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The surface area can be calculated by integrating the length of the normal vector to the surface over the appropriate region D in the parametric uv plane: = | |. Although this formula provides a closed expression for the surface area, for all but very special surfaces this results in a complicated double integral , which is typically evaluated ...
A parametric equation for the sphere with radius > and center (,,) can be ... of all solids having a given surface area, the sphere is the one having the greatest ...
A sphere of radius r has surface area 4πr 2.. The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies. [1] The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with ...
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.
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 ...
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 (,,) = can be solved (at least implicitly) for x, y or z. But in general the solution may not be made explicit.
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 :
The following passage from page 673 shows how Hamilton uses biquaternion algebra and vectors from quaternions to produce hyperboloids from the equation of a sphere: ... the equation of the unit sphere ρ 2 + 1 = 0, and change the vector ρ to a bivector form, such as σ + τ √ −1. The equation of the sphere then breaks up into the system of ...