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In most cases, including even simple curves, there are no closed-form solutions for arc length and numerical integration is necessary. Numerical integration of the arc length integral is usually very efficient. For example, consider the problem of finding the length of a quarter of the unit circle by numerically integrating the arc length integral.
The same terminology applies. A regular solution is a solution at which the Jacobian is full rank (). A singular solution is a solution at which the Jacobian is less than full rank. A regular solution lies on a k-dimensional surface, which can be parameterized by a point in the tangent space (the null space of the Jacobian).
The curve-shortening flow is an example of a geometric flow, and is the one-dimensional case of the mean curvature flow. Other names for the same process include the Euclidean shortening flow, geometric heat flow, [1] and arc length evolution. As the points of any smooth simple closed curve move in this way, the curve remains simple and smooth ...
The goat problems do not yield any new mathematical insights; rather they are primarily exercises in how to artfully deconstruct problems in order to facilitate solution. Three-dimensional analogues and planar boundary/area problems on other shapes, including the obvious rectangular barn and/or field, have been proposed and solved. [ 1 ]
The arc length functional has as its domain the vector space of rectifiable curves – a subspace of ([,],) – and outputs a real scalar. This is an example of a non-linear functional. This is an example of a non-linear functional.
The arc length, from the familiar geometry of a circle, is s = θ R {\displaystyle s={\theta }R} The area a of the circular segment is equal to the area of the circular sector minus the area of the triangular portion (using the double angle formula to get an equation in terms of θ {\displaystyle \theta } ):
A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist.
This is a 2-d problem in span{^, ^}, which will be solved with the help of the arc length formula above. If the arc length, s 12 {\displaystyle s_{12}} is given then the problem is to find the corresponding change in the central angle θ 12 {\displaystyle \theta _{12}} , so that θ 2 = θ 1 + θ 12 {\displaystyle \theta _{2}=\theta _{1}+\theta ...