Ovals, ellipses and flat ovals

“Oval, 1,800 × 900” sounds like a complete description, and it is not. At least three different shapes answer to it in sheet metal work, and they differ in girth by more than a quarter of a metre. A lid made for one will not fit a vessel made to another, a bottom cut for one will leave gaps around another, and a capacity worked out for one will be wrong for the others. Before any oval is laid out, the shop has to know which oval it is.

This chapter sets out the three: the true ellipse, the four-centre oval and the flat oval. It compares their girths and areas at the running case’s size, explains why sheet metal work almost always uses arcs rather than a true ellipse, introduces the running case, a flared oval stock-water trough, and ends with a lid made for the wrong oval.

The true ellipse

An ellipse is the set of points whose distances from two fixed points, the foci, add up to a constant. It is the shape a circle appears to have when seen at an angle, and the shape a plane cuts from a cylinder on a slant. Its curvature changes continuously all the way round, tightest at the ends of its long axis and flattest at the ends of its short axis.

That continuous change is what makes the ellipse awkward in sheet metal. A rolled or formed piece of an ellipse has a different radius at every point, so it cannot be rolled on a set of rolls at one setting, a template for it is a curve that must be plotted point by point, and a flared elliptical vessel’s wall is not part of any cone or cylinder, so its pattern must be triangulated. Ellipses do appear in sheet metal, most often as the opening where a cylinder is cut on a slant, but as the plan shape of a vessel they are the exception.

The four-centre oval

A four-centre oval approximates an ellipse with four circular arcs. Two small arcs, struck from centres on the long axis, form the ends; two large arcs, struck from centres on the short axis extended, form the sides. At each of the four points where an end arc meets a side arc, the two arcs share a tangent, so the outline is smooth to the eye and to the hand.

Every part of it is a circle. The ends can be rolled at one setting and the sides at another; a template is two arcs; and, as Chapter 5 shows, a flared four-centre oval’s walls are parts of right cones, which develop by radial lines like any other cone. That is why the oval tinware of a century ago, and most oval vessels, hoods and troughs made today, are four-centre ovals, whatever the drawing calls them.

The curvature does jump where the arcs meet. In the running case, the end radius is 300 mm and the side radius 1,575 mm, so the curvature changes by a factor of 5.25 at each joint. On a polished surface that change shows as a slight kink in reflections; on galvanised sheet it is invisible.

The flat oval

A flat oval is the limiting case of a four-centre oval in which the side arcs become straight: two semicircular ends joined by two straight sides. It is the shape of flat-oval ductwork, of many tanks and troughs, and of anything made by splitting a round vessel and inserting straight panels. Its geometry is the simplest of the three: its ends are half circles, its sides flat sheets.

At 1,800 × 900 mm, a flat oval’s ends are semicircles of 900 mm diameter and its straight sides are 1,800 − 900 = 900 mm long. It is the fullest of the three shapes, with the largest area and the longest girth, because its sides do not curve in at all.

Figure 1A four-centre oval and its centres

Same label, three girths

At 1,800 × 900 mm, measured on the mean line, the three shapes compare like this:

• Ellipse: perimeter about 4,359.8 mm by Ramanujan’s approximation, area π × 900 × 450 = 1.2723 m².
• Four-centre oval with a 300 mm end radius: perimeter 4,383.7 mm, area 1.2754 m², as Chapters 3 and 4 calculate.
• Flat oval: perimeter π × 900 + 2 × 900 = 4,627.4 mm, area π × 450² + 900 × 900 = 1.4462 m².

The ellipse and the four-centre oval are close: 23.9 mm in girth and 0.24% in area. The flat oval is far from both, 243.7 mm longer than the four-centre oval and 13% larger in area. A sheet metal worker who treats them as interchangeable is fine with the first two for a rough estimate and badly wrong with the third.

Figure 2Three shapes called 1,800 × 900 oval

Shape	Perimeter, mm	Area, m²	Made from
True ellipse	4,359.8	1.2723	plotted points; triangulated if flared
Four-centre oval, end R300	4,383.7	1.2754	four arcs; part cones if flared
Flat oval	4,627.4	1.4462	two semicircles and two flats

Inside, outside and mean ovals

Sheet has thickness, and an oval vessel has an inside outline, an outside outline and a mean line between them. Here the four-centre oval has a property that the ellipse lacks, and it matters every time a mating part is made.

Offset a four-centre oval inward or outward by any distance, and the result is another four-centre oval struck from the same four centres: every arc’s radius simply grows or shrinks by the offset. The running case’s 1.6 mm sheet puts its inside outline 0.8 mm inside the mean line, with an end radius of 299.2 mm and a side radius of 1,574.2 mm, and its outside outline 0.8 mm outside, at 300.8 and 1,575.8 mm. Each outline’s girth differs from the mean line’s by exactly 2π × 0.8 = 5.0 mm, just as for a circle.

Offset an ellipse, and the result is not an ellipse. It is a smooth curve with no simple equation, slightly fuller at the ends than any ellipse of its size. For a thin lid or a skirt that must fit an elliptical vessel, that difference is usually too small to see; for a thick rim or a machined flange it is not, and the mating part must be plotted point by point from the offset.

The same property makes the flared trough possible. Its bottom is the top oval offset inward by the 75 mm flare, so it is struck from the same four centres with every radius 75 mm smaller: end radius 225 mm, side radius 1,500 mm. Every horizontal section of the trough between top and bottom is a four-centre oval from those same centres, which is why each wall is part of a cone with its axis through one of them.

