Q. Circular section “hyperboloid of one sheet” radius

Q: What is the standard equation of a hyperboloid of one sheet?

The standard centered form is \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 . Its symmetry axis is the z -axis; horizontal z = \text{const} sections are ellipses (or circles when a = b ).

Q: How do I get the radius of a circular horizontal cross-section z = z_0 when a = b?

For a = b, the horizontal section is a circle of radius r(z_0) = a \sqrt{1 + \frac{z_0^2}{c^2}}. The minimal “waist” radius is r(0) = a.

Q: How do I test whether the intersection with an arbitrary plane ux + vy + wz = d is a circle?.

Solve for one variable (e.g., z = (d - ux - vy)/w) and substitute into x^2/a^2 + y^2/b^2 - z^2/c^2 = 1. Form the quadratic Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0; it is a circle iff B = 0 and A = C (after completing squares).

Q: How do I find the radius of the circle in an arbitrary plane once I know it’s a circle?

After substitution and completing the square, rewrite as (x - h)^2 + (y - k)^2 = R^2 . Then R^2 = (constant term after translation) / (common quadratic coefficient). Compute numerically from the substituted coefficients.

Q: What is a convenient parametrization and how does it give radii of circular parallels?

Parametrization: x = a \cosh u \cos v, y = b \cosh u \sin v, z = c \sinh u. For a = b, fixing u gives a circle radius a \cosh u; relate z by z = c \sinh u so radius = a \sqrt{1 + z^2/c^2}.

Q: Example: what are the semi-axes for the horizontal section z = z_0 when a \ne b ?

The intersection is the ellipse x^2/a^2 + y^2/b^2 = 1 + z0^2/c^2, so semi-axes are a \sqrt{1 + z0^2/c^2} and b \sqrt{1 + z0^2/c^2}. It's a circle only if a = b..

Answer

For an axisymmetric hyperboloid of one sheet with equation
\[
\frac{x^2}{a^2}+\frac{y^2}{a^2}-\frac{z^2}{c^2}=1,
\]
a horizontal cross section at height z is
\[
\frac{x^2+y^2}{a^2}=1+\frac{z^2}{c^2},
\]
hence the circular radius is
\[
r(z)=a\sqrt{1+\frac{z^2}{c^2}}.
\]
(For a=b=1,c=1 this gives r(z)=\sqrt{1+z^2}.)

Detailed Explanation

We are asked to find the radius of a circular section of a hyperboloid of one sheet. I will assume the standard axisymmetric form (so the horizontal cross sections are circles) and proceed step by step with detailed explanation.

Write the equation of the hyperboloid of one sheet in its axisymmetric form. Use parameters a>0 and c>0 where a controls the horizontal scale and c controls the vertical scale. The surface is

\[ \frac{x^{2}}{a^{2}} \;+\; \frac{y^{2}}{a^{2}} \;-\; \frac{z^{2}}{c^{2}} \;=\; 1. \]

Because the coefficients of x^2 and y^2 are equal, horizontal cross sections (planes parallel to the xy-plane) will be circles.
Intersect the surface with a horizontal plane z = z_{0}. Substitute z = z_{0} into the surface equation to obtain the equation of the intersection curve in the plane z = z_{0}:

\[ \frac{x^{2}}{a^{2}} \;+\; \frac{y^{2}}{a^{2}} \;-\; \frac{z_{0}^{2}}{c^{2}} \;=\; 1. \]
Rearrange this equation to isolate the quadratic form in x and y. Move the z_{0}^{2}/c^{2} term to the right-hand side:

\[ \frac{x^{2}}{a^{2}} \;+\; \frac{y^{2}}{a^{2}} \;=\; 1 \;+\; \frac{z_{0}^{2}}{c^{2}}. \]

Factor the left-hand side as (x^{2}+y^{2})/a^{2}:

\[ \frac{x^{2}+y^{2}}{a^{2}} \;=\; 1 \;+\; \frac{z_{0}^{2}}{c^{2}}. \]
Multiply both sides by a^{2} to obtain the standard circle equation in the plane z=z_{0}:

\[ x^{2}+y^{2} \;=\; a^{2}\!\left(1 \;+\; \frac{z_{0}^{2}}{c^{2}}\right). \]

