New PDF release: 2D Curves

By Wassenaar J.

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Perimeter, contour. 6) The ultimate proof was given not earlier than in the second part of the 19th century, by Weierstrass. 7) To be compared with the sine. ) = two. 9) Davis 1981 p. 100. 10) Ku−klux is derived from the Greek word for circle: kyklos. ellipse conic section last updated: 2003−05−20 The word ellipse is derived from the Greek word elleipsis (defect). The same defect do we find in the linguistic ellipse: omitting a part of a sentence, in situations that the part can be thought easily.

In the parabola curve the parabola (with its vertex oriented downwards) is being repeated infinitely. Some properties of the parabola: • the parabola is the involute of the semi−cubic parabola • a path of a parabola is formed when the involute of a circle rolls over a line • the curve is a specimen of the sinusoidal spiral • it is the pedal of the Tschirnhausen's cubic Besides, the following curves can be derived from the parabola: kind of derived curve catacaustic rays perpendicular to the axis isoptic orthoptic curve Tschirnhausen's cubic hyperbola the parabola's directrix cardioid polar inverse focus as the center of inversion polar inverse cissoid vertex as the center of inversion roulette a parabola rolls over another (equal) parabola roulette a parabola rolls over a line cissoid: the path of the vertex catenary: the path of the focus Pedals of the parabola are given by: a pedal point pedal of the parabola 0 vertex cissoid (of Diocles) 1 foot of − intersection of (right) axis and − directrix strophoid 3 reflection of focus in trisectrix of directrix MacLaurin − on directrix oblique strophoid − focus line The Italian Luca Valerio determined the area of a parabola, in 1606; is was called the quadrature of the parabola.

Where do we see the hyperbola in our world? Well, take a cistern, filled with water, and place two oscillators, with same frequency and amplitude. Then you see an extinction in the form of hyperbolas, as result of the constant difference in distance between the two foci (oscillators), on a hyperbola. Above that, we find the hyperbola in various branches of science as variables, that are inversely proportional related. It appears that the attention for the recent past is inversely proportional to the passed 3 time ).

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2D Curves by Wassenaar J.

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