is a cone. Under ASME Y14.5, when a conical feature has surface data, the toleranced feature is constructed in the following way:
First, a fit type is selected, based on the feature math type (DEFAULT or LSQ). The LSQ math type always does a least-squares best fit. The DEFAULT math type does an inscribed or circumscribed best fit, chosen to be external to the material. Thus, the DEFAULT math type produces the unrelated actual mating envelope (UAME). Since traditional inscribed and circumscribed fits are notoriously unstable, PC-DMIS uses a constrained least squares algorithm to compute the inscribed or circumscribed fit in a stable way. In all of these fits, the cone angle is allowed to optimize from the nominal angle.
Second, an extrapolation is selected. It is based on whether sample hits are available:
When there is no sample plane, the best-fit axis is extrapolated to the nominal end-faces of the cone.
When there is a sample plane, the extrapolation starts at the measured start plane.
The extrapolated best-fitted axis is the toleranced feature.
The toleranced feature is an axis when the considered feature is a cone. Under ISO 1101, when a cylindrical feature has surface data, the toleranced feature is constructed in the following way:
When the feature math type is DEFAULT, PC-DMIS decides that a generalization of ISO 17450-3 : 2016 applies. If the surface data were measured in cross sections, the toleranced feature is an imperfect axis. Each cross section has a least-squares circle fitted to it. The vector of each circle is the vector of the least-squares axis of the entire cone. The center-points of the circles form the toleranced feature. This process matches closely with the specification in ISO 17450-3. When the surface data were not measured in cross sections, it is not possible to adhere so closely to the description in ISO 17450-3, and so an approximation is employed. Specifically, the toleranced feature is the axis of the least-squares cylinder, extrapolated to the end-points of the measured surface data.
When the feature math type is LSQ, PC-DMIS decides that ISO 17450-3 : 2016 does not apply. A least-squares fit is computed to produce a least-squares axis. The cone angle is allowed to optimize from the nominal angle. Next, an extrapolation is selected, based on whether sample hits are available:
When there is no sample plane, the best-fit axis is extrapolated to the nominal end-faces of the cone.
When there is a sample plane, the extrapolation starts at the measured start plane.
The extrapolated best-fitted axis is the toleranced feature.
The toleranced feature is a plane when the considered feature is a width. Please note all PC-DMIS width features have surface data. Under ASME Y14.5, when the considered feature is a width, the toleranced feature is constructed in the following way:
First, a fit type is selected, based on the feature math type (DEFAULT or LSQ) and on the material modifier. The LSQ math type always does a least-squares best fit. The DEFAULT math type does an inscribed or circumscribed best fit. When the material modifier is RFS (no material modifier) or MMC, the inscribed or circumscribed fit is chosen to be external to the material. When the material modifier is LMC, the inscribed or circumscribed fit is chosen to be internal to the material. Thus, the DEFAULT math type usually produces the unrelated actual mating envelope (UAME) unless the modifier is LMC. In that case, the math type produces the unrelated actual minimum material envelope (UAMME). Since traditional inscribed and circumscribed fits are notoriously unstable, PC-DMIS uses a constrained least squares algorithm to compute the inscribed or circumscribed fit in a stable way.
Second, the surface points are all projected to the center-plane of the fitted width. The toleranced feature is then the convex polygon that describes the perimeter of those projected surface points. Mathematically, the toleranced feature is the convex hull of the projected surface points.
The toleranced feature is a plane when the considered feature is a width. Please note that all PC-DMIS width features have surface data. Under ISO 1101, when the considered feature is a width, the toleranced feature is constructed in the following way:
First, PC-DMIS decides decided whether ISO 17450-3 : 2016 applies or not. To PC-DMIS, ISO 17450-3 : 2016 applies when there is no material modifier and the feature math type is DEFAULT.
When ISO 17450-3 applies, the toleranced feature is an imperfect plane. The toleranced feature is the center-points of the opposed two point sizes as described in ISO 17450-3 and ISO 14405-1. This process matches closely with the specification in ISO 17450-3.
