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By Wassenaar J.

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They studied the 'figure numbers' (or 'polygon numbers'), constructed from a regular polygon. In the following way, from a regular triangle: and so on. This results in the triangular numbers 1, 3, 6, 10, 15, and so on. In Pascal's triangle we find these same numbers als binomial coefficients (n over k) 1) : 1 1 1 1 1 1 1 2 3 4 5 1 3 6 10 4 10 1 1 5 1 These coefficients are often used in probability theory and also in formulae for the th th n power and the n derivative. In probability theory the triangle diagram, is used, a way to express the a function of three variables, in a triangle grid.

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. ' Tschirnhaus(en)'s cubic or Two of the three real roots are equal. The curve has two forms, the first is more common in curves' literature.

Sometimes a distribution can not be matched with a hyperbola, the Mandelbrot−Zipf −a 5) . law makes a generalization towards the power function: y = x In another math region we can find the hyperbolic geometry (or geometry of Lobachevsky), one of the non−Euclidean geometries, based on the axiom that there exist at least two lines through a point P that are parallel to a line not through P. The hyperbolic functions have not much in common with the hyperbola. Their name is based on the fact that two hyperbolic functions can be drawn geometrically in a hyperbola.

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2d Curves by Wassenaar J.


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