Equiangular spiral

Parametric equations

An equiangular spiral (a logarithmic spiral) can be written:

In Cartesian coordinates:

Here $a>0$. This figure uses $b>0$, and the angle between the tangent and a ray from the origin is the constant $\arctan (1/b)$. If $b=0$, the curve degenerates to a circle.

Interaction

• Growth coefficient b: control how fast the spiral opens outward
• Maximum angle θ: limit how much of the spiral is shown; the bright arc runs out along the angle, and a faint copy keeps the full range
• Rotation speed: rotate the whole figure slowly about the origin

What to notice

• The radius $r=a{e}^{b\theta }$ keeps the angle between the curve and any ray from the origin constant.
• For $b>0$, increasing $\theta$ spreads successive turns by an exponential factor.
• Rotating the view does not change that angle. It only changes the direction you are looking from.

Related

• Fibonacci spiral
• Rotation and scaling, composed

Further reading

• Logarithmic spiral (Wikipedia)