Where ovals earn their place

An oval costs more to make than a circle of the same area: two rolling settings instead of one, four seams or tangent points to align, and patterns that are not single sectors. It earns that cost where its shape does a job a circle cannot.

• Along a line: a trough or feeder that sits along a fence, a wall or a bench, with access along both sides.
• Under a height limit: a duct or tank that must carry a large area within a shallow ceiling space or a vehicle’s low deck, keeping its centre of gravity low.
• Around a body or a machine: baths, sinks and hoods whose users or machines are longer than they are wide.
• For appearance: covers, lamp housings and architectural features where an oval simply looks right.

In each case, the oval’s proportion and end radius come from the use, and the fabricator’s job is to make exactly the oval specified, with every mating part made to the same one.

A family, not three shapes

The four-centre oval is really a family. With the length and width fixed, choosing the end radius sets everything else. A small end radius gives sharp ends and broad, flat sides, close to a pointed shape; an end radius of half the width gives the flat oval, whose sides are straight. In between lie all the ovals that look elliptical. Chapter 2 shows how to calculate the side radius from the end radius, and how to choose an end radius that suits the job.

So the drawing of any oval vessel needs three numbers, not two: length, width and end radius, or an explicit statement that the shape is an ellipse or a flat oval. A fourth, whether the dimensions are inside, outside or on the mean line, matters for the same reasons as on any rolled work.

The running case

This is an illustration. A rural fabricator is making galvanised steel stock-water troughs, oval so that cattle drink along both sides and the trough sits along a fence. The specification:

1. Top: four-centre oval, 1,800 × 900 mm on the mean line, end radius 300 mm.
2. Depth: 450 mm, top to bottom on the mean line.
3. Walls flare outward by 75 mm all round, so the bottom is 1,650 × 750 mm, a four-centre oval with an end radius of 225 mm struck from the same centres.
4. Material: 1.6 mm galvanised steel sheet.
5. Walls in four pieces, two ends and two sides, with vertical seams where they meet; flat oval bottom plate; rim rolled over a 10 mm steel rod.
6. Fill line 50 mm below the rim.

Figure 3The running case in numbers

4,383.7 mmGirth at the topChapter 3

439.7 LCapacity to the fill lineChapter 4

9.46°Wall flare from verticalChapter 5

36 kgWalls and bottom in 1.6 mm sheetChapter 9

The end radius of 300 mm was the fabricator’s choice, not the customer’s. A smaller end radius would have made sharper ends, harder for cattle to reach into and harder to clean; a larger one, approaching the 450 mm of a flat oval, would have made the sides nearly straight and the trough look like a split tank. Three hundred millimetres, a third of the width, gives an outline close to an ellipse of the same size, with ends a hand and a brush can reach round, and side arcs gentle enough to roll easily in 1.6 mm sheet. Chapter 2 shows how the side radius follows from that choice.

The error: a lid for the wrong oval

This is an illustration. A customer who had bought a trough asked for a sheet metal lid to keep debris out between uses, and gave the size as “1,800 × 900 oval”. The fabricator’s apprentice, who had made flat-oval duct, made a flat-oval lid: semicircular ends of 450 mm radius and straight sides 900 mm long, with a skirt to fit over the rim.

The lid’s girth was 243.7 mm longer than the trough’s rim, and its shape was wrong all round: too full at the four tangent points, where the flat oval bulges beyond the four-centre oval, and too long at the ends. It sat on the trough’s rim but hung loose by tens of millimetres at the sides and could not be clipped.

The fault was found at the trial fit. The cure was in the job sheet: the shop now records every oval as “four-centre, L × W, end R”, “ellipse, L × W” or “flat oval, L × W”, and a size without the type is sent back for clarification. The lid was remade as a four-centre oval with a 300 mm end radius, from the trough’s own drawing, plus the skirt’s clearance. Because the skirt fits outside the rim, its outline is the rim’s outline offset outward by the clearance, which for a four-centre oval means the same four centres with every radius increased by the same amount, as the section above explained.

Figure 4Before laying out any oval

• Do: Shape named: ellipse, four-centre oval or flat ovalfour-centre
• Do: Length and width, and on which line1,800 × 900 on the mean line
• Do: End radius for a four-centre oval300 mm
• Do: Flare or straight walls, and how much75 mm all round
• Do: Mating parts made to the same ovallid, bottom, rim
• Avoid: Size given as "L × W oval" without a typethe flat-oval lid

Exercise: three shapes at a smaller size

A smaller trough is 1,200 × 600 mm. Find the perimeters of the ellipse and the flat oval of that size, and compare them. Ramanujan’s approximation for an ellipse with semi-axes a and b is π[3(a + b) − √((3a + b)(a + 3b))].

Worked answer. With a = 600 and b = 300 mm:

• Ellipse: π × [3 × 900 − √(2,100 × 1,500)] = π × [2,700 − 1,774.8] = 2,906.5 mm.
• Flat oval: semicircular ends of 600 mm diameter and straight sides of 1,200 − 600 = 600 mm, so π × 600 + 2 × 600 = 3,085.0 mm.

The flat oval’s girth is 178.4 mm longer, the same 6.1% as at the running case’s size, because both shapes have the same 2 : 1 proportion. A four-centre oval of 1,200 × 600 with an end radius of 200 mm would be the running case’s oval scaled by two thirds, with a girth of two thirds of 4,383.7, about 2,922.5 mm, between the two but close to the ellipse.