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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Basket-handle arch | 3/3 | https://en.wikipedia.org/wiki/Basket-handle_arch | reference | science, encyclopedia | 2026-05-05T13:59:06.280548+00:00 | kb-cron |
The table provided allows for the straightforward construction of a basket-handle arch with any specified opening using five, seven, or nine centers, eliminating the need for extensive calculations. The only stipulation is that the drop must match one of the values proposed by Michal. For instance, to draw a curve with seven centers, a 12-meter opening, and a 3-meter slope corresponding to a drop of one-quarter (or 0.25), the first and second radii can be calculated as follows: 6×0.265 and 6×0.419, resulting in values of 1.594 meters and 2.514 meters, respectively. To inscribe the curve within a rectangle labeled ABCD, one would start by describing a semicircle on line segment AB, which serves as the diameter, and divide it into seven equal parts. Chords Aa, ab, bc, and cd are then traced, with chord cd representing a half-division. On the AB axis, from point A, a length of 1.590 meters is measured to establish the first center, labeled m1. A parallel line with radius Oa is drawn through this point, intersecting chord Aa at point n, marking the endpoint of the first arc. From point n, a length of nm2 equal to 2.514 meters is measured to identify the second center, m2. A parallel line with radius Ob is drawn from point m2, while a parallel line to chord ab is drawn from point n. The intersection of these two parallels at point n′ defines the endpoint of the second arc. Continuing this process, a parallel is drawn through point n′ to chord bc, and from point E, a parallel is drawn to chord cd. The intersection of these two lines at point n′′ is used to draw a parallel to radius Oc. The points m3 and m4, where this line intersects the extensions of radius n′m2 and the vertical axis, become the third and fourth centers. The final three centers, m5, m6, and m7, are positioned symmetrically relative to the first three centers m1, m2, and m3. As illustrated in the figure, the arcs An, nn′, n′n′′, etc., subtend equal angles at their centers, specifically 51° 34' 17" 14'. Moreover, constructing a semi-ellipse with AB as the major axis and OE as the minor axis reveals that the arcs of the semi-ellipse, contained within the same angles as the circular arcs, possess a radius of curvature equal to that of the arcs themselves. This method demonstrates the ease with which curves can be constructed with five, seven, or nine centers.
=== The Lerouge method === Following Mr. Michal's contributions, the subject was further explored by Mr. Lerouge, the chief engineer of the Ponts et Chaussées. Lerouge developed tables for constructing curves with three, five, seven, and even up to fifteen centers. His approach diverges from Michal's methodology by stipulating that the successive radii must increase according to an arithmetic progression. This requirement means that the angles formed between the radii do not necessarily need to be equal, allowing for greater flexibility in the design of the curves.
== References ==
=== Bibliography === "Notice sur les courbes en anse de panier employées dans la construction des ponts". Annales des ponts et chaussées. Mémoires et documents relatifs à l'art des constructions et au service de l'ingénieur (in French). Zoroastre Alexis Michal. Paris: Carilian-Goeury. 1831. 49-61.{{cite book}}: CS1 maint: others (link) "Mémoire sur les voûtes en anse de panier". Annales des ponts et chaussées. Mémoires et documents relatifs à l'art des constructions et au service de l'ingénieur (in French). Pierre-Jacques Lerouge. Paris: Carilian-Goeury. 1839. pp. 335–362.{{cite book}}: CS1 maint: others (link) Degrand, Eugène; Resal, Jean (1887). Ponts en maçonnerie (in French). Vol. 2. Paris: Baudry. Séjourné, Paul (1913). Grandes Voûtes : Partie 1 – voûtes inarticulées (in French). Vol. 1. Bourges: Imprimerie Vve Tardy-Pigelet et fils. Séjourné, Paul (1913). Grandes Voûtes : voûtes inarticulées (suite) (in French). Vol. 3. Bourges: Imprimerie Vve Tardy-Pigelet et fils. Ministère des Transports, Direction des routes (1982). Les ponts en maçonnerie (PDF) (PDF) (in French). Bagneux. Prade, Marcel (1986). Les Ponts, Monuments historiques : inventaire, description, histoire des ponts et ponts-aqueducs de France protégés au titre des monuments historiques (in French). Brissaud. ISBN 978-2902170548. Prade, Marcel (1988). Ponts et Viaducs au xixe siècle (in French). Brissaud. ISBN 978-2902170593. Prade, Marcel (1990). Les Grands Ponts Du Monde. Ponts Remarquable d'Europe (in French). Brissaud. ISBN 9782902170654. Woodman, Francis; Bloom, Jonathan M. (2003). "Arch". Oxford Art Online. Oxford University Press. doi:10.1093/gao/9781884446054.article.t003657. ISBN 978-1-884446-05-4. American Technical Society (1920). "Basket-Handle Arch". Cyclopedia of Civil Engineering. Retrieved 2024-04-05. Baker, I.O. (1889). "Basket-Handle Arch". A Treatise on Masonry Construction. J. Wiley & sons. Retrieved 2024-04-05. Hourihane, C. (2012). "Arch". The Grove Encyclopedia of Medieval Art and Architecture. Vol. 1. Oxford University Press. pp. 129–134. ISBN 978-0-19-539536-5. Retrieved 2024-12-24.