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The curve was first proposed and studied by René Descartes in 1638. [1] Its claim to fame lies in an incident in the development of calculus.Descartes challenged Pierre de Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines.
Plane curves can be represented in Cartesian coordinates (x, y coordinates) by any of three methods, one of which is the implicit equation given above. The graph of a function is usually described by an equation y = f ( x ) {\displaystyle y=f(x)} in which the functional form is explicitly stated; this is called an explicit representation.
The result corresponds to 256 / 81 (3.16049...) as an approximate value of π. [3] Book 3 of Euclid's Elements deals with the properties of circles. Euclid's definition of a circle is: A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal.
A detailed consideration shows: The midpoints of the circles lie on the perimeter of the fixed generator circle. (The generator circle is the inverse curve of the parabola's directrix.) This property gives rise to the following simple method to draw a cardioid: Choose a circle and a point on its perimeter, draw circles containing with centers ...
For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof. [1] [2] [3]
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 ...
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 ):
A curve, which intersects any curve of a given pencil of (planar) curves by a fixed angle is called isogonal trajectory. Between the slope η ′ {\displaystyle \eta '} of an isogonal trajectory and the slope y ′ {\displaystyle y'} of the curve of the pencil at a point ( x , y ) {\displaystyle (x,y)} the following relation holds: