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In 1636, Fermat wrote a letter [3] to Marin Mersenne which contains the following special case: Let φ 1 = 0, φ 2 = 2π; then the area of the black region (see diagram) is A 0 = a 2 π 2, which is half of the area of the circle K 0 with radius r(2π). The regions between neighboring curves (white, blue, yellow) have the same area A = 2a 2 π 2 ...
Construction of the limaçon r = 2 + cos(π – θ) with polar coordinates' origin at (x, y) = ( 1 / 2 , 0). In geometry, a limaçon or limacon / ˈ l ɪ m ə s ɒ n /, also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius.
Definition of slope angle and sector Animation showing the constant angle between an intersecting circle centred at the origin and a logarithmic spiral. The logarithmic spiral r = a e k φ , k ≠ 0 , {\displaystyle r=ae^{k\varphi }\;,\;k\neq 0,} has the following properties (see Spiral ):
From the above equation, it can thus be stated: position of the particle from point of start is proportional to angle θ as time elapses. Archimedes described such a spiral in his book On Spirals . Conon of Samos was a friend of his and Pappus states that this spiral was discovered by Conon.
This image shows a graphical trifolium curve using its Cartesian Equation. The trifolium curve (also three-leafed clover curve, 3-petaled rose curve, and paquerette de mélibée) is a type of quartic plane curve. The name comes from the Latin terms for 3-leaved, defining itself as a folium shape with 3 equally sized leaves. It is described as
The commonly-used diagram for the Borromean rings consists of three equal circles centered at the points of an equilateral triangle, close enough together that their interiors have a common intersection (such as in a Venn diagram or the three circles used to define the Reuleaux triangle).
All these curves are roulettes with a circle rolled along another curve of uniform curvature. The cycloid, epicycloids, and hypocycloids have the property that each is similar to its evolute . If q is the product of that curvature with the circle's radius, signed positive for epi- and negative for hypo-, then the similitude ratio of curve to ...
The same relation between Cartesian coordinates would describe a hyperbola, and the hyperbolic spiral was first discovered by applying the equation of a hyperbola to polar coordinates. [2] Hyperbolic spirals can also be generated as the inverse curves of Archimedean spirals, [3] [4] or as the central projections of helixes. [5]