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With the Cartesian equation it is easier to check whether a point lies on the circle or not. With the parametric version it is easier to obtain points on a plot. In some contexts, parametric equations involving only rational functions (that is fractions of two polynomials) are preferred, if they exist.
Equivalently, in polar coordinates (r, θ) it can be described by the equation = with real number b. Changing the parameter b controls the distance between loops. From the above equation, it can thus be stated: position of the particle from point of start is proportional to angle θ as time elapses.
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:
In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitization. [1] "
A parametric equation for a curve expresses the coordinates of the points of the curve as functions of a variable, called a parameter. [8] [9] For example, = = are parametric equations for the unit circle, where t is the parameter. Together, these equations are called a parametric representation of the curve.
Using the hyperbolic sine and cosine functions,, a parametric representation of the hyperbola = can be obtained, which is similar to the parametric representation of an ellipse: (, ), , which satisfies the Cartesian equation because =
Equation. An animated construction gives an idea of the complexity of the curve ... The curve is given by the following parametric equations: [2]
The scale factors for the parabolic coordinates (,) are equal = = + Hence, the infinitesimal element of area is = (+) and the Laplacian equals = + (+) Other differential operators such as and can be expressed in the coordinates (,) by substituting the scale factors into the general formulae found in orthogonal coordinates.