Sagittal curve

SAGITTAL CURVE

Curve defined by Aubry in 1893 (Journal de mathématiques spéciales p. 187 et 203)
Homemade name.

The sagittal curve with base [AB] is constructed in the following way: an isosceles triangle with vertex S₁ is drawn on [AB]. The ratio of its height S₁I₁ (rather called here the "sagitta") and the half-base AI₁ is some number r₁.
The same process is applied to both the other sides, the sagittas having the same ratio r₂ with respect to the half-base.
Then, this process is applied an infinite number of times with the sequence of ratios sagitta/half-base (r_n).
The sagittal curve is the limit curve, if it exists.

For example, if the r_n are all equal to 1, then the resulting curve is none other than the Lévy fractal.

If the r_n are all equal to r < 1, we still get a fractal, attractor of two affine contractions, the shape of which reminds of the blancmange curve, which has a similar construction.

But this method was used, at first, to trace, thanks to ropes, circular junctions between two lines.
Once the vertex S₁ is fixed, a calculation shows that for the vertices S_n to be on the arc of a circle passing by A,S₁,B, one only has to take .
The limit curve is then this arc of a circle.

As, in the previous sequence, we have , if we take , we get a curve very close to the arc of a circle.

If, in the case of the arc of a circle, the sagitta and the half-base at the step n are written respectively and , then we have and so . Therefore, if we construct a curve for which the sagitta is simply the quarter of the previous one (which gives ), we get again the approximation of the arc of a circle. It is this curve, with a simple practical construction, that was referred to as "curve of proportional sagittas" in the railway construction manuals in the 19th century.
Note opposite that the approximation is far worse than in the previous case, but the defect becomes less and less pronounced as the initial angle is smaller.

Here is what Aubry thought of this last curve in 1893:
"Is this limit polygon a continuous curve? It is unlikely, but to be sure, one would need to be able to give a definition of it leading to an equation, which does not seem possible.
This curve is certainly continuous in the geometrical - or maybe rather physical - sense of the word; but in the analytical sense, it appears to us with no doubt that it is infinitely discontinuous or dotted.
Does it have tangents? The answer depends on what will be made of the issues raised above.
(Editor's note: the Koch curve dates from 1904).
It is remarkable that this pseudo-curve (it may, indeed, not be a curve, but the set of infinitely many points not linked by any fixed law) is used daily for simple designs of roads, to junctions of straight lines."

And, in 2015, can we find an equation for this curve?

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The sagittal curve with base [AB] is constructed in the following way: an isosceles triangle with vertex S₁ is drawn on [AB]. The ratio of its height S₁I₁ (rather called here the "sagitta") and the half-base AI₁ is some number r₁. The same process is applied to both the other sides, the sagittas having the same ratio r₂ with respect to the half-base. Then, this process is applied an infinite number of times with the sequence of ratios sagitta/half-base (r_n). The sagittal curve is the limit curve, if it exists.
For example, if the r_n are all equal to 1, then the resulting curve is none other than the Lévy fractal.
If the r_n are all equal to r < 1, we still get a fractal, attractor of two affine contractions, the shape of which reminds of the blancmange curve, which has a similar construction.
But this method was used, at first, to trace, thanks to ropes, circular junctions between two lines. Once the vertex S₁ is fixed, a calculation shows that for the vertices S_n to be on the arc of a circle passing by A,S₁,B, one only has to take . The limit curve is then this arc of a circle.
As, in the previous sequence, we have , if we take , we get a curve very close to the arc of a circle.
If, in the case of the arc of a circle, the sagitta and the half-base at the step n are written respectively and , then we have and so . Therefore, if we construct a curve for which the sagitta is simply the quarter of the previous one (which gives ), we get again the approximation of the arc of a circle. It is this curve, with a simple practical construction, that was referred to as "curve of proportional sagittas" in the railway construction manuals in the 19th century. Note opposite that the approximation is far worse than in the previous case, but the defect becomes less and less pronounced as the initial angle is smaller.