Spiral of Archimedes

Summary: §526. The simplest spiral, $z=as$ in polar coordinates: distance proportional to angle. Tangent to the axis $AC$ at $C$; intersects $AC$ and $BC$ at right angles infinitely many times as it spirals outward. $BC{B}^{-1}$ is the curve’s diameter. Drawn as figure 109.

Sources: chapter21 §526, figures109-110 (figure 109).

Last updated: 2026-05-12

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The equation (§526)

$z=as.$

The distance $CM$ from the center $C$ is proportional to the arc $s$ on a unit circle measuring $\angle ACM$. Just as $y=ax$ would give a straight line in rectangular coordinates, this equation gives an infinitely-coiled spiral in polar.

Multi-valued $s$

For a fixed direction of $CM$, the angle $s$ can be $s,2\pi +s,4\pi +s,\dots$ or $-(2\pi -s),-(4\pi -s),\dots$ Substituting all of these:

$z\in \{as,a(2\pi +s),a(4\pi +s),\dots \}\cup \{-a(\pi +s),-a(3\pi +s),\dots ,a(\pi -s),a(3\pi -s),\dots \}.$

Each gives a point on the spiral, distinct, separated by $2\pi a$ along the same ray. Hence the spiral has infinitely many turns and meets any radius from $C$ in infinitely many points, going out to infinity.

Figure 109 features (§526)

• $C$ = center.
• $AC$ = axis, tangent to the spiral at $C$.
• $BC$ = perpendicular axis.
• Spiral intersects $AC$ at $A,A^1,A^2,A^3,\dots$ at distances $a\pi ,2a\pi ,3a\pi ,\dots$ on one side and $A^{-1},A^{-2},\dots$ on the other.
• Spiral intersects $BC$ at $B,B^1,B^2,\dots$ at distances $a\pi /2,3a\pi /2,\dots$
• Crossing $AC$ or $BC$ is at right angles.
• $BC{B}^{-1}$ is a diameter — the spiral is symmetric across it. (The line through $C$ perpendicular to the tangent direction.)

Discoverer

This curve is usually called by the name of its discoverer, the spiral of Archimedes. (source: chapter21, §526)

Once the equation $z=as$ is known, all intersections of the spiral with any straight line (through $C$ or elsewhere) become calculable: substitute the line’s polar equation and solve.

Place in the chapter

Archimedes’ spiral is to the spiral family what the straight-line-equation is to the algebraic-curve family: the simplest case, where one of the two polar variables is a linear function of the other. The §527 hyperbolic spiral $z=a/s$ is the next-simplest algebraic-in-$z,s$ relation, by analogy with the hyperbola.

Figures

Figures 109–110

Related pages

• chapter-21-on-transcendental-curves
• spirals — overview.
• hyperbolic-and-lemniscate-spirals — sibling examples.
• logarithmic-spiral — the genus’ transcendental equation.
• curves-from-polar-coordinates — chapter 17 polar setup.
• straight-line-equation — rectangular analogue ($y=ax$ vs. $z=as$).