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Roundness is the measure of how closely the shape of an object approaches that of a mathematically perfect
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equivalently it is the curve traced out by a point that moves in a pl ...
. Roundness applies in two dimensions, such as the cross sectional circles along a cylindrical object such as a shaft or a cylindrical roller for a bearing. In geometric dimensioning and tolerancing, control of a cylinder can also include its fidelity to the longitudinal axis, yielding cylindricity. The analogue of roundness in
three dimensions Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''par ...
(that is, for
sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathematics), ball (viz., analogous to the circular objects in two d ...
s) is sphericity. Roundness is dominated by the shape's gross features rather than the definition of its edges and corners, or the surface roughness of a manufactured object. A smooth
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the sp ...
can have low roundness, if its
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off- center, in geometry * Eccentricity (graph theory) of a ...
is large.
Regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex polygon, c ...
s increase their roundness with increasing numbers of sides, even though they are still sharp-edged. In
geology Geology (from the Ancient Greek γῆ, ''gē'' ("earth") and -λoγία, ''-logia'', ("study of", "discourse")) is an Earth science concerned with the solid Earth, the rock (geology), rocks of which it is composed, and the processes by which th ...
and the study of
sediment Sediment is a naturally occurring material that is broken down by processes of weathering and erosion, and is subsequently sediment transport, transported by the action of wind, water, or ice or by the force of gravity acting on the particles. Fo ...
s (where three-dimensional particles are most important), roundness is considered to be the measurement of surface roughness and the overall shape is described by sphericity.

# Simple definitions

The definition of roundness is based on the ratio between the inscribed and the circumscribed circles, i.e. the maximum and minimum sizes for circles that are just sufficient to fit inside and to enclose the shape.

## Diameter

Having a constant diameter, measured at varying angles around the shape, is often considered to be a simple measurement of roundness. This is misleading. Although constant diameter is a necessary condition for roundness, it is not a sufficient condition for roundness: shapes exist that have constant diameter but are far from round. Mathematical shapes such as the Reuleaux triangle and, an everyday example, the Fifty pence (British coin), British 50p coin demonstrate this.

Roundness does not describe radial displacements of a shape from some notional centre point, merely the overall shape. This is important in manufacturing, such as for crankshafts and similar objects, where not only the roundness of a number of bearing journals must be measured, but also their alignment on an axis. A bent crankshaft may have perfectly round bearings, yet if one is displaced sideways, the shaft is useless. Such measurements are often performed by the same techniques as for roundness, but also considering the centre position and its relative position along an additional axial direction.

# Calculation in two dimensions

A single trace covering the full rotation is made and at each equally spaced angle, $\theta_i$, a measurement, $R_i$, of the radius or distance between the center of rotation and the surface point. A least-squares fit to the data gives the following estimators of the parameters of the circle:Roundness measurements
at NIST
:$\hat = \frac\sum\limits_^N R_i$ :$\hat = \frac \sum\limits_^N R_i \cos$ :$\hat = \frac \sum\limits_^N R_i \sin$ The deviation is then measured as: :$\hat = R_i - \hat - \hat \cos - \hat \sin$

# Roundness measurements

Roundness measurement is very important in metrology. It includes measurement of a collection of points.

# Methods

For this two fundamental methods are followed:

## Intrinsic datum method

#The round object is placed over a flat plate and the point of contact is taken as the datum point. Again a dial gauge is placed over the round object and the object is rotated keeping the datum at constant position. Thus the error in roundness can be directly known by comparing the peak height as measured by the dial gauge. #Alternatively a V shaped base can be used instead of a flat plate. Two datum points will exist instead of one since the base is V-shaped. The error in roundness can be measured similar to the previous method. #Also a cylindrical body can be clamped between two axle centres. Here also the dial gauge is mounted over the cylindrical body and thus the roundness is measured by similar procedure as above.

## Extrinsic datum method

The intrinsic method is limited to small deformations only. For large deformations extrinsic method has to be followed. In this case the datum is not a point or set of points on the object, but is a separate precision bearing usually on the measuring instrument. The axis of the object or part of the object to be measured is aligned with the axis of the bearing. Then a stylus from the instrument is just made to touch the part to be measured. A touch sensor connected to the tip of the stylus makes sure that the stylus just touches the object. A minimum of three readings are taken and an amplified polar plot is drawn to get the required error.

# Roundness error definitions

*Least square circle (LSC): It is a circle which separates the roundness profile of an object by separating the sum of total areas of the inside and outside it in equal amounts. The roundness error then can be estimated as the difference between the maximum and minimum distance from this reference circle *Minimum Zone circle (MZC): Here two circles are used as reference for measuring the roundness error. One circle is drawn outside the roundness profile just as to enclose the whole of it and the other circle is drawn inside the roundness profile so that it just inscribes the profile. Both circles, however, have the same center point. The roundness error here is the difference between the radius of the two circles. *Minimum circumscribed circle (MCC): It is defined as the smallest circle which encloses whole of the roundness profile. Here the error is the largest deviation from this circle *Maximum inscribed circle (MIC): It is defined as the largest circle that can be inscribed inside the roundness profile. The roundness error here again is the maximum deviation of the profile from this inscribed circle. *A common definition used in digital image processing (image analysis) for characterizing 2-D shapes is: Circularity = Perimeter^2 / (4 * pi * Area). This ratio will be 1 for a circle and greater than 1 for non-circular shapes. Another definition is the inverse of that: Circularity = (4 * pi * Area) / Perimeter^2, which is 1 for a perfect circle and goes down as far as 0 for highly non-circular shapes.