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In the theory of quadratic forms, the parabola is the graph of the quadratic form x 2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x 2 + y 2 (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form x 2 − y 2. Generalizations to more variables yield ...
Successive parabolic interpolation is a technique for finding the extremum (minimum or maximum) of a continuous unimodal function by successively fitting parabolas (polynomials of degree two) to a function of one variable at three unique points or, in general, a function of n variables at 1+n(n+3)/2 points, and at each iteration replacing the "oldest" point with the extremum of the fitted ...
If two of these three lengths are known, this equation can be used to calculate the third. A more complex calculation is needed to find the diameter of the dish measured along its surface . This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of material, usually metal, which is the right size to be ...
Graph of Johnson's parabola (plotted in red) against Euler's formula, with the transition point indicated. The area above the curve indicates failure. The Johnson parabola creates a new region of failure. In structural engineering, Johnson's parabolic formula is an empirically based equation for calculating the critical buckling stress of a column.
The equation for a conic section with apex at the origin and tangent to the y axis is + (+) = alternately = + (+) where R is the radius of curvature at x = 0. This formulation is used in geometric optics to specify oblate elliptical ( K > 0 ), spherical ( K = 0 ), prolate elliptical ( 0 > K > −1 ), parabolic ( K = −1 ), and hyperbolic ( K ...
The zero level set F(t 0,(x,y)) = 0 gives the equation of the tangent line to the parabola at the point (t 0,t 0 2). The equation t 2 – 2tx + y = 0 can always be solved for y as a function of x and so, consider + = Substituting = / gives the ODE
Muller's method fits a parabola, i.e. a second-order polynomial, to the last three obtained points f(x k-1), f(x k-2) and f(x k-3) in each iteration. One can generalize this and fit a polynomial p k,m (x) of degree m to the last m+1 points in the k th iteration. Our parabola y k is written as p k,2 in this notation.
While a parabolic arch may resemble a catenary arch, a parabola is a quadratic function while a catenary is the hyperbolic cosine, cosh(x), a sum of two exponential functions. One parabola is f(x) = x 2 + 3x − 1, and hyperbolic cosine is cosh(x) = e x + e −x / 2 . The curves are unrelated.