Is there a term for the shape shown above: a smooth, continuous curve with three axes of symmetry? And if anyone could share the equation for this curve, I would be very grateful.
Chris Daniel
Glan Conwy, Conwy, UK
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The plate appears to be a smoothed Reuleaux triangle. A Reuleaux triangle, named after 19th-century German engineer Franz Reuleaux, has a constant width and is formed of three intersecting arcs, each one being centred on the opposite vertex of an equilateral triangle. The vertices of the triangle are points.
A smoothed Reuleaux triangle – that is, one with rounded vertices – can be constructed by adding a small circle to each vertex of an equilateral triangle and sweeping out arcs from each one.
Similar shapes, though not necessarily with constant width, can be generated by tracing a point linked to one circle that is rolled around another circle without slipping. They are called epitrochoids, from the Greek epi (over) and trochos (wheel). A specific case of an epitrochoid is an epicycloid, created by tracing a point on the edge of a circle that rotates around another circle. Similarly, a circle that is rolled inside another circle generates hypotrochoids and a hypocycloid. Curves like this can be produced using a Spirograph.
The shape shown isn't quite a Reuleaux triangle, since that has sharp not smooth corners. It is also not, mercifully, a wankel
Epitrochoids can be made from a base circle of radius a and a rolling circle of radius b, where b = a/q. The point traced around the circle is distance d from the centre of the rolling circle, where d = kb. The values q and k are variables that affect the shape. When q = 3 and k = 1/8, a shape similar to that of the plate is generated.
Equations for similar shapes can be found at .
Robin Maguire
Hobart, Tasmania, Australia
Starting with the name, the shape shown isn’t quite a Reuleaux triangle, since that has sharp rather than smooth corners. For the same reason, mercifully, it shouldn’t be called a Wankel (after the shape of the engine rotor named for Felix Wankel). How about a tricorn, after the 18th-century hat that has a similar shape in plan?
Regarding the formula, it makes sense to get away from Cartesian coordinates based on right angles. Radial coordinates (r, ɵ) make matters simple. A circle may be defined by r = 1, and the tricorn can be depicted by superimposing three cycles of sine wave on the circumference of the circle. This gives r = 1 + s × sin(3ɵ), where s is a shape factor that determines the prominence of the corners.
Going Cartesian (aligning the x axis with the radial datum), we get x = r × cos(ɵ) = cos(ɵ) × (1 + s × sin(3ɵ)), y = r × sin(ɵ) = sin(ɵ) × (1 + s × sin(3ɵ)). See below for a plot of this with s = 0.07, which is a good match for the pictured plate.
Sandy Chadwick
Bodmin, Cornwall, UK
I think this shape is an n-ellipse, one of a family of elliptical figures with more than two foci. This particular member of the family has three foci, located at the apexes of an equilateral triangle (so should perhaps be called an equilateral 3-ellipse). Other names proposed for a curve of this type are trifocal ellipse, polyellipse and egglipse.
An n-ellipse is the locus of points where the sum of distances to the foci is a constant. These, and other related figures, were the subject of James Clerk Maxwell’s first scientific paper, presented at the Royal Society of Edinburgh in 1846, when he was 14 years old.
A simple way to draw this very pleasing curve is to position three pins at the vertices of an equilateral triangle, then make a loose loop of string around the three pins and, taking up the slack with a pencil point, trace around the loop. A variety of shapes can be produced depending on the shape of the triangle and the size of the loop.
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