Analytic geometry

Most of the situations encountered in capillarity involve figures of revolution, and for these it is possible to write down explicit expressions for and R2 by choosing plane 1 so that it passes through the axis of revolution. As shown in Fig. II-7n, R then swings in the plane of the paper, i.e., it is the curvature of the profile at the point in question. R is therefore given simply by the expression from analytical geometry for the curvature of a line... 

The exact treatment of capillary rise must take into account the deviation of the meniscus from sphericity, that is, the curvature must correspond to the AP = Ap gy at each point on the meniscus, where y is the elevation of that point above the flat liquid surface. The formal statement of the condition is obtained by writing the Young-Laplace equation for a general point (x, y) on the meniscus, with R and R2 replaced by the expressions from analytical geometry given in... 

Johnson, R. E., and F. L. Kiokemeister. Calculus with Analytic Geometry, 4th ed., Allyn and Bacon, Boston (1969). 

Levy, H. Analytic Geometry, Harcourt, Brace World, New York (1969). 

Shenk, A. Calculus and Analytic Geometry. Goodyear Publishing Company, Santa Monica, CA (1977). 

Swokowsld, E. W. Calculus with Analytic Geometry. 2d ed., Prindle, Weber Schmidt, Boston (1981). 

Analytic geometry uses algebraic equations and methods to study geometric problems. It also permits one to visualize algebraic equations in terms of geometric curves, which frequently clarifies abstract concepts. 

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

Thomas, G. B., and Finney, R. L., Calculus and Analytic Geometry, 5th Edition, Addison Wesley, Reading, 1979. 

The barrier to EE between n and c, where the two parabolas cross, is the point with one-fourth the vertical energy gap A, owing to the analytical geometry of a parabola. This value of was incorporated in Eq. (10-57) and is explicitly treated in Problem 10-22. Reference should be made to Fig. 10-11. 

Something that is not obvious from the simple sag equation is how the local radii of curvature change with radial distance, p. Using analytical geometry one can show that the local radius of curvature in the radial direction goes as... 

From analytical geometry follows that the total curvature of the plumb line is... 

Fuller, Gordon and Parker, Robert M., Analytic Geometry and Calculus, D. Van Nostrand Company, Princeton, New Jersey (1964). 

Coordinate Systems The basic concept of analytic geometry is the establishment of a one-to-one correspondence between the points of the plane and number pairs (x, y). This correspondence may be done in a number of ways. The rectangular or cartesian coordinate system consists of two straight lines intersecting at right angles (Fig. 3-12). A point is designated by (x, y), where x (the abscissa) is the distance of the point from the y axis measured parallel to the x axis,... 

---