|
The superparabola is a geometric curve defined in the Cartesian coordinate system as a set of points withwhere ''p'', ''a'', and ''b'' are positive integers. The equation defines an open curve in the rectangle The superparabola can vary in shape from a rectangular function }} , to a semi-ellipse ( }}, to a parabola }}, to a pulse function . == Mathematical properties == Without loss of generality we can consider the canonical form of the superparabola}} When the function describes a continuous differentiable curve on the plane. The curve can be described parametrically on the complex plane as The area under the curve is given by where is a global function valid for all , The area under a portion of the curve requires the indefinite integralwhere is the Gauss hypergeometric function. An interesting property is that any superparabola raised to a power is just another superparabola, thusThe centroid of the area under the curve is given by where the -component is zero by virtue of symmetry. Thus, the centroid can be expressed as one-half the ratio of the area of the square of the curve to the area of the curve. The ''n''th (mathematical) moment is given by The arc length of the curve is given by In general, integrals containing cannot be found in terms of standard mathematical functions. Even numerical solutions can be problematic for the improper integrals that arise when is singular at . Two instances of exact solutions have been found. For the semicircle , and the parabola , . The arc length is for both and has a minimum value of at . The area under the curve decreases monotonically with increasing . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Superparabola」の詳細全文を読む スポンサード リンク
|