Cylinder

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First let's have a quick review of circles. Here are two circles of radius $r$, one centered at the origin $(0, 0)$ and the other centered at coordinates $(h, k)$:

The equation of the circle centered at the origin is

$$x^2 + y^2 = r^2$$

The dashed circle is the same, just translated from the origin $h$ units to the right and $k$ units upward. Its equation (the general equation of a circle centered at $(h, k)$ and of radius $r$) is

$$(x - h)^2 + (y - k)^2 = r^2$$

The cylinder is just a stretching of the circle along the axis orthogonal to the plane containing the circle, in this case, the $z$ axis:

Ellipsoid

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The equation of a sphere of radius $r$ is

$$\frac{x^2}{r^2} + \frac{y^2}{r^2} + \frac{z^2}{r^2} = 1.$$

If we modify that equation so that there is a different "radius" associated with each direction $(x, y, z)$, then we have an ellipsoid:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \tag{1}$$

where $a, \; b$ and $c$ are the radii along each of the $x, \; y$ and $z$ axes, respectively.

In a more general way, we can locate an ellipsoid with its center at any 3D coordinates $(h, k, l)$ with the equation

$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} + \frac{(z-l)^2}{c^2} = 1,$$

In most cases, we'll locate our ellipsoid at the origin of our coordinate system, so equation (1) is the most important.

Paraboloid

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The paraboloid is just an extension of the 2D parabola into three dimensions. There are two flavors of paraboloid, elliptical paraboloid, which resembles a bowl, and the hyperbolic paraboloid, which ... well, it's like a paraboloid turned inside-out.

Elliptical

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An elliptical paraboloid with vertex $(0, 0, 0)$ has the form

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = z,$$

where $a$ and $b$ are parameters. This is the equation of a paraboloid with its vertex at the origin and with horizontal (parallel to the $x-y$ plane) cross sections that are ellipses (circles if $a = b$). Here's an example, the paraboloid

$$x^2 + y^2 = z$$

We can show that the cross sections parallel to the $x-y$ plane are circles (ellipses if $a \ne b$) if we set $z$ equal to a constant, $k$: $x^2 + y^2 = k$ is the quation of a circle.

Likewise, if we set $x$ or $y$ equal to zero, we get the trace of the function in the $x-z$ or $y-z$ plane, respectively:

$$\frac{x^2}{a^2} = z \phantom{00000} \frac{y^2}{b^2} = z$$

Hyperbolic

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The equation of a hyperbola centered at the origin of a 2D coordinate system is

$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text{ or } \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$$

The first opens in the left-right direction and the second upward and downward:

The hyperbolic paraboloid is just a 2D hyperbola,

Hyperboloid

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The hyperboloid comes in two forms, hyperboloids of one sheet and of two sheets. Here's an example of the former, the hyperboloid

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$$

Here is a 3D graph of the hyperboloid $x^2 + y^2 - z^2 = 1$. The horizontal traces (parallel to the $xy$ plane) are ellipses, or in this case, in which $a = b$, circles. The traces in the $xz$ and $yz$ planes are hyperbolas. You can prove that to yourself by setting $x$ or $y$ equal to zero and finding the resulting 2D equation.

Elliptical cone

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The equation of a paraboloid is

$$z = \frac{x^2}{a^2} + \frac{y^2}{b^2}$$

If we square the $z$, we effectively linearize this equation to form a cone — actually two cones because of the squaring. Here is the graph of the simple elliptical cone, $z^2 = x^2 + y^2$

The general form of an elliptical cone is

$$\frac{z^2}{c^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2}$$

The openings of the opposing cones lie along the axis with its variable alone on one side of the equal sign.

For example, the elliptical cone

$$\frac{x^2}{a^2} = \frac{z^2}{c^2} + \frac{y^2}{b^2}$$

opens along the $x$-axis.