## Level curves

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A level curve of $$f(x,y)$$ is a curve on the domain that satisfies $$f(x,y) = k$$. It can be viewed as the intersection of the surface $$z = f(x,y)$$ and the horizontal plane $$z = k$$ projected onto the domain. The following diagrams shows how the level curves $f(x,y) = \dfrac{1}{\sqrt{1-x^2-y^2}} = k$ changes as $$k$$ changes. Note that the level curves are circles given by $x^2 + y^2 = 1-\dfrac{1}{k^2}$ for $$k \ge 1$$, for which the radii never exceed 1. For $$k < 1$$, the plane $$z = k$$ does not intersect with the surface and hence there are no level curves.

To view the interactive graph:

1. Make sure you have the latest version of Java 7 installed in your computer. Tablets and smartphones are not supported.