Search results
Results From The WOW.Com Content Network
The asymptotic directions are the same as the asymptotes of the hyperbola of the Dupin indicatrix through a hyperbolic point, or the unique asymptote through a parabolic point. [ 1 ] An asymptotic direction is a direction along which the normal curvature is zero: take the plane spanned by the direction and the surface's normal at that point.
Similarly, the separated equations for the Laplace equation can be obtained by setting = in the above. Each of the separated equations can be cast in the form of the Baer equation . Direct solution of the equations is difficult, however, in part because the separation constants α 2 {\displaystyle \alpha _{2}} and α 3 {\displaystyle \alpha _{3 ...
A complete Fermat's spiral (both branches) is a smooth double point free curve, in contrast with the Archimedean and hyperbolic spiral. Like a line or circle or parabola, it divides the plane into two connected regions. Definition of sector (light blue) and polar slope angle α
An elliptic paraboloid is shaped like an oval cup and has a maximum or minimum point when its axis is vertical. In a suitable coordinate system with three axes x , y , and z , it can be represented by the equation [ 1 ] z = x 2 a 2 + y 2 b 2 . {\displaystyle z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}.} where a and b are constants that ...
The curve of the chains of a suspension bridge is always an intermediate curve between a parabola and a catenary, but in practice the curve is generally nearer to a parabola due to the weight of the load (i.e. the road) being much larger than the cables themselves, and in calculations the second-degree polynomial formula of a parabola is used.
In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form y 2 − a 2 x 3 = 0 {\displaystyle y^{2}-a^{2}x^{3}=0} (with a ≠ 0 ) in some Cartesian coordinate system .
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.
A three-dimensional version of parabolic coordinates is obtained by rotating the two-dimensional system about the symmetry axis of the parabolas. Parabolic coordinates have found many applications, e.g., the treatment of the Stark effect and the potential theory of the edges.