Roundness

The ISO roundness of square is ${\displaystyle {\frac {1}{\sqrt {2}}}\simeq 0.7}$, while the roundness of octagon is ${\displaystyle ={\frac {\sqrt {2+{\sqrt {2}}}}{2}}\simeq 0.92}$.

The ISO definition of roundness is the ratio of the radii of inscribed and circumscribed circles, i.e. the maximum and minimum sizes for circles that are just sufficient to fit inside and to enclose the shape.[1][2]

A single trace covering the full rotation is made and at each equally spaced angle, ${\displaystyle \theta _{i}}$, a measurement, ${\displaystyle R_{i}}$, 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:[4]

${\displaystyle {\hat {R}}={\frac {1}{N}}\sum \limits _{i=1}^{N}R_{i}}$

${\displaystyle {\hat {a}}={\frac {2}{N}}\sum \limits _{i=1}^{N}R_{i}\cos {\theta _{i}}}$

${\displaystyle {\hat {b}}={\frac {2}{N}}\sum \limits _{i=1}^{N}R_{i}\sin {\theta _{i}}}$

The deviation is then measured as:

${\displaystyle {\hat {\Delta }}=R_{i}-{\hat {R}}-{\hat {a}}\cos {\theta _{i}}-{\hat {b}}\sin {\theta _{i}}}$

Roundness measurement

Roundness measurement is very important in metrology. It includes measurement of a collection of points. For this two fundamental methods exist: intrinsic or extrinsic.

• 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:

Roundness = Perimeter²/4π × Area.

This ratio will be 1 for a circle and greater than 1 for non-circular shapes. Another definition is the inverse of that:

Roundness = 4π × Area/Perimeter²,

which is 1 for a perfect circle and goes down as far as 0 for highly non-circular shapes.

• Compactness measure of a shape
• Eccentricity (mathematics), how much a conic section (e.g., ellipse) deviates from being circular
• Flattening
• Geometric dimensioning and tolerancing
• Surface roughness
• Sphericity