What is "radius of curvature"

Radius \Ra"di*us\ (r[=a]"d[i^]*[u^]s), n.; pl. L. Radii (r[=a]"d[i^]*[imac]); E. Radiuses (r[=a]"d[i^]*[u^]s*[e^]z). [L., a staff, rod, spoke of a wheel, radius, ray. See Ray a divergent line.]

1. (Geom.) A right line drawn or extending from the center of a circle to the periphery; the semidiameter of a circle or sphere.

2. (Anat.) The preaxial bone of the forearm, or brachium, corresponding to the tibia of the hind limb. See Illust. of Artiodactyla.

   Note: The radius is on the same side of the limb as the thumb, or pollex, and in man it is so articulated that its lower end is capable of partial rotation about the uln

3. (Bot.) A ray, or outer floret, of the capitulum of such plants as the sunflower and the daisy. See Ray, 2.

4. pl. (Zo["o]l.)

   1. The barbs of a perfect feather.

   2. Radiating organs, or color-markings, of the radiates.

5. The movable limb of a sextant or other angular instrument.
   --Knight.

   Radius bar (Mach.), a bar pivoted at one end, about which it swings, and having its other end attached to a piece which it causes to move in a circular arc.

   Radius of curvature. See under Curvature.

Curvature \Cur"va*ture\ (k?r"v?-t?r; 135), n. [L. curvatura. See Curvate.]

1. The act of curving, or the state of being bent or curved; a curving or bending, normal or abnormal, as of a line or surface from a rectilinear direction; a bend; a curve.
   --Cowper.

   The elegant curvature of their fronds.
   --Darwin.

2. (Math.) The amount of degree of bending of a mathematical curve, or the tendency at any point to depart from a tangent drawn to the curve at that point.

   Aberrancy of curvature (Geom.), the deviation of a curve from a circular form.

   Absolute curvature. See under Absolute.

   Angle of curvature (Geom.), one that expresses the amount of curvature of a curve.

   Chord of curvature. See under Chord.

   Circle of curvature. See Osculating circle of a curve, under Circle.

   Curvature of the spine (Med.), an abnormal curving of the spine, especially in a lateral direction.

   Radius of curvature, the radius of the circle of curvature, or osculatory circle, at any point of a curve.

n. (context mathematics for any point on a curve English) The radius of the osculating circle at the point on the curve

n. the radius of the circle of curvature; the absolute value of the reciprocal of the curvature of a curve at a given point

• If the vertex lies to the left of the center of curvature, the radius of curvature is positive.
• If the vertex lies to the right of the center of curvature, the radius of curvature is negative.

Thus when viewing a biconvex lens from the side, the left surface radius of curvature is positive, and the right surface has a negative radius of curvature.

Note however that in areas of optics other than design, other sign conventions are sometimes used. In particular, many undergraduate physics textbooks use an alternate sign convention in which convex surfaces of lenses are always positive. Care should be taken when using formulas taken from different sources.

In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof.