In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane . Hyperbolic coordinates take values in the hyperbolic plane defined as: . These coordinates in HP are useful for studying logarithmic comparisons of direct proportion in Q and measuring deviations from direct proportion. For in take and . The parameter u is the hyperbolic angle to (x, y) and v is the geometric mean of x and y. The inverse mapping is . The function is a continuous mapping, but not an analytic function. Alternative quadrant metric Since HP carries the metric space structure of the Poincaré half-plane model of hyperbolic geometry, the bijective correspondence brings this structure to Q. It can be grasped using the notion of hyperbolic motions. Since geodesics in HP are semicircles with centers on the boundary, the geodesics in Q are obtained from the correspondence and turn out to be rays from the origin or petal-shaped curves leaving and re-entering the origin. And the hyperbolic motion of HP given by a left-right shift corresponds to a squeeze mapping applied to Q. Since hyperbolas in Q correspond to lines parallel to the boundary of HP, they are horocycles in the metric geometry of Q. If one only considers the Euclidean topology of the plane and the topology inherited by Q, then the lines bounding Q seem close to Q. Insight from the metric space HP shows that the open set Q has only the origin as boundary when viewed through the correspondence. Indeed, consider rays from the origin in Q, and their images, vertical rays from the boundary R of HP. Any point in HP is an infinite distance from the point p at the foot of the perpendicular to R, but a sequence of points on this perpendicular may tend in the direction of p. The corresponding sequence in Q tends along a ray toward the origin. The old Euclidean boundary of Q is no longer relevant. Applications in physical science Fundamental physical variables are sometimes related by equations of the form k = x y. For instance, V = I R (Ohm's law), P = V I (electrical power), P V = k T (ideal gas law), and f λ = v (relation of wavelength, frequency, and velocity in the wave medium). When the k is constant, the other variables lie on a hyperbola, which is a