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Euler's spiral is a type of superspiral that has the property of a monotonic curvature function. [4] The Euler spiral has applications to diffraction computations. They are also widely used in railway and highway engineering to design transition curves between straight and curved sections of railways or roads.
The normal curvature, k n, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, k g, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τ r, measures the rate of change of the surface ...
Polynomial curves fitting points generated with a sine function. The black dotted line is the "true" data, the red line is a first degree polynomial, the green line is second degree, the orange line is third degree and the blue line is fourth degree. The first degree polynomial equation = + is a line with slope a. A line will connect any two ...
A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane. The curvature and the torsion of a helix are constant. Conversely, any space curve whose curvature and torsion are both constant and ...
When a line of curvature has a local extremum of the same principal curvature then the curve has a ridge point. These ridge points form curves on the surface called ridges. The ridge curves pass through the umbilics. For the star pattern either 3 or 1 ridge line pass through the umbilic, for the monstar and lemon only one ridge passes through. [3]
Below it, the red surface is the graph of a level set function determining this shape, and the flat blue region represents the X-Y plane. The boundary of the shape is then the zero-level set of φ {\displaystyle \varphi } , while the shape itself is the set of points in the plane for which φ {\displaystyle \varphi } is positive (interior of ...
However, the implicit function theorem gives conditions under which an implicit curve locally is given by the graph of a function (so in particular it has no self-intersections). If the defining relations are sufficiently smooth then, in such regions, implicit curves have well defined slopes, tangent lines, normal vectors, and curvature.
The Gaussian curvature K = κ 1 κ 2 and the mean curvature H = (κ 1 + κ 2)/2 can be computed as follows: =, = + (). Up to a sign, these quantities are independent of the parametrization used, and hence form important tools for analysing the geometry of the surface.