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A parametric equation for the sphere with radius > and center (,,) can be ... is also the equation of a sphere for arbitrary values of the parameters s and t.
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 if 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, called a parametric surface.
The sphere is a rational surface. ... Let the parametric surface be given by the equation = (,), where is a vector-valued function of the ...
Parametric equations of surfaces are often irregular at some points. For example, all but two points of the unit sphere, are the image, by the above parametrization, of exactly one pair of Euler angles (modulo 2 π). For the remaining two points (the north and south poles), one has cos v = 0, and the longitude u may take any values. Also, there ...
Such a parametric equation completely determines the curve, without the need of any interpretation of t as time, and is thus called a parametric equation of the curve (this is sometimes abbreviated by saying that one has a parametric curve). One similarly gets the parametric equation of a surface by considering functions of two parameters t and u.
The 3-sphere is the boundary of a -ball in four-dimensional space. The -sphere is the boundary of an -ball. Given a Cartesian coordinate system, the unit -sphere of radius can be defined as:
For a plane, a sphere, and a torus there exist simple parametric representations. This is not true for the fourth example. The implicit function theorem describes conditions under which an equation F ( x , y , z ) = 0 {\displaystyle F(x,y,z)=0} can be solved (at least implicitly) for x , y or z .
With help of this parametric representation it is easy to prove the statement: The area of the half sphere (containing Viviani's curve) minus the area of the two windows is . The area of the upper right part of Viviani's window (see diagram) can be calculated by an integration :