Hyperbolas

Let there be two points F and F and a strictly positive value 2a. The locus of all points P such that the absolute difference between PF and PF = 2a will be a hyperbola. 

Parametric equations of hyperbola. Take in a plane two lines 1 and m with respective equations 

to obtain an equation of the curve, we eliminate the parameter t from the two equations. Eliminating t from Equation 1.76a and Equation 1.76b, we get 

Tangent to a point P of a hyperbola. The general equation of a hyper- 

The slope of the tangent line can be obtained by differentiating implicitly 

STRUCTURE semi axis semi ima axis center CASE d OF 

This is then normalised to produce a, the hyperbola is then defined for any real values of u  

The reference point is approximately a vertex position, used to define a major axis direction on the chosen branch of the hyperbola. The parameter range is infinite. 

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We can also visualise the subsurface as being made up of an enormous number of point scatterers or diffractors. (Fig b). Each contributes a diffraction curve (hyperbola) to the reflection section. Migration focuses the energy in these curves to a single point. 

Hyperbola Exponential curve Geometric curve Modified exponential curve Modified geometric curve... 

It is interesting to note that the BET equation is equivalent to the difference between the upper branches of two rectangular hyperbolae, as may be seen by breaking up the right-hand side of Equation (2.12) into partial fractions ... 

The graph of n/n against plp° will thus be obtained as the difference between the two hyperbolae represented by the equations... 

Conic Sections The cui ves included in this group are obtained from plane sections of the cone. They include the circle, ehipse, parabola, hyperbola, and degeneratively the point and straight line. A conic is the locus of a point whose distance from a fixed point called the focus is in a constant ratio to its distance from a fixea line, called the directrix. This ratio is the eccentricity e. lie = 0, the conic is a circle if 0 < e < 1, the conic is an ellipse e = 1, the conic is a parabola ... 

The Michaelis-Menten equation is, like Eq. (3-146), a rectangular hyperbola, and it can be cast into three linear plotting forms. The double-reciprocal form, Eq. (3-152), is called the Lineweaver-Burk plot in enzyme kinetics. ... 

FIGURE 14.7 Substrate saturation curve for au euzyme-catalyzed reaction. The amount of enzyme is constant, and the velocity of the reaction is determined at various substrate concentrations. The reaction rate, v, as a function of [S] is described by a rectangular hyperbola. At very high [S], v= Fnax- That is, the velocity is limited only by conditions (temperature, pH, ionic strength) and by the amount of enzyme present becomes independent of [S]. Such a condition is termed zero-order kinetics. Under zero-order conditions, velocity is directly dependent on [enzyme]. The H9O molecule provides a rough guide to scale. The substrate is bound at the active site of the enzyme. 

The Michaelis-Menten equation (14.23) describes a curve known from analytical geometry as a rectangular hyperbola. In such curves, as [S] is increased,... 

In this form, Alhas the units of torr.) The relationship defined by Equation (A15.4) plots as a hyperbola. That is, the MbOg saturation curve resembles an enzyme substrate saturation curve. For myoglobin, a partial pressure of 1 torr for jbOg is sufficient for half-saturation (Figure A15.1). We can define as the partial pressure of Og at which 50% of the myoglobin molecules have a molecule of Og bound (that is, F= 0.5), then... 

Symmetry 50. Intercepts 50. Asymptotes 50. Equations of Slope 51. Tangents 51. Equations of a Straight Line 52. Equations of a Circle 53. Equations of a Parabola 53. Equations of an Ellipse of Eccentricity e 54. Equations of a Hyperbola 55. Equations of Three-Dimensional Coordinate Systems 56. Equations of a Plane 56. Equations of a Line 57. Equations of Angles 57. Equation of a Sphere 57. Equation of an Ellipsoid 57. Equations of Hyperboloids and Paraboloids 58. Equation of an Elliptic Cone 59. Equation of an Elliptic Cylinder 59. 

Volume is inversely proportional to pressure. Figure 5.4 shows a typical plot of volume (V) versus pressure (P). Notice that Vdecreases as P increases. The graph is a hyperbola. The general relation between the two variables is... 

FIGURE 2.9 Amplification of stimulus through successive rectangular hyperbolae. The output from the first function (P = 0.3) becomes the input of a second function with the same coupling efficiency (P = 0.3), to yield a more efficiently coupled overall function (P = 0.069). Arrows indicate the potency for input to yield 50% maximal output for the first function and the series functions. 

It can be seen from Equation 2.2 that for positive non-zero values of p2, ptotai < pi- Therefore, the location parameter of the rectangular hyperbola of the composite set of reactions in series is shifted to the left (increased... 

FIGURE 2.16 Effects of successive rectangular hyperbolae on receptor stimulus, (a) Stimulus to three agonists, (b) Three rectangular hyperbolic stimulus-response functions in series. Function 1 ((3 = 0.1) feeds function 2 ((3 = 0.03), which in turn feeds function 3 ((3 = 0.1). (c) Output from function 1. (d) Output from function 2 (functions 1 and 2 in series), (e) Final response output from function 3 (all three functions in series). Note how all three are full agonists when observed as final response. 

Series hyperbolae can be modeled by a single hyperbolic function (2.11.1)... 

Series Hyperbolae Can Be Modeled by a Single Hyperbolic Function... 

The function f is usually hyperbolic, which introduces the nonlinearity between receptor occupancy and response. A common experimentally observed relationship between receptor stimulus and response is a rectangular hyperbola (see Chapter 2). Thus, response can be thought of as a hyperbolic function of stimulus ... 

A hyperbola of the form Response = Stimulus/ (Stimulus + [3) translates stimulus to response. Under these circumstances, response is given as... 

Hyperbola (hyperbolic), a set of functions defining nonlinear relationships between abscissae and ordinates. This term is used loosely to describe nonlinear relationships between the initial interaction of molecules and receptors and the observed response (i.e., stimulus-response cascades of cells). 

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