Pin-and-String Parabola Drawing - a Small Improvement

Using pins and string is a popular way to demonstrate drawing a parabola, an important curve in maths and physics. The string is fixed at one end (the Focus) and free to slide along a line (Rod) at the other. Here we demonstrate how to facilitate this technique and improve drawing accuracy by spinning the Rod with a small motor.

Straight, rigid rod or tube, 4-10 mm diameter, long enough to span the drawing

Plastic or metal washers which fit on the rod

DC motor and power supply (batteries, power-block, phone charger etc.)

Support board or table-top that accepts push-pins

Push-pins and a sheet they can be pushed into if not the table

Plastic block as a bearing for the rod

Modelling clay or hot-melt glue to fix motor and bearing

Pens or pencils and paper

Something to join rod and motor shaft, e.g. silicone tubing

Adhesive tape.

The parabola is an influential mathematical curve, one of the “conic-sections” of classical geometry. In physics it describes projectile motion, with fundamental use in aerospace and sports science. When rotated about an axis it defines a paraboloid, whose sound and electromagnetic wave focussing qualities are essential in the design of satellite dishes, astronomical telescopes, headlights and solar-ovens. Hence it is of interest to be able to draw parabolas by hand and investigate their properties.

In the internet you will find many “pin-and-string” drawing techniques for the parabola. To understand these we begin with a definition of the parabola, namely “the locus of a point in a plane that moves so that its distances from a fixed point and a fixed line in the plane are equal”. The fixed point and line are named Focus and Directrix respectively. (Analytic Geometry – E.S. Smith, M. Salkover, H.K. Justice 1947). See Figure 1. P is one point on the red parabola; by the above definition u and v are equal. The point at the intersection of the parabola and its line of symmetry is called the Vertex. This is midway between the Directrix and the Focus, a distance a from both.

This definition makes possible drawing with ruler and compasses, but isn’t much help with pin-and-string construction. Instead we draw another line parallel to the Directrix at an arbitrary distance L away. From the figure, L is obviously a constant length, equal to v + w. But as u = v , L is also equal to u + w. So if u + w is of fixed length, it can be drawn with a fixed length of string. This approach was nicely described in https://www.youtube.com/watch?v=S2zzr-yX1Ak . The green line becomes a rigid Rod along which the Slider at Q must slide. The problem is that the string part represented by w must be perpendicular to the Rod at all points. As you trace out the parabola by moving the point P under string tension from left to right, the Slider Q should slide along the line, but it tends to stick and slip due to friction, and must be monitored while drawing.

Friction and sticking can be minimised by correct choice of materials, for example a PTFE polymer Slider on polished steel or glass. Another improvement, the one used in this Instructable, is to spin the rod. This allows the Slider to move more easily. Any non-perpendicularity of the string at Q will apply a force along the Rod, allowing the point of contact to adjust itself for a perpendicular string w. In other words, Q dutifully and smoothly walks along and quickly catches up with any movement of P, maintaining the 90° angle. If you move point P too quickly, Q will still lag behind. Either move P more slowly, or increase the motor voltage to speed catch-up.

The setup consists of a 400 x 500 mm support board overlaid with an A4 sized rubber sheet which accepts a push-pin for the Focus. Wood or foam-board should also work as support. A sheet of paper is taped on top. The 4 mm diameter polished steel Rod was cut from an old BBQ grill! This is supported at one end by fixing to the DC motor shaft with a 10 mm length of silicone tubing, tight on both shafts. 5 V on the motor spins the Rod at a couple of hundred RPM. Nothing is critical here. The motor should be fixed to the support board, for instance with hot-melt glue. The other end of the Rod is supported by a plastic block, drilled 4 mm clearance. A 5 mm steel or nylon washer is threaded onto the Rod and tied to the string. The other end of the string is tied in a loop and placed over the Focus pin. With light tension where the string pivots on a pencil, ball-point or other pen, the string closely wraps around the pen.

If the push-pin is small and the Rod is slightly above the drawing surface, the string can pass over it to allow both left and right side of the parabola to be drawn in one go. This gives a small error as the paper is not in the plane of Focus and Rod. On the second half of the parabola the string wraps further around the pen, giving a small asymmetry. If this error is too large, you will have to change the string wrapping and draw both sides separately.

With suitable placement of the Focus you can draw "off-axis" parabolas too, which are more convenient for some applications where the Focus point needs its own support; domestic satellite dishes are designed like this.

When I say “string”, I really mean fine thread. This must be thin (<1 mm), low friction and with low stretch. Fishing tackle shops provide the perfect material. Nylon monofilament is cheap and works OK, but has a bit too much stretch. I used a 30 lb Dacron braided fly line backing. 30 lb is of course huge overkill, so get the finest you can find. Modern braided polymer (UHMWPE) or metal wire fishing lines are incredibly strong, fine and flexible. Some are quite expensive for a full reel, but you may be able to “borrow” a few meters from a fisher friend.

It is very important that the pen is not simply tapered to a point, as tension will pull the pen over the string and be lost. Some pens already have grooves near the point which perfectly guide the string. If not, either carve a tiny groove near the writing point, or attach a small piece of tube, washer or similar with hot-melt glue to define the string position. We have not tried it, but the pen could perhaps be replaced with a rotating scalpel or focussed diode-laser engraver.

In order to minimise error due to finite pen diameter, our videos used a ball-point pen refill with a 4 mm length of pink silicone tubing pushed on. This retains the string perfectly during drawing.

The only variable to choose to draw the parabola is a. Having defined the Directrix line, place the Focus push-pin, so fixing a as half the distance to the Directrix. Small values of a give deep parabolas. The larger is a, the flatter the parabola. String length L can be any convenient length, as long as the Rod is placed clear of the drawing area. The conventional formula for the parabola is y = x²/4a, which can be compared with what you draw so you can check that everything is working correctly. In the video a is about 100 mm, L about 230 mm.

Sure, it is a trivial matter to plot out a parabola y = x²/4a using Excel, Python or a scripted drawing app such as Coreldraw and print it out. However, the hands-on pin-and-string method gives better feeling for the conic sections and is easily scalable to dimensions much bigger than A4 printers. The spinning guide-rod removes the need to continuously monitor the perpendicularity of the string, freeing you to maintain string tension, hold the pen vertically and guide the pen as it draws.