A conchoid delineator traces a curve using a slotted arm, a fixed pole pin and a pin guided by a straight rail. A stylus fixed on the arm stays a constant distance from the rail pin. The animation shows those physical constraints while the calculator reports the stylus coordinates.
Conchoid Delineator Interactive Calculator
Move the arm through the conchoid construction. The fixed pole pin runs in the arm slot, while the arm pin travels in the straight rail. A stylus stays at the selected offset from that rail pin. The angle control and coordinate outputs follow the live animation; move any slider to pause and inspect an exact setting.
Equation Used
- The rail pin is fixed on the arm and moves only along x = a.
- The fixed pole pin can rotate and slide relative to the slotted arm.
- The stylus is clamped a fixed signed distance sb from the rail pin.
- Finite width, pin collisions and pencil-holder clearances are not solved.
- Autoplay sweeps the arm smoothly between −60 and +60 degrees; output coordinates follow the shown pose.
Ideal plan-view construction, with schematic fittings and separated layers. No pin clearance, manufacturing accuracy or physical collision envelope is calculated. The displayed curve covers only −60° to +60°.
Two guides constrain the moving arm
The historical instrument has a fixed T-shaped frame, a slot along the moving arm and a traverse pin in the straight head rail. The pole pin runs in the arm slot. This matters: the arm must slide past the pole as well as turn about it, because the distance from the pole to the rail pin changes with angle.
In this view the rail is vertical at x = a and the pole O is at the origin. The rail pin Q is fixed to the arm. A stylus P is clamped at a distance b from Q, on either the positive or negative side selected by the branch control.
The blue line shows the selected conchoid branch over the displayed angle range. The moving stylus remains on that line. The angle slider and coordinate outputs follow the actual shown pose; changing a control pauses playback for inspection.
A mechanical curve-drawing construction
Hiscox describes the conchoid delineator as a draughting instrument and mentions architectural column outlines. Modern demonstrations use the same constraints to explain the conchoid of Nicomedes and its coordinate equations.
The illustration is a kinematic construction, not a precision-instrument specification. It does not assign a universal tolerance, accuracy or service life to a real drawing device.
Conchoid coordinates
Let O = (0,0), let the rail be x = a, and measure θ from the positive horizontal axis. The arm meets the rail at Q = (a, a tan θ), a distance a/cos θ from O.
Moving a signed distance sb along the arm gives r = a/cos θ + sb, where s is +1 or −1. The stylus coordinates are x = r cos θ and y = r sin θ. Equivalently, P = Q + sb(cos θ, sin θ).
When r is negative, the point lies beyond the pole on the opposite side from the positive angular ray. That is valid signed polar notation, not a negative physical distance. The absolute distance OP is |r|. The calculator is limited to angles from −60° to +60°, away from the cosine singularity at ±90°.
Checking one stylus position
Take a = 100 mm, b = 75 mm, θ = 0° and branch +1. The rail pin is Q = (100,0) mm, the signed radius is 175 mm, and the stylus is P = (175,0) mm.
At θ = 60° the rail pin is Q = (100,173.21) mm. The pole-to-rail distance along the arm is 200 mm, so r = 275 mm and P is approximately (137.5,238.16) mm. The distance QP remains 75 mm.
For the negative branch with a = 100 mm, b = 150 mm and θ = 0°, r = −50 mm and P = (−50,0) mm. The updated view includes this point left of the pole. At b = 0, the stylus coincides with the rail pin and the curve reduces to the directrix.
Ideal geometry and a physical instrument
The slot removes the changing-distance constraint that would prevent a simple fixed-pivot arm from following this motion. A practical instrument still needs adequate slot length, rail travel, layer separation and secure stylus adjustment.
The animation includes mathematical limit cases such as a stylus crossing the pole or coinciding with the rail pin. Those cases explain the curve but may require a different pencil-holder arrangement to avoid physical interference. No manufacturing clearance is inferred from the drawn line widths.
The full curve extends beyond the displayed ±60° range. The frame and viewport are sized for the shown sweep; they do not claim that a finite instrument can trace an infinite asymptote.
Conchoid delineator questions
Why does the arm need a slot?
The distance from the pole to the rail pin changes as the arm turns. The slot allows the arm to slide relative to the fixed pole pin.
Which distance stays constant?
The stylus stays b from the rail pin, measured along the arm. The pole-to-rail distance along the arm is not constant.
What does a negative radius mean?
The stylus lies in the direction opposite the positive ray for that angle. The Cartesian coordinates retain the correct position.
Why do the result values move during autoplay?
They now use the displayed angle, keeping the calculator and the animation on the same pose. Changing a slider pauses the animation.
Does the picture check mechanical collisions?
No. It is an ideal plan-view construction. Real fittings and pencil holders need clearance checks.
Instrument and equation references
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances (1901), page 364, figure 1520 depicts the T-frame, slotted arm, traverse pin and pencil of the conchoid delineator.
Oliver Knill and Michael Teodorescu, Harvard: Conchoid of Nicomedes gives the polar construction r = 1/cos θ + c. This calculator restores dimensional parameter a and signed offset sb.
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