If the curve is given parametrically by functions x(t) and y(t), then the radius of curvature is
Heuristically, this result can be interpreted as
In n dimensions
If γ : ℝ → ℝn is a parametrized curve in ℝn then the radius of curvature at each point of the curve, ρ : ℝ → ℝ, is given by
As a special case, if f(t) is a function from ℝ to ℝ, then the radius of curvature of its graph, γ(t) = (t, f(t)), is
Let γ be as above, and fix t. We want to find the radius ρ of a parametrized circle which matches γ in its zeroth, first, and second derivatives at t. Clearly the radius will not depend on the position γ(t), only on the velocity γ′(t) and acceleration γ″(t). There are only three independent scalars that can be obtained from two vectors v and w, namely v · v, v · w, and w · w. Thus the radius of curvature must be a function of the three scalars |γ′(t)|2, |γ″(t)|2 and γ′(t) · γ″(t).
The general equation for a parametrized circle in ℝn is
where c ∈ ℝn is the center of the circle (irrelevant since it disappears in the derivatives), a,b ∈ ℝn are perpendicular vectors of length ρ (that is, a · a = b · b = ρ2 and a · b = 0), and h : ℝ → ℝ is an arbitrary function which is twice differentiable at t.
The relevant derivatives of g work out to be
If we now equate these derivatives of g to the corresponding derivatives of γ at t we obtain
These three equations in three unknowns (ρ, h′(t) and h″(t)) can be solved for ρ, giving the formula for the radius of curvature:
An ellipse (red) and its evolute (blue). The dots are the vertices of the ellipse, at the points of greatest and least curvature.
For a semi-circle of radius a in the lower half-plane
The circle of radius a has a radius of curvature equal to a.
In an ellipse with major axis 2a and minor axis 2b, the vertices on the major axis have the smallest radius of curvature of any points, R = b2/a; and the vertices on the minor axis have the largest radius of curvature of any points, R = a2/b.
The ellipse's radius of curvature, as a function of θ
Stress in the semiconductor structure involving evaporated thin films usually results from the thermal expansion (thermal stress) during the manufacturing process. Thermal stress occurs because film depositions are usually made above room temperature. Upon cooling from the deposition temperature to room temperature, the difference in the thermal expansion coefficients of the substrate and the film cause thermal stress.
Intrinsic stress results from the microstructure created in the film as atoms are deposited on the substrate. Tensile stress results from microvoids (small holes, considered to be defects) in the thin film, because of the attractive interaction of atoms across the voids.
The stress in thin film semiconductor structures results in the buckling of the wafers. The radius of the curvature of the stressed structure is related to stress tensor in the structure, and can be described by modified Stoney formula. The topography of the stressed structure including radii of curvature can be measured using optical scanner methods. The modern scanner tools have capability to measure full topography of the substrate and to measure both principal radii of curvature, while providing the accuracy of the order of 0.1% for radii of curvature of 90 meters and more.