|
Fish Curve
A fish curve is an ellipse negative pedal curve that is shaped like a fish. In a fish curve, the pedal point is at the focus for the special case of the squared eccentricity e^2=\tfrac. The parametric equations for a fish curve correspond to those of the associated ellipse. Equations For an ellipse with the parametric equations :\textstyle , the corresponding fish curve has parametric equations :\textstyle . When the origin is translated to the node (the crossing point), the Cartesian equation A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ... can be written as: :\left(2x^2+y^2\right)^2-2 \sqrt ax\left(2x^2-3y^2\right)+2a^2\left(y^2-x^2\right)=0. Area The area of a fish curve is given by: :A=\frac \left, \int\ :=\frac a^2\left, \int\, so the area of the tail and head a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fish Curve
A fish curve is an ellipse negative pedal curve that is shaped like a fish. In a fish curve, the pedal point is at the focus for the special case of the squared eccentricity e^2=\tfrac. The parametric equations for a fish curve correspond to those of the associated ellipse. Equations For an ellipse with the parametric equations :\textstyle , the corresponding fish curve has parametric equations :\textstyle . When the origin is translated to the node (the crossing point), the Cartesian equation A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ... can be written as: :\left(2x^2+y^2\right)^2-2 \sqrt ax\left(2x^2-3y^2\right)+2a^2\left(y^2-x^2\right)=0. Area The area of a fish curve is given by: :A=\frac \left, \int\ :=\frac a^2\left, \int\, so the area of the tail and head a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negative Pedal Curve
In geometry, a negative pedal curve is a plane curve that can be constructed from another plane curve ''C'' and a fixed point ''P'' on that curve. For each point ''X'' ≠ ''P'' on the curve ''C'', the negative pedal curve has a tangent that passes through ''X'' and is perpendicular to line ''XP''. Constructing the negative pedal curve is the inverse operation to constructing a pedal curve. Definition In the plane, for every point ''X'' other than ''P'' there is a unique line through ''X'' perpendicular to ''XP''. For a given curve in the plane and a given fixed point ''P'', called the pedal point, the negative pedal curve is the envelope of the lines ''XP'' for which ''X'' lies on the given curve. Parameterization For a parametrically defined curve, its negative pedal curve with pedal point (0; 0) is defined as :X ,y\frac :Y ,y\frac Properties The negative pedal curve of a pedal curve with the same pedal point is the original curve. See also *Fish curve A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fish
Fish are aquatic, craniate, gill-bearing animals that lack limbs with digits. Included in this definition are the living hagfish, lampreys, and cartilaginous and bony fish as well as various extinct related groups. Approximately 95% of living fish species are ray-finned fish, belonging to the class Actinopterygii, with around 99% of those being teleosts. The earliest organisms that can be classified as fish were soft-bodied chordates that first appeared during the Cambrian period. Although they lacked a true spine, they possessed notochords which allowed them to be more agile than their invertebrate counterparts. Fish would continue to evolve through the Paleozoic era, diversifying into a wide variety of forms. Many fish of the Paleozoic developed external armor that protected them from predators. The first fish with jaws appeared in the Silurian period, after which many (such as sharks) became formidable marine predators rather than just the prey of arthropods. Mos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Focus (geometry)
In geometry, focuses or foci (), singular focus, are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an ''n''-ellipse. Conic sections Defining conics in terms of two foci An ellipse can be defined as the locus of points for which the sum of the distances to two given foci is constant. A circle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be more simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as the circle of Apollonius, in terms of two different foci, as the locus of points having a fixed ratio of distances to the two foci. A parabola is a li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eccentricity (geometry)
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. More formally two conic sections are similar if and only if they have the same eccentricity. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: * The eccentricity of a circle is zero. * The eccentricity of an ellipse which is not a circle is greater than zero but less than 1. * The eccentricity of a parabola is 1. * The eccentricity of a hyperbola is greater than 1. * The eccentricity of a pair of lines is \infty Definitions Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the eccentricity, commonly denoted as . The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section. If the cone is or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parametric Equation
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization (alternatively spelled as parametrisation) of the object. For example, the equations :\begin x &= \cos t \\ y &= \sin t \end form a parametric representation of the unit circle, where ''t'' is the parameter: A point (''x'', ''y'') is on the unit circle if and only if there is a value of ''t'' such that these two equations generate that point. Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors: :(x, y)=(\cos t, \sin t). Parametric representations are generally nonunique (see the "Examples in two dimensions" section belo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity (mathematics), eccentricity e, a number ranging from e = 0 (the Limiting case (mathematics), limiting case of a circle) to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola). An ellipse has a simple algebraic solution for its area, but only approximations for its perimeter (also known as circumference), for which integration is required to obtain an exact solution. Analytic geometry, Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is: : \frac+\frac = 1 . Assuming a \ge b, the foci are (\pm c, 0) for c = \sqrt. The standard parametric e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation Of Axes
In mathematics, a translation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x'y'''-Cartesian coordinate system in which the ''x''' axis is parallel to the ''x'' axis and ''k'' units away, and the ''y''' axis is parallel to the ''y'' axis and ''h'' units away. This means that the origin ''O''' of the new coordinate system has coordinates (''h'', ''k'') in the original system. The positive ''x''' and ''y''' directions are taken to be the same as the positive ''x'' and ''y'' directions. A point ''P'' has coordinates (''x'', ''y'') with respect to the original system and coordinates (''x''', ''y''') with respect to the new system, where or equivalently In the new coordinate system, the point ''P'' will appear to have been translated in the opposite direction. For example, if the ''xy''-system is translated a distance ''h'' to the right and a distance ''k'' upward, then ''P'' will appear to have been translated a distance ''h'' to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Equation
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference coordinate line is called a ''coordinate axis'' or just ''axis'' (plural ''axes'') of the system, and the point where they meet is its ''origin'', at ordered pair . The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, ''n'' Cartesian coordinates (an element of real ''n''-space) specify the point in an ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
