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15 Φεβ 2022 · Pyramids, cubes, and spheres have often been used in the genre of science fiction, fantasy, and horror. Pyramids hold a special power especially when they are large and imposing in the world of...
5 ημέρες πριν · A cone with elliptical cross section. The parametric equations for an elliptic cone of height h, semimajor axis a, and semiminor axis b are x = a (h-u)/hcosv (1) y = b (h-u)/hsinv (2) z = u, (3) where v in [0,2pi) and u in [0,h]. The elliptic cone is a quadratic ruled surface, and has volume V=1/3piabh.
23 Ιουν 2024 · Conic sections are generated by the intersection of a plane with a cone (Figure 11.1.2). If the plane is parallel to the axis of revolution (the y -axis), then the conic section is a hyperbola. If the plane is parallel to the generating line, the conic section is a parabola.
4 Ιαν 2019 · An ellipse is a slice of a cone at an angle. This means it's the intersection of a plane ($ax+by+cz-d=0$) and a cone ($x^2 + y^2 - z^2=0$), which begets the equation $Ax^2 + Bxy+Cy^2+Dx+Ey+F=0$ for the case that $B^2-4AC<0$.
The curves known as conic sections, the ellipse, hyperbola, and parabola, were investigated intensely in Greek mathematics. The most famous work on the subject was the Conics, in eight books by Apollonius of Perga, but conics were also studied earlier by Euclid and Archimedes, among others.
An ellipse is a conic section, that resembles an oval, but is formally characterized by the following property: there exist two points \(F_1\) and \(F_2\) inside the ellipse (called focal points) such that for every point \(P\) on the ellipse, the quantity \(PF_1 + PF_2\) is constant \((\)where \(PF_i\) denotes the distance from \(P\) to \(F_i).\)
Learn all about ellipses in this video. The standard form for an ellipse centered at the origin is x²/a² + y²/b² = 1. The semi-major axis is the longest radius and the semi-minor axis is the shortest radius. The video also explains how to shift an ellipse.