When ISO 17450-3 does not apply, a fit type is selected, based on the feature math type (DEFAULT or LSQ) and on the material modifier. The LSQ math type always does a least-squares best fit. The DEFAULT math type does an inscribed or circumscribed best fit. When the material modifier is RFS (no material modifier) or MMC, the inscribed or circumscribed fit is chosen to be external to the material. When the material modifier is LMC, the inscribed or circumscribed fit is chosen to be internal to the material. Since traditional inscribed and circumscribed fits are notoriously unstable, PC-DMIS uses a constrained least squares algorithm to compute the inscribed or circumscribed fit in a stable way.
When ISO 17450-3 does not apply, and after a fit has been computed, the surface points are all projected to the fitted plane. The toleranced feature is then the convex polygon that describes the perimeter of those projected surface points. Mathematically, the toleranced feature is the convex hull of the projected surface points.
Several types of considered features do not have surface data (for information, see "Feature Types With and Without Surface Data"). When the considered feature does not have surface data, the feature math type is unavailable in the geometric tolerance command. In most cases, you should not use features that do not have surface data. This is because the geometric tolerance command is unable to construct the toleranced feature from the surface data in a way that complies with the ASME Y14.5 or ISO 1101. Instead, you are responsible to define the toleranced feature according to any applicable standards.
For axial or linear features without surface data, the toleranced feature is the line segment from the measured start point to the measured end point. For circular, spherical, and point features without surface data, the toleranced feature is the measured centroid of the feature.
3D constructed BF lines have more complex handing. PC-DMIS interprets them as features without surface data. Instead it interprets the input points as circle centers of circular cross sections. Under ISO 1101, this interpretation complies with ISO 17450-3 : 2016, and the toleranced feature is the set of centroids. However, under ASME Y14.5, PC-DMIS interprets 3D constructed BF lines in the same way as other axial or linear features without surface data. In that case, the toleranced feature is the start-to-end line segment (except for straightness of an axis tolerances which use all the input centroids).
There is only one planar feature without surface data that is allowed as a considered feature: the mid-plane. For the most part, you should use 3D widths instead of mid-planes (and 2D widths instead of mid-lines, and 1D widths instead of mid-points). The mid-plane command is still supported so that old programs continue to work after they migrate to PC-DMIS 2020.2 or later. Since PC-DMIS still supports mid-plane for these legacy applications, its interpretation in the geometric tolerance command is similar to XactMeasure's interpretation. Specifically, PC-DMIS interprets mid planes to have four corners that lie in the mid-plane, and the toleranced feature consists of the rectangle between those four corners.
Slots and notches are treated as 2D widths without surface data. That is, the toleranced feature is a line centered on the feature's centroid. For slots, users can choose whether the slot is considered widthwise or lengthwise, as discussed in "Lengthwise_versus_Widthwise_Slots":
For a widthwise slot, the size of the slot is the slot's width, and the tolerance zone controls the position in the width direction. That means the toleranced feature line is parallel to the slot's length and is as long as the slot's length.
For a lengthwise slot, the size of the slot is the slot's length, and the tolerance zone controls the position in the length direction. That means the toleranced feature line is parallel to the slot's width and is as long as the slot's width.
For notches, the size of the notch is the notch's length, and the tolerance zone controls the position in the length direction. That means the toleranced feature line is parallel to the notch's width and is as long as the notch's width.
Most of the time, for plane-type considered features, the toleranced feature is the considered feature's surface data. However, the tangent plane modifier makes the toleranced feature different from the surface data. Angularity, parallelism, and perpendicularity tolerances on planes are allowed to use the tangent plane modifier. The toleranced feature is derived in the following way.
First, an external-to-material constrained least squares plane is fitted in a way that removes the influence of surface voids. This is the same way primary datum planes are fitted under ASME Y14.5 with DEFAULT datum math, and the same way primary datum planes are fitted under ISO 1101 with CL2 datum math. We use this math because (1) it is external to the material, (2) it mimics the behavior of a surface plate quite well, and (3) it is stable compared to other external-to-material fits.
Next, the surface points are all projected to the constrained least squares plane. The toleranced feature is then the convex polygon that describes the perimeter of those projected surface points. Mathematically, the toleranced feature is the convex hull of the projected surface points.