Lesson plan
hyperbolic coordinates
In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane : \{(x, y) \ :\ x > 0,\ y > 0\ \} = Q . Hyperbolic coordinates take values in the hyperbolic plane defined as: : HP = \{(u, v) : u \in \mathbb{R}, v > 0 \} . These coordinates in HP are useful for studying logarithmic comparisons of direct proportion in Q and measuring deviations from direct proportion. For (x,y) in Q take : u = \ln \sqrt{\frac{x}{y}} and : v = \sqrt{xy} . The parameter u is the hyperbolic angle to (x, y) and v is the geometric mean of x and y. The inverse mapping is : x = v e^u ,\quad y = v e^{-u} . The function Q \rarr HP is a continuous mapping, but not an analytic function.
Article summary
In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane : \{(x, y) \ :\ x > 0,\ y > 0\ \} = Q . Hyperbolic coordinates take values in the hyperbolic plane defined as: : HP = \{(u, v) : u \in \mathbb{R}, v > 0 \} . These coordinates in HP are useful for studying logarithmic comparisons of direct proportion in Q and measuring deviations from direct proportion. For (x,y) in Q take : u = \ln \sqrt{\frac{x}{y}} and : v = \sqrt{xy} . The parameter u is the hyperbolic angle to (x, y) and v is the geometric mean of x and y. The inverse mapping is : x = v e^u ,\quad y = v e^{-u} . The function Q \rarr HP is a continuous mapping, but not an analytic function.
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