Hyperbolic and Lemniscate Spirals

Summary: §527. Two spirals related to the spiral-of-archimedes by analogy with the conic curves. The hyperbolic spiral $z=a/s$ approaches a straight-line asymptote at infinite distance — John Bernoulli’s name, after the hyperbola–asymptote relationship. The lemniscate spiral $z=a\sqrt{n^2-s^2}$ is bounded; it has the figure-eight shape $ACBCA$ tangent to two lines making angle $n$ with the axis. Drawn as figure 110.

Sources: chapter21 §527, figures109-110 (figure 110).

Last updated: 2026-05-12

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Hyperbolic spiral (§527)

$z=a/s.$

The equation $z=a/s$, which is similar to an equation for the hyperbola related to its asymptotes, gives a spiral which John Bernoulli called a hyperbolic spiral. (source: chapter21, §527)

Bernoulli’s analogy: the rectangular hyperbola $xy=a$ has asymptotes parallel to the axes; the polar relation $zs=a$ produces a curve where:

• As $s\to 0^{+}$, $z\to +\infty$ — infinitely many turns close to the axis, at infinite distance.
• As $s\to \infty$, $z\to 0$ — spirals inward to the center, never reaching it.

The asymptote. After infinitely many turns, the curve approaches a straight line $AA$ as asymptote (the line in the asymptotic direction $s\to 0$, at infinite distance). This is the polar counterpart to the hyperbola’s rectilinear asymptote.

A square-root variant

Euler also considers $z=a\sqrt{s}$:

For negative angles $s$ there are no corresponding real distances $z$. For each positive angle $s$ we obtain a pair of values for $z$, one positive and the other negative. The curve spirals about $C$ an infinite number of times. (source: chapter21, §527)

The double-valuedness gives a paired-spiral structure: two arms emerging from the center, each spiraling outward.

Lemniscate spiral (§527, figure 110)

$z=a\sqrt{n^2-s^2}.$

For $|s|>n$, $z$ is imaginary — no real points. For $|s|\le n$, the spiral is bounded by $z\le an$.

If the two straight lines $EF$ and $EF$ make an angle equal to $n$ with the axis $ACB$ through the center $C$ then they will be tangent to the curve at $C$ and the curve has the form of the lemniscate $ACBCA$. (source: chapter21, §527)

Reading figure 110: a horizontal axis $ACB$ with center $C$; two diagonal tangent lines $EF$ crossed at $C$ making angle $\pm n$ with the axis; the curve is a figure eight, both lobes passing through $C$, tangent to the diagonals there. The full path traces $A\to C\to B\to C\to A$.

This is the lemniscate of Bernoulli shape (though Euler’s parametrization is not the classical $r^2=a^2\cos 2\theta$ form).

Place in the chapter

These two spirals, together with the Archimedes spiral, illustrate the four basic algebraic-in-$z,s$ relations: $z=as$ (line), $z=a/s$ (hyperbola), $z=a\sqrt{s}$ (parabola-like), $z=a\sqrt{n^2-s^2}$ (ellipse-like, hence bounded). Just as the chapter-6 classification-of-conics organized $y^2=\alpha +\beta x+\gamma x^2$ by sign of $\gamma$, the polar quadratic in $s$ gives ellipses/parabolas/hyperbolas of spirals.

In a similar way we obtain an infinite number of other transcendental curves which it would take too long to develop. (source: chapter21, §527)

Figures

Figures 109–110

Related pages

• chapter-21-on-transcendental-curves
• spirals — general spiral setup.
• spiral-of-archimedes — the linear sibling.
• logarithmic-spiral — the transcendental-equation case.
• hyperbola — Bernoulli’s motivating analogy.
• classification-of-conics — the rectangular-coordinate parallel.