By Wassenaar J.
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Extra resources for 2d Curves
When we set the distance of the focus to the line to 1, we can write for a conic section with eccentricity e the polar equation at top of this page. In this equation it is quite clear that the polar inverse is the limaçon. The circular conic section is the circle. The orbit of a comet can in fact take the form of each of the conic sections. ) = cone circle conic section last updated: 2003−05−25 The circle is the ellipse of which the two axes are equal in length. Because of its symmetry the circle is considered as the perfect shape.
In 1690 Jacob Bernoulli found that the given curve is the semi−cubic parabola (change x and y in above formula). Also Huygens found this solution. You can also find this curve when considering an object as cause of the gravitation of, easy to imagine, the earth. The corresponding equation for the distance s(t) as function of the elapsed time t is s" = − g M / s ². It is easy to verify that the semicubic 2) parabola t(s) is a solution, isn't it? The curve is the evolute of the parabola, and the catacaustic of Tschirnhausen's cubic.
In the case of a chord without lamps, instead of a parabola, a catenary is formed. Jungius (1669) proved that Galileo was wrong to state this curve being a parabola. On holidays at the Dutch North Sea beach, you can see dunes in the form of a parabola (parabola dune), formed by the sand, thrown by the wind, but hold by the dune plants. When experimenting, one will often encounter quadratic relations between variables. So is the kinetic energy of a body proportional with the square velocity. In another branch of mathematics we find the iterated parabola, studied for the first time (in the 60s) by Myrberg.
2d Curves by Wassenaar J.