Reflection across a line is well familiar, reflection across a circle is the inversion, the point at a distance $d$ from the center is reflected into a point on the same ray through the center, but at the distance $R^2/d$, where $R$ is the radius. Under the inversion shapes get distorted in funhouse ways.
But apparently there is a general notion of reflection across an analytic curve. Thinking of the curve as the image of $\mathbb{R}$ under an invertible holomorphic function $\gamma$ the Schwarz reflection across $\gamma(\mathbb{R})$ is defined as $R_\gamma=\gamma\circ R\circ\gamma^{-1}$, where $R$ is the usual reflection across $\mathbb{R}$. This seems too analytic and non-visual.
Kasner writes:"The function-theoretic definition of Schwarz may be stated in purely geometric language as follows: two points are symmetric with respect to a given curve provided the pairs of minimal lines determined by the points intersect on the given curve." But what are the "pairs of minimal lines"? Even if it means that point and its reflection should be on the same perpendicular to the curve (which is the case for lines and circles) it still does not tell us how far the reflected point should be.
Can someone explain what reflection across a curve means geometrically? In particular, what is the reflection of the real line $y=0$ across the parabola $y=x^2$?