Profilometry of toroidal surfaces with an improved Ronchi test

Santiago Royo, Josep Arasa, and Carles Pizarro

Santiago Royo, Josep Arasa, and Carles Pizarro

^{}The authors are with the Center for Development of Sensors, Instrumentation and Systems (CD6), Technical University of Catalonia, Violinista Vellsolà 37, E-08222 Terrassa, Spain.

An implementation of the well-known Ronchi test technique, which
allows for the profilometric measurement of nonrotationally symmetrical
surfaces, is presented and applied to the measurement of toroidal
surfaces. Both the experimental setup and the data-processing
procedures are described, and parameters such as the radius of
curvature of the sample surface, the orientation of its principal
meridians, and the position of its vertex are measured by means of the
values of the local normal to the surface obtained at a set of sampling
points. Integration of these local normal values allows for the
reconstruction of the three-dimensional profile of the toroidal surface
considered with micrometric accuracy, and submicrometric surface
details may be calculated by use of surface-fitting procedures. The
density of sampling points on the surface may be tailored to fit test
requirements, within certain limits that depend on selection of
experimental setup.

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C_{
J
} stands for the
curvature at the meridian considered, K_{
J
} for the
independent term of the linear fit, and R_{
J
} for
the radius of curvature. J may be either the base or the
cross curves of the surface, and r^{2} is the
correlation coefficient of the linear fit.

Table 2

Orientation G60: Results of the Three-Dimensional
Fitting of a Tilted and Decentered Spherocylindrical Surface (Eq.
3) to the Measured Profile

The position of the vertex was set
manually, so no reference value is
provided. R_{
B
} stands for the radius of the
base curve, R_{
C
} for the radius of the cross
curve, θ for the orientation of the principal meridians,
x_{0} and y_{0} for the
coordinates of the vertex of the surface, and r^{2}
for the correlation coefficient of the surface fit.

Table 3

Results for the Sample in the Three Additional
Orientations Measured

N stands for the number of
sampling points, A for the area sampled,
R_{
B
}^{LINEAR} for the radius of the base
curve obtained with linear fitting,
R_{
C
}^{LINEAR} for the radius of the cross
curve obtained with linear fitting,
r_{
B
}^{2(LINEAR)} and
r_{
B
}^{2(LINEAR)} for the correlation
coefficients of the linear fits for the base and the cross curves,
R_{
B
}^{3D} for the radius of the base
curve obtained with surface fitting,
R_{
C
}^{3D} for the radius of the cross
curve obtained with surface fitting, θ for the orientation of the
principal meridians, x_{0} and
y_{0} for the coordinates of the vertex of the
surface, r^{2(3D)} for the correlation
coefficient of the surface fit.

Tables (3)

Table 1

G60 Orientation: Results of Fitting the Plots in Fig. 6 to Eq. (5)

C_{
J
} stands for the
curvature at the meridian considered, K_{
J
} for the
independent term of the linear fit, and R_{
J
} for
the radius of curvature. J may be either the base or the
cross curves of the surface, and r^{2} is the
correlation coefficient of the linear fit.

Table 2

Orientation G60: Results of the Three-Dimensional
Fitting of a Tilted and Decentered Spherocylindrical Surface (Eq.
3) to the Measured Profile

The position of the vertex was set
manually, so no reference value is
provided. R_{
B
} stands for the radius of the
base curve, R_{
C
} for the radius of the cross
curve, θ for the orientation of the principal meridians,
x_{0} and y_{0} for the
coordinates of the vertex of the surface, and r^{2}
for the correlation coefficient of the surface fit.

Table 3

Results for the Sample in the Three Additional
Orientations Measured

N stands for the number of
sampling points, A for the area sampled,
R_{
B
}^{LINEAR} for the radius of the base
curve obtained with linear fitting,
R_{
C
}^{LINEAR} for the radius of the cross
curve obtained with linear fitting,
r_{
B
}^{2(LINEAR)} and
r_{
B
}^{2(LINEAR)} for the correlation
coefficients of the linear fits for the base and the cross curves,
R_{
B
}^{3D} for the radius of the base
curve obtained with surface fitting,
R_{
C
}^{3D} for the radius of the cross
curve obtained with surface fitting, θ for the orientation of the
principal meridians, x_{0} and
y_{0} for the coordinates of the vertex of the
surface, r^{2(3D)} for the correlation
coefficient of the surface fit.