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The curve of the chains of a suspension bridge is always an intermediate curve between a parabola and a catenary, but in practice the curve is generally nearer to a parabola due to the weight of the load (i.e. the road) being much larger than the cables themselves, and in calculations the second-degree polynomial formula of a parabola is used.
Successive parabolic interpolation is a technique for finding the extremum (minimum or maximum) of a continuous unimodal function by successively fitting parabolas (polynomials of degree two) to a function of one variable at three unique points or, in general, a function of n variables at 1+n(n+3)/2 points, and at each iteration replacing the "oldest" point with the extremum of the fitted ...
The average distance between a center of a unit square and a point on the square's boundary is . If we uniformly sample every point on the perimeter of the square, take line segments (drawn from the center) corresponding to each point, add them together by joining each line segment next to the other, scaling them down, the curve obtained is a ...
Low-order polynomials tend to be smooth and high order polynomial curves tend to be "lumpy". To define this more precisely, the maximum number of inflection points possible in a polynomial curve is n-2, where n is the order of the polynomial equation. An inflection point is a location on the curve where it switches from a positive radius to ...
For quadratic Bézier curves one can construct intermediate points Q 0 and Q 1 such that as t varies from 0 to 1: Point Q 0 (t) varies from P 0 to P 1 and describes a linear Bézier curve. Point Q 1 (t) varies from P 1 to P 2 and describes a linear Bézier curve. Point B(t) is interpolated linearly between Q 0 (t) to Q 1 (t) and describes a ...
"The spiral itself is not drawn: we see it as the locus of points where the circles are especially close to each other." [1] An osculating circle is a circle that best approximates the curvature of a curve at a specific point. It is tangent to the curve at that point and has the same curvature as the curve at that point. [2]
Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification. For a rectifiable curve these approximations don't get arbitrarily large (so the curve has a finite length).
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.