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The area enclosed between a parabola and a chord (see diagram) is two-thirds of the area of a parallelogram that surrounds it. One side of the parallelogram is the chord, and the opposite side is a tangent to the parabola. [16] [17] The slope of the other parallel sides is irrelevant to the area. Often, as here, they are drawn parallel with the ...
The unit hyperbola is a special case of the rectangular hyperbola, with a particular orientation, location, and scale. As such, its eccentricity equals [1] The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction of spacetime as a pseudo ...
This is the equation of an ellipse (<) or a parabola (=) or a hyperbola (>). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).
Conic sections of varying eccentricity sharing a focus point and directrix line, including an ellipse (red, e = 1/2), a parabola (green, e = 1), and a hyperbola (blue, e = 2). The conic of eccentricity 0 in this figure is an infinitesimal circle centered at the focus, and the conic of eccentricity ∞ is an infinitesimally separated pair of lines.
A Lissajous curve / ˈlɪsəʒuː /, also known as Lissajous figure or Bowditch curve / ˈbaʊdɪtʃ /, is the graph of a system of parametric equations. which describe the superposition of two perpendicular oscillations in x and y directions of different angular frequency ( a and b). The resulting family of curves was investigated by Nathaniel ...
Paraboloid. In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plane section of a paraboloid by a plane parallel to the axis of symmetry is a parabola.
A radial hyperbolic trajectory is a non-periodic trajectory on a straight line where the relative speed of the two objects always exceeds the escape velocity. There are two cases: the bodies move away from each other or towards each other. This is a hyperbolic orbit with semi-minor axis = 0 and eccentricity = 1.
Diagram of forces acting on a segment of a catenary from c to r. The forces are the tension T 0 at c, the tension T at r, and the weight of the chain (0, −ws). Since the chain is at rest the sum of these forces must be zero. A differential equation for the curve may be derived as follows. [50]