I haven't been able to find an English translation for arco capaz, though I've been searching for it myself. I will, however, attempt to give a definition of it in English.
Given segment AB and angle λ, the arco capaz is a major (outside) arc AB. From any point P on this arc, the angle between secant lines PA and PB -- APB -- must equal λ. The problem is constructing this arc by finding its center C, and thereby also its radius CA=CB.
Here is how to construct it:
Draw segment AB, then describe angle ABF (It doesn't matter where F lies, so long as angle ABF be the given one). Since AB is a chord of the circle containing our arc, its center will be located along the perpendicular bisector of AB. Now construct a line perpendicular to AF, intersecting A. The intersection of these two lines is the arc's center C. The radius is CA or CB. The major arc with these characteristics is the arco capaz.
Does anyone know if there is an English term for this?
Arco capaz = "The most capacious arc, meaning the most efficient arc containing the desired space."
An industrial engineer in Rhode Island, who wishes to be anonymous, kindly researched this for us! His source is: The Harper Collins Dictionary ofMathematics (E. J. Borowski & J. M. Borwein)c.1991, pp.23-24.
LPFR, I think that your definition is excellent and concise. However, we were looking for an English term to describe this idea. From my experience, I don't think that the English-speaking world gives sufficient importance to this idea to have come up with any word for it. None of the specialized people I have asked from the U.S., or even from Mexico or Argentina, has been familiar with the construction. It seems never to have emigrated from Europe.
I accept "capacious arc" as a good direct translation of arco capaz,. Google yields only two results for this phrase, however, and neither is in the geometrical context. I think it will do though, if we are to coin a term. Thank-you, Nomi.
Otherwise I don't think that "the most efficient arc containing the desired space" is necessarily incorrect , but neither is it descriptive enough to serve as a definition. The locus LPFR describes in his definition, of course, is a set of points comprising an arc; such could possibly be described as "the most efficient arc", so that is why I say it is not necessarily incorrect.
I'm curious what this concept is called in French, LPFR.
As for your last question, in French it is called "arc capable".
I am not sure that the word in English for this geometrical concept has to be explicit. In Spanish and French it is not. The words "capaz" and "capable" mean usually "able". Only in very rare cases (all "lettered") it means "capacious".
I do not know the origin of the name, but I suspect that the terms "capaz" and "capable" apply, not to the arc itself, but to the angles drawn from the points of the arc. Then trying to bind the word "capacious" to the geometrical properties of the arc is useless.
On the other hand, the "arco capaz" is a geometrical concept useless in current life. The only use that I know was in sailing before the GPS. It was used to situate very accurately a vessel near three visible objects whose position was accurately known. This needed to use a sextant to measure the angles. Nowadays it is just a geometrical concept without practical uses.
It a specialist concept and its name do not need to mean anything.
As for the definition "The most capacious arc, meaning the most efficient arc containing the desired space", it is bad. First "efficient" is meaningless. You do not know how to decide if an arc is "more efficient" than another one. Also there is only one circle arc that shares its extremities with a segment and that "contains the desired space". Efficient is useless.
If you want to define "arco capaz" this way you could rather say: "it is the circle arc which encloses a given surface between itself and the cord." But it is a bad definition: all arcs enclose a surface. It does not describe the main property of the name: the angle of view of the cord.
If you want an English name I would call it "spanning arc", because of the "span" of the angles formed by the lines drawn to the cord.
"Ângulo Capaz" is not simply an angle but a locus. The best translation to English is the "Inscribed Angle". In a circle, if 2*theta is the central angle of any secant, any inscribed angle to AB (draw from any point of the circle in the same side of the center) has angle=theta, and therefore it is constant at any point of the arc. The Inscribed angle theorem, then, says that the locus of points that subtend the same angle theta on a segment AB is a circle in which the segment AB is a secant to the circle whose central angle is 2*theta.