A circle in R^2 is a set of points in a two-dimensional Euclidean space that are all equidistant from a fixed point, known as the center. The distance from the center to any point on the circle is called the radius, and this can be expressed mathematically as the set of points satisfying the equation $$x^2 + y^2 = r^2$$ for a circle centered at the origin, where 'r' is the radius. This geometric shape serves as a fundamental example of a 1-dimensional embedded submanifold in a higher-dimensional space.
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