This is the equation of a circle centered at the origin of the plane z=z_{0} with radius squared equal to the right-hand side.
Take the positive square root to get the radius r as a function of z_{0}:

\[ r(z_{0}) \;=\; a \,\sqrt{\,1 \;+\; \frac{z_{0}^{2}}{c^{2}}\,}. \]

Thus the radius of the circular cross-section at height z_{0} is a times the square root of 1 plus (z_{0}^{2}/c^{2}). If you prefer to write the radius as a function of z, replace z_{0} by z:

\[ r(z) \;=\; a \,\sqrt{\,1 \;+\; \frac{z^{2}}{c^{2}}\,}. \]
Remark on the general (noncircular) case: If the hyperboloid has the general form

\[ \frac{x^{2}}{a^{2}} \;+\; \frac{y^{2}}{b^{2}} \;-\; \frac{z^{2}}{c^{2}} \;=\; 1, \]

then the horizontal cross section at z=z_{0} is an ellipse with semi-axes

\[ a\,\sqrt{\,1+\frac{z_{0}^{2}}{c^{2}}\,} \quad\text{and}\quad b\,\sqrt{\,1+\frac{z_{0}^{2}}{c^{2}}\,}. \]

Only when a=b does this ellipse become a circle, recovering the formula above.

Final answer: For the axisymmetric hyperboloid of one sheet \[ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{a^{2}}-\dfrac{z^{2}}{c^{2}}=1, \] the radius of the circular cross-section at height z is

\[ r(z)=a\,\sqrt{1+\dfrac{z^{2}}{c^{2}}}\,. \]

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Geometry FAQs

What is the standard equation of a hyperboloid of one sheet?

The standard centered form is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \). Its symmetry axis is the \( z \)-axis; horizontal \( z = \text{const} \) sections are ellipses (or circles when \( a = b \)).

How do I get the radius of a circular horizontal cross-section \(z = z_0\) when \(a = b\)?

For \(a = b\), the horizontal section is a circle of radius \(r(z_0) = a \sqrt{1 + \frac{z_0^2}{c^2}}\). The minimal “waist” radius is \(r(0) = a\).

What are the conditions for a cross-section to be a circle?

After substituting the plane into the quadric, the resulting conic in x,y must be of the form \( (x - h)^2 + (y - k)^2 = R^2 \). Equivalently, the \(x^2\) and \(y^2\) coefficients must be equal and the \(xy\) coefficient zero (after coordinate translation/rotation as needed). .

How do I test whether the intersection with an arbitrary plane \(ux + vy + wz = d\) is a circle?.

Solve for one variable (e.g., \(z = (d - ux - vy)/w\)) and substitute into \(x^2/a^2 + y^2/b^2 - z^2/c^2 = 1\). Form the quadratic \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\); it is a circle iff \(B = 0\) and \(A = C\) (after completing squares).

How do I find the radius of the circle in an arbitrary plane once I know it’s a circle?

After substitution and completing the square, rewrite as \( (x - h)^2 + (y - k)^2 = R^2 \). Then \( R^2 = \) (constant term after translation) / (common quadratic coefficient). Compute numerically from the substituted coefficients.

Can a hyperboloid of one sheet contain tilted (non-horizontal) circles even when \(a \ne b\)?

What is a convenient parametrization and how does it give radii of circular parallels?

Parametrization: \(x = a \cosh u \cos v\), \(y = b \cosh u \sin v\), \(z = c \sinh u\). For \(a = b\), fixing \(u\) gives a circle radius \(a \cosh u\); relate \(z\) by \(z = c \sinh u\) so radius \(= a \sqrt{1 + z^2/c^2}\).

Example: what are the semi-axes for the horizontal section \( z = z_0 \) when \( a \ne b \) ?

The intersection is the ellipse \(x^2/a^2 + y^2/b^2 = 1 + z0^2/c^2\), so semi-axes are \(a \sqrt{1 + z0^2/c^2}\) and \(b \sqrt{1 + z0^2/c^2}\). It's a circle only if \(a = b\)..

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