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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.
The Fermat spiral with polar equation = can be converted to the Cartesian coordinates (x, y) by using the standard conversion formulas x = r cos φ and y = r sin φ.Using the polar equation for the spiral to eliminate r from these conversions produces parametric equations for one branch of the curve:
However, the origin satisfies the Cartesian equation given above, so the graph of this equation has an acnode or isolated point. When b > 2 a {\displaystyle b>2a} , the area bounded by the curve is convex, and when a < b < 2 a {\displaystyle a<b<2a} , the curve has an indentation bounded by two inflection points .
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 Kampyle of Eudoxus (Greek: καμπύλη [γραμμή], meaning simply "curved [line], curve") is a curve with a Cartesian equation of x 4 = a 2 ( x 2 + y 2 ) , {\displaystyle x^{4}=a^{2}(x^{2}+y^{2}),}
The equation of a circle is (x − a) 2 + (y − b) 2 = r 2 where a and b are the coordinates of the center (a, b) and r is the radius. Cartesian coordinates are named for René Descartes, whose invention of them in the 17th century revolutionized mathematics by allowing the expression of problems of geometry in terms of algebra and calculus.
In geometry, a strophoid is a curve generated from a given curve C and points A (the fixed point) and O (the pole) as follows: Let L be a variable line passing through O and intersecting C at K. Now let P 1 and P 2 be the two points on L whose distance from K is the same as the distance from A to K (i.e. KP 1 = KP 2 = AK ).
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.