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The discussion of plane sections can be performed for the unit hyperboloid of two sheets with equation : + = which can be generated by a rotating hyperbola around one of its axes (the one that cuts the hyperbola) A plane with slope less than 1 (1 is the slope of the asymptotes of the generating hyperbola) intersects either in an ellipse or in a ...
A hyperboloid of one sheet is a doubly ruled surface: it can be generated by either of two families of straight lines. Four images of hyperboloid towers. The Shukhov Tower in Polibino , the world's first hyperboloid structure , a water tower by Vladimir Shukhov at the All-Russian Exposition in Nizhny Novgorod , Russia
A 1910 paper [38] by Geiger, The Scattering of the α-Particles by Matter, describes an experiment to measure how the most probable angle through which an alpha particle is deflected varies with the material it passes through, the thickness of the material, and the velocity of the alpha particles. He constructed an airtight glass tube from ...
The vectors v ∈ R n+1 such that Q(v) = -1 form an n-dimensional hyperboloid S consisting of two connected components, or sheets: the forward, or future, sheet S +, where x 0 >0 and the backward, or past, sheet S −, where x 0 <0. The points of the n-dimensional hyperboloid model are the points on the forward sheet S +.
Hyperbola: the midpoints of parallel chords lie on a line. Hyperbola: the midpoint of a chord is the midpoint of the corresponding chord of the asymptotes. The midpoints of parallel chords of a hyperbola lie on a line through the center (see diagram). The points of any chord may lie on different branches of the hyperbola.
A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points (a, 1/a) and (b, 1/b) on the rectangular hyperbola xy = 1, or the corresponding region when this hyperbola is re-scaled and its orientation is altered by a rotation leaving the center at the origin, as with the unit hyperbola.
This problem is known as the primitive circle problem, as it involves searching for primitive solutions to the original circle problem. [9] It can be intuitively understood as the question of how many trees within a distance of r are visible in the Euclid's orchard , standing in the origin.
The Three Dunce Caps Theorem then says that P 1, P 2, and P 3 all lie on the same line. [4] Proof: Construct a sphere on top of each circle and then construct a plane tangent to these three spheres. The plane intersects the plane that the circles lies on at a straight line containing P 1, P 2, and P 3. These points are also the centers of ...