Conchoid — Definition, Formula & Examples

A conchoid is a classical curve generated by marking two points at a fixed distance along a line drawn from a fixed point (the pole) through each point on a given straight line. The result is a shell-shaped curve that appears on both sides of the reference line.

Given a fixed point $O$ (the pole), a fixed line $\ell$ at perpendicular distance $d$ from $O$ , and a constant $k > 0$ , the conchoid of Nicomedes is the locus of all points $P$ such that $P$ lies on a ray from $O$ through a point $Q$ on $\ell$ and $|PQ| = k$ . In polar coordinates with $O$ at the origin and $\ell$ perpendicular to the initial ray at distance $d$ , the equation is $r = \frac{d}{\cos\theta} \pm k$ .

Key Formula

r = \frac{d}{\cos\theta} \pm k

Where:

$r$ = Distance from the pole O to a point on the curve
$d$ = Perpendicular distance from the pole O to the reference line ℓ
$k$ = Fixed distance measured along each ray from the point on ℓ
$\theta$ = Angle of the ray measured from the line perpendicular to ℓ through O

How It Works

To construct the conchoid, pick any point

Q

on the reference line

\ell

. Draw the ray from the pole

O

through

Q

. Along that ray, mark two points—one at distance

k

beyond

Q

and one at distance

k

before

Q

(toward

O

). As

Q

slides along

\ell

, these marked points trace the two branches of the conchoid. When

k > d

, the inner branch loops through the pole, forming a visible loop. When

k = d

, the inner branch has a cusp at

O

. When

k < d

, both branches stay on the same side as

\ell

relative to

O

Worked Example

Problem: Find the polar equation of the conchoid of Nicomedes when the pole is at the origin, the reference line is at perpendicular distance

d = 3

, and the fixed marking distance is

k = 2

. Then find

r

for both branches when

\theta = 60°

Write the equation: Substitute the given values into the conchoid formula.

r = \frac{3}{\cos\theta} \pm 2

Evaluate at θ = 60°: Since

\cos 60° = 0.5

, compute

\frac{3}{0.5} = 6

r = 6 + 2 = 8 \quad \text{or} \quad r = 6 - 2 = 4

Answer: The outer branch gives

r = 8

and the inner branch gives

r = 4

when

\theta = 60°

Why It Matters

The conchoid of Nicomedes was one of the first curves studied for solving classical Greek construction problems, specifically trisecting an angle and doubling the cube. Understanding how curves are generated by geometric loci connects directly to polar coordinates and parametric equations in precalculus and calculus courses.

Common Mistakes

Mistake: Forgetting that the formula produces two branches (the ± term) and only plotting one.

Correction: Always account for both

r = \frac{d}{\cos\theta} + k

and

r = \frac{d}{\cos\theta} - k

. The inner branch can produce a loop, cusp, or smooth curve depending on whether

k > d

k = d

, or

k < d