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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 (geometry), focus for the special case of the squared eccentricity (geometry), 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 translation of axes, translated to the node (the crossing point), the Cartesian equation 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. Properties Area The area of a fish curve is given by: \begin A &= \frac \left, \int\ \\ &= \frac a^2\left, \int\, \end so the area of the tail and head are given by: \begin A_ &= \left(\frac -\frac \right)a^2, \\ A_ &= \left(\frac +\frac \right)a^2, \end giving the overall area for the fish as: A = \frac a^2. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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''. 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 Examples The negative pedal curve of a line is a parabola. The negative pedal curves of a circle are an ellipse if ''P'' is chosen to be inside t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fish
A fish (: fish or fishes) is an aquatic animal, aquatic, Anamniotes, anamniotic, gill-bearing vertebrate animal with swimming fish fin, fins and craniate, a hard skull, but lacking limb (anatomy), limbs with digit (anatomy), digits. Fish can be grouped into the more basal (phylogenetics), basal jawless fish and the more common jawed fish, the latter including all extant taxon, living cartilaginous fish, cartilaginous and bony fish, as well as the extinct placoderms and acanthodians. In a break to the long tradition of grouping all fish into a single Class (biology), class (Pisces), modern phylogenetics views fish as a paraphyletic group. Most fish are ectotherm, cold-blooded, their body temperature varying with the surrounding water, though some large nekton, active swimmers like white shark and tuna can hold a higher core temperature. Many fish can communication in aquatic animals#Acoustic, communicate acoustically with each other, such as during courtship displays. The stud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Focus (geometry)
In geometry, focuses or foci (; : 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, ''n''-ellipse. Conic sections Defining conics in terms of two foci An ellipse can be defined as the locus (mathematics), 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 Circles of Apollonius, circle of Apollonius, in terms of two different foci, as the locus of points having a fixed ratio of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Eccentricity (geometry)
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. 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 0. * The eccentricity of a non-circular ellipse is between 0 and 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. Two conic sections with the same eccentricity are similar. 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 oriented with its axis vertical, the eccentricity is : e = \frac, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Parametric Equation
In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters. In the case of a single parameter, parametric equations are commonly used to express the trajectory of a moving point, in which case, the parameter is often, but not necessarily, time, and the point describes a curve, called a parametric curve. In the case of two parameters, the point describes a Surface (mathematics), surface, called a parametric surface. In all cases, the equations are collectively called a parametric representation, or parametric system, or parameterization (also spelled parametrization, 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 is the parameter: A point is on the unit circle if and only if there is a value of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 for Perimeter of an ellipse, its perimeter (also known as circumference), Integral, integration is required to obtain an exact solution. The largest and smallest diameters of an ellipse, also known as its width and height, are typically denoted and . An ellipse has four extreme points: two ''Vertex (geometry), vertices'' at the endpoints of the major axis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cartesian Equation
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular oriented lines, called ''coordinate lines'', ''coordinate axes'' or just ''axes'' (plural of ''axis'') of the system. The point where the axes meet is called the ''origin'' and has as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame. Similarly, the position of any point in three-dimensional space can be specified by three ''Cartesian coordinates'', which are the signed distances from the point to three mutually perpendicular planes. More generally, Cartesian coordinates specify the point in an -dimensional Euclidean space for any dimension . These coordinates are the signed distances from the point to mutually perpendicular fixed hyperpl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Tangential Angle
In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the -axis. (Some authors define the angle as the deviation from the direction of the curve at some fixed starting point. This is equivalent to the definition given here by the addition of a constant to the angle or by rotating the curve.) Equations Parametric If a curve is given parametrically by , then the tangential angle at is defined (up to a multiple of ) by : \frac = (\cos \varphi,\ \sin \varphi). Here, the prime symbol denotes the derivative with respect to . Thus, the tangential angle specifies the direction of the velocity vector , while the speed specifies its magnitude. The vector :\frac is called the unit tangent vector, so an equivalent definition is that the tangential angle at is the angle such that is the unit tangent vector at . If the curve is parametrized by arc length , so , then the de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
