The butterfly curve is a transcendental plane curve discovered by Temple H. Fay of University of Southern Mississippi in 1989. [1]
The butterfly curve is a transcendental plane curve discovered by Temple H. Fay of University of Southern Mississippi in 1989. [1]
The curve is given by the following parametric equations: [2]
or by the following polar equation:
The sin term has been added for purely aesthetic reasons, to make the butterfly appear fuller and more pleasing to the eye. [1]
In 2006, two mathematicians using Mathematica analyzed the function, and found variants where leaves, flowers or other insects became apparent. [3]
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
A tautochrone curve or isochrone curve is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point on the curve. The curve is a cycloid, and the time is equal to π times the square root of the radius over the acceleration of gravity. The tautochrone curve is related to the brachistochrone curve, which is also a cycloid.
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of which starts with trigonometric functions:
In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line, which may not intersect the generatrix. The surface created by this revolution and which bounds the solid is the surface of revolution.
A surface of revolution is a surface in Euclidean space created by rotating a curve one full revolution around an axis of rotation . The volume bounded by the surface created by this revolution is the solid of revolution.
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions, under suitably restricted domains. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
In geometry, a limaçon or limacon, 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. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called centered trochoids; more specifically, they are epitrochoids. The cardioid is the special case in which the point generating the roulette lies on the rolling circle; the resulting curve has a cusp.
In geometry, a cardioid is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion. A cardioid can also be defined as the set of points of reflections of a fixed point on a circle through all tangents to the circle.
In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an epicycle—which rolls without slipping around a fixed circle. It is a particular kind of roulette.
In mathematics, a rose or rhodonea curve is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates. Rose curves or "rhodonea" were named by the Italian mathematician who studied them, Guido Grandi, between the years 1723 and 1728.
In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle.
In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation It has two cusps and is symmetric about the y-axis.
In mathematics, a Clélie or Clelia curve is a curve on a sphere with the property:
In geometry, a strophoid is a curve generated from a given curve C and points A and O as follows: Let L be a variable line passing through O and intersecting C at K. Now let P1 and P2 be the two points on L whose distance from K is the same as the distance from A to K. The locus of such points P1 and P2 is then the strophoid of C with respect to the pole O and fixed point A. Note that AP1 and AP2 are at right angles in this construction.
In mathematics, Viviani's curve, also known as Viviani's window, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder that is tangent to the sphere and passes through two poles of the sphere. Before Viviani this curve was studied by Simon de La Loubère and Gilles de Roberval.
In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z).
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted as and .
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the Cartesian product of two circles S1
a and S1
b. It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3. To see why R4 is necessary, note that if S1
a and S1
b each exists in its own independent embedding space R2
a and R2
b, the resulting product space will be R4 rather than R3. The historically popular view that the Cartesian product of two circles is an R3 torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y.
In geometry, a limaçon trisectrix is the name for the quartic plane curve that is a trisectrix that is specified as a limaçon. The shape of the limaçon trisectrix can be specified by other curves particularly as a rose, conchoid or epitrochoid. The curve is one among a number of plane curve trisectrixes that includes the Conchoid of Nicomedes, the Cycloid of Ceva, Quadratrix of Hippias, Trisectrix of Maclaurin, and Tschirnhausen cubic. The limaçon trisectrix a special case of a sectrix of Maclaurin.
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