What is the implicit , explicit and parametric equations for ellipsoid?

Answer 1

The equation for an ellipsoid in standard form is given by:

(x - h)²/a² + (y - k)²/b² + (z - p)²/c² = 1

This ellipsoid has center (h,k,p) and three semi-axes a, b, and c.

The parametric equations of the ellipsoid are:

x = h + a cosθ sinφ
y = k + b sinθ sinφ
z = p + c cosφ

The angle θ is called the azimuthal angle. It measures the angle from the positive x-axis counterclockwise around the xy plane. The angle φ is called the polar angle. It measures the angle away from the positive z-axis.
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Let's look at the standard equation again.

(x - h)²/a² + (y - k)²/b² + (z - p)²/c² = 1

If two of the axes a, b, and c are equal we have a surface of revolution which can be obtained by rotating an ellipse around its major or minor axis. This surface or revolution is called a spheroid.

Spheroids come in two varieties. If you rotate the ellipse around its minor axis you get a "squashed" spheroid for which the equatorial radius is greater than the polar radius. This is called an "oblate spheroid". If you rotate the ellipse around its major axis you get a "pointy" spheroid for which the polar radius is greater than the equatorial radius. This is called a "prolate spheroid".

If all three axes a, b, and c are equal you get a sphere.

These comments are, in part, excerpted and edited from the link below:

http://mathworld.wolfram.com/Ellipsoid.html

Answer 2

Equation Of An Ellipsoid

Answer 3

Explicit, under your definition. Although atheism means one thing, there are many philosophies that are inherently atheistic. I always encourage atheists to stop reading just the pop authors of our time (Dawkins, Harris, Hitchens, etc.) who offer little that's important and nothing that's original. Instead, I suggest people see just how different the people have been in the history of atheism--people like Robert J. Ingersoll. AJ Ayers. John Stuart Mill. Karl Popper (I think many atheists here would resonate with his views on the philosophy of science) Friedrich Nietzche. John-Paul Sartre. George H. Smith (one of my personal favorites). Theodorus of Cyrene. Or the polar opposite pair of Karl Marx and Ayn Rand. Atheists include secular humanists, communists (which is more than just a political philosophy), Objectivists, apatheists, and (almost always) monists, and many others. Everybody who lives, ever lived, and ever will live has, had, or will have a philosophy--a way of understanding the world and acting within it. Some of those philosophies are atheistic. Atheism means only one thing, but some people get so caught up in that one thing that they miss the important part: the things we hold as true.

Answer 4

Let a, b, c = semiaxes lengths, then

Volume, V, = (88/21)abc .