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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.
The classical convex polytopes may be considered tessellations, or tilings, of spherical space. Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less.
A curve is called equichordal when it has an equichordal point. [1] Such a curve may be constructed as the pedal curve of a curve of constant width. [2] For instance, the pedal curve of a circle is either another circle (when the center of the circle is the pedal point) or a limaçon; both are equichordal curves.
Carathéodory's theorem (convex hull) - If a point x of R d lies in the convex hull of a set P, there is a subset of P with d+1 or fewer points such that x lies in its convex hull. Choquet theory - an area of functional analysis and convex analysis concerned with measures with support on the extreme points of a convex set C .
The inner loop of the limaçon trisectrix has the desirable property that the trisection of an angle is internal to the angle being trisected. [6] Here, we examine the inner loop of r = 1 + 2 cos θ {\displaystyle r=1+2\cos \theta } that lies above the polar axis, which is defined on the polar angle interval π ≤ θ ≤ 4 π / 3 ...
In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ε-δ definition of uniform convexity as the modulus of continuity does to the ε-δ definition of continuity.
However, for locally convex, closed space curves, one can define tangent turning sign as (), where is the turning number of the stereographic projection of its tangent indicatrix. Its two values correspond to the two non-degenerate homotopy classes of locally convex curves.
From this it appears that, with Q = distance of the generating point from the rolling circle's center and with R being the radius of both circles (in particular of the rolling circle), the limaçon: is convex if Q/R < 1/2; is dimpled if 1/2 < Q/R < 1; is a cardioid if Q/R = 1 (with the generating point on the circumference of the rolling circle ...