The evolutes of regular curves in the Euclidean plane are given by the caustics of regular curves. In this paper, we define the generalized evolutes of planar curves which are spatial curves, and the projection of generalized evolutes along a fixed direction are the evolutes. We also prove that the generalized evolutes are the locus of centers of slant circles of the curvature of planar curves. Moreover, we define the generalized parallels of planar curves and show that the singular points of generalized parallels sweep out the generalized evolute. In general, we cannot define the generalized evolutes at the singular points of planar curves, but we can define the generalized evolutes of fronts by using moving frames along fronts and curvatures of the Legendre immersion. Then we study the behaviors of generalized evolutes at the singular points of fronts. Finally, we give some examples to show the generalized evolutes.

中文翻译：

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平面曲线的广义演化

欧几里得平面中规则曲线的演化由规则曲线的焦散给出。本文将平面曲线的广义演化定义为空间曲线，广义演化沿固定方向的投影即为演化。我们还证明了广义演化线是平面曲线曲率的斜圆心的轨迹。此外，我们定义了平面曲线的广义平行线，并表明广义平行线的奇异点扫除了广义演化曲线。一般来说，我们不能在平面曲线的奇异点处定义广义演化，但我们可以使用沿锋面的移动框架和勒让德浸没的曲率来定义锋面的广义演变。然后我们研究了广义进化体在锋面奇异点处的行为。最后，我们给出了一些例子来展示广义的进化。