MHB Elliptic Cylinder: What Determines Radius & Height?

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The formula for an elliptic cylinder is represented by the equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), where \(a\) and \(b\) correspond to the semi-major and semi-minor axes of the ellipse, not a radius. The center of the ellipse is at the origin, and if centered at \((h,k)\), the equation adjusts accordingly. The height of the cylinder can be defined by restricting the \(z\) coordinate, such as \(0 \le z \le h\). Therefore, different cylinders with varying radii and heights will have equations that reflect their specific axis lengths and height constraints. Understanding these parameters is crucial for accurately describing the geometry of elliptic cylinders.
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Hello all,

I have a theoretical question. This is the formula for an elliptic cylinder (right?)

\[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\]

I understand that the point (a,b) represent the center of the basic ellipse of the cylinder. What in the formula represent the radius of the basic ellipse ? What determines the height of the cylinder ?

In other words, if I have two cylinders, let's say both with a base centered at (1,1), but one has a radius of 2 and height 1, and the other radius 5 with height 4, how will the equations differ ?

Many thanks !
 
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The ellipse does not have a radius, but rather major and minor axes. The center of the cross-sections of the cylinder you cite is at the origin, not $(a,b)$. An ellipse in two dimensions centered at $(h,k)$ is given by:

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

To denote the height of the cylinder, you could place a restriction on $z$, such as $0\le z\le h$.
 
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