Steepest Descent as a Tool to Find Critical Points of ∫ k2 Defined on Curves in the Plane with Arbitrary Types of Boundary Conditions
Steepest Descent as a Tool to Find Critical Points of ∫ k2 Defined on Curves in the Plane with Arbitrary Types of Boundary Conditions
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最速下降法作为寻找具有任意类型边界条件的平面曲线上定义的 ∫ k2 临界点的工具
DOI：
10.1007/978-1-4613-9711-3_14
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发表时间：
1991
期刊：
影响因子：
--
通讯作者：
Anders Linnér
中科院分区：
文献类型：
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作者：
Anders Linnér
我们将显式计算满足任意类型边界条件的固定或可变长度参数化曲线空间上的总平方曲率泛函的梯度。我们展示了如何把这种曲线的空间变成一个无限维的子流形的内积空间。最陡的下降将是沿着这个流形中负梯度向量场的积分曲线。我们将推导出梯度和相应的流动方程。最后，我们使用计算机图形来说明这个过程，通过遵循一个这样的轨迹开始接近一个不稳定的临界点，并在一个稳定的临界点结束。
We will explicitly compute the gradient of the total squared curvature functional on a space of parametrized curves of fixed or variable length satisfying arbitrary types of boundary conditions. We show how to turn the space of such curves into an infinite dimensional submanifold of an inner product space. The steepest descent will then be along the integral curves of the negative gradient vector field in this manifold. We will derive the gradients and the corresponding flow equations. In conclusion we use computer graphics to illustrate this process by following one such trajectory starting close to an unstable critical point and ending at a stable critical point.