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Radius of curvature and center of curvature. 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 ...
Radius of curvature sign convention for optical design. Radius of curvature (ROC) has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located either along or decentered from the system local optical axis. The vertex of the lens surface is located on the local optical axis.
The mean curvature is an extrinsic measure of curvature equal to half the sum of the principal curvatures, k 1 + k 2 / 2 . It has a dimension of length −1. Mean curvature is closely related to the first variation of surface area. In particular, a minimal surface such as a soap film has mean curvature zero and a soap bubble has
Where degree of curvature is based on 100 units of arc length, the conversion between degree of curvature and radius is Dr = 18000/π ≈ 5729.57795, where D is degree and r is radius. Since rail routes have very large radii, they are laid out in chords, as the difference to the arc is inconsequential; this made work easier before electronic ...
The product k 1 k 2 of the two principal curvatures is the Gaussian curvature, K, and the average (k 1 + k 2)/2 is the mean curvature, H. If at least one of the principal curvatures is zero at every point, then the Gaussian curvature will be 0 and the surface is a developable surface. For a minimal surface, the mean curvature is zero at every ...
Deep blue ray refers the radius of curvature and the red line segment is the sagitta of the curve (black). In optics and especially telescope making, sagitta or sag is a measure of the glass removed to yield an optical curve. It is approximated by the formula (),
The parabola has fourth order contact with its osculating circle there. For large t the radius of curvature increases ~ t 3, that is, the curve straightens more and more.
For a spherically-curved mirror in air, the magnitude of the focal length is equal to the radius of curvature of the mirror divided by two. The focal length is positive for a concave mirror, and negative for a convex mirror. In the sign convention used in optical design, a concave mirror has negative radius of curvature, so