Lines {curve, circles} can vary curvature radius. Curves {regular arc} {regular curve} can be in intervals.
Surfaces have metric elements {arc length, curve}: ds^2 = dx^2 + dy^2 + dz^2. Locally, arc length is invariant. Arc-length curvature and functions {torsion, curve} can determine space-curve properties, except position.
Curve order and class relate to simple singularities {Plucker formula}.
Curvature depends only on surface parameters {Gauss characteristic equation}. Curvature is product of principal curvatures. Linear curvature is tangent-angle change divided by arc length. It is radial-length change times two divided by arc length squared. It is curved-surface solid angle divided by surface area. To measure curvature around point, use regular hexagon and measure angles.
All surface points have a perpendicular {binormal} to osculating plane.
In three dimensions, circles {coaxial circle} can share axis.
Curves {higher curve} {higher plane curve} can have equations with degree greater than two. Defining degree-n curves requires n * (n + 3) / 2 points. Higher plane curves have inflection points, multiple points, cusps, conjugate points, genuses, and branches. One degree-m curve and one degree-n curve can intersect at m*n points.
A point and two points on a curve near the point define a circle with curvature equal to curve {osculating curve}| curvature.
Rotation about, and translation along, line makes space curve {screw curve}.
Non-intersecting curves {simple closed curve} can enclose regions. Simple closed curves {simply connected region} can surround only region points.
Curves {skew curve} {twisted curve} can go outside of plane.
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Date Modified: 2022.0225