"Ovoide" construction

Question

In my country, mandatory exercises in the courses of technical drawings are the construction of "ovoides" ( "ovoid" in English ? ). The definition of "ovoide" in wikipedia (translated from wikipedia spanish page, no English one exists) is:

The ovoid is a closed curve symmetric with respect to its concave axis towards it, and conformed by four arcs of circumference: one of them is a semi-circumference and other two are equal and symmetrical.

It seems not a concrete definition. What I understand is that an ovoide has:

1. One semi-circunference of some radius.
2. One of its diameters is called "minor axis". The diameter perpendicular to it and extended by one side till some point is called "major axis"
3. Two arcs of some radius, with centers over the line of the minor axis, and tangent to the semi-circunference touching it in the ends of the minor axis.
4. A fourth arc tangent to the two previous arcs and center in the major axis, passing by its end.
5. QUESTION 1: Something else ???

The 3 usual problems are create the ovoide given:

• the major axis; or
• the minor axis; or
• both axis

The are lots of www pages and videos explaining how to do it (look for "dibujo tecnico ovoide"). All they explains the same three methods. See by example this one or this one (sorry, no english one found). Here is a draw that resumes these methods:

However, trying to understand the methods:

• it seems contradictory that if an ovoide is defined and can be constructed given only its major axis, it was also possible construct it given both axis. Several solutions for first case must exists to allow addition of a second restriction.
• the free parameters of the draw seems to be 3 radius and one distance between the centers (QUESTION 2: is this statement true ?). That gives infinite solutions for each one of the three problems (QUESTION 3: is this statement true?).

There are hundreds of pages explaining the same three methods, but none found that clarifies these doubts.

(I'm wondering if we are in front of one of these things that are repeated decade after decade in school without any criticism).

Answer 1

There are four parameters in the construction: the major axis $M$ of the ovoid, the radius $r$ of the upper half-circle, the radius $R$ of the middle arcs, the radius $\rho$ of the lower arc. These are related, however, so that we end with three free parameters.

To find the relation between them, let $\alpha$ be the center angle of the middle arcs and $M$ the major axis of the ovoid. We have then: $$ M=r+(R-r)\tan\alpha+\rho. $$ But on the other hand: $$ R-\rho={R-r\over\cos\alpha}. $$ Eliminating $\alpha$ we obtain then the relation: $$ M=r+\rho+\sqrt{2Rr-r^2-2R\rho+\rho^2}. $$ You can invert this, to obtain an expression for $\rho$: $$ \rho={M^2+2r^2-2Rr-2Mr\over2(M-R-r)}. $$

I presume the constructions you mention use some other implicit "aesthetical" assumptions.

Of course the solution above works as long as the expression for $\rho$ gives a positive result: this is a constraint on the possible values of $M$, $R$ and $r$.

Answer 2

The figure has three degrees of freedom (such as the three radii). If you impose the two axis, one DOF remains.

But on many figures that I see, the aperture of the small arc is a right angle. If this holds, there are only 2 DOFs.

Answer 3

Your issue has indeed three degrees of freedom and I show that this number is based on a hidden assumption of smoothness that brings an explainable constraint on all these constructions.

I show it in a first part.

In a second part, I give a personal opinion to the questionning in your last sentence and the interest to repeat ad libitum old techniques.

1) Scientific answer : The hidden smoothness constraint is given on Fig. 1. It is based on the following fact :

If two circular arcs centered in $C_1, C_2$ are juxtaposed, they constitute a "smooth curve" (no angular point) if and only if $C_1, C_2$ are aligned with the connection point $M$.

The proof is straightforward : indeed, the tangent at a point $M$ in a circle centered in $O$ is orthogonal to radius $OM$.

Fig. 1 : Smoothness constraint at connection point $M$ : $C_1, C_2, M$ must be aligned. (Please note that in our issue, the second type of connection giving an inflection point doesn't occur).

Let us take as reference your third figure. The general construction of an oval can be described in the following way, being given an orthogonal axes system centered in $O_1$ :

• Choose (first degree of freedom) a point $T_2$ on the horizontal axis and its symmetrical point $T_3$ in order that $T_2T_3$ is a diameter of the upper half-circle.

• Then choose, again on the horizontal axis a center $O_2$ (2nd degree of freedom) outside the open line segment $(T_2T_3)$.

• Finally, draw in the fourth quadrant a circular arc centered in $O_2$, starting at $T_3$ with endpoint $T_4$ wherever you want (as long as the arc remains in the fourth quadrant) : this the 3rd degree of freedom.

There are no other degrees of freedom. Indeed, the last (bottom) circular arc :

a) its center $O_4$ has to be at the intersection of line $O_2T_4$ with the vertical axis. Why that ? Because the smoothness constraint we have seen upwards imposes alignment for $O_2,O_4,T_4$.

b) its radius must be $O_4T_4$.

Remarks : of course, one completes the drawing by symmetrizing the second arc with respect to the vertical axis. In particular, one can check that the connection of circular arcs in $T_3$ is smooth because $O_1, O_2$ and $T_3$ are aligned.

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Besides, there are several similar constructions (this could be interpreted in favor of what you say : why this technique should be learned instead of this other one ?).

Here are two of them (Fig. 2) : it is construction of "ovals" which are cousin curves of "ovoids" (with 2 symmetry axes, and rather close to ellipses). These constructions has been first published by Serlio (an italian architect active in the middle of the XVI-th century ; see reference below). A common feature with your ovoids is that they are obtained by connecting in a smooth manner four circular arcs.

Fig. 1 : Serlio"s constructions : On the left figure, the arcs centered in $A$ and $B$ with resp. radii $d/2$ and $d$ where $d$ is the diagonal of the squares (the two other arcs are symmetrical to these ones) ; on the right figure, the first two arcs are centered in C and D with resp. radii $s$ and $2s$ where $s$ is the sidelength of the small equilateral triangles. Both figures are based on "pure" shapes (in the "Platonic sense") : squares on the left, equilateral triangles on the right.

2) A personal opinion about repeating old techniques

I have a mitigated attitude towards learning in any domain by beginning by "copying the antiques". I think that the interest of having been strongly acquainted with old techniques much depends on the arts (in a wide sense of the word) : I do believe that in music, architecture, painting, drawing, repeating the old masters and their techniques is necessary, but not till you are fed up, even more, disgusted... In the "art of learning" mathematics, in particular geometry, it is a rather good idea to devote some time to the classics, through the history of mathematics, but also a little technical drawing. There is also an algorithmic profit (do that precisely, then that precisely, then...) which is good and is appealing to learners ("Daddy, we have learned today how to do a long division !"). But, there is a certain stage where explanations must come in the forefront. If you are 18 and all the teching you receive is to repeat like a parrot a list of techniques without them being discussed, you fall into scolastism. As a teacher, I have been the witness of such sclerotic teaching ; I remember having asked to some former colleagues "why do we still teach that ?" ; answer : "because it is easy to ask exam questions on this subject...".

An excellent reference for Serlio is :

https://pdfs.semanticscholar.org/148a/d9806fee3d1009cc72e4807c9d7aa01fac4b.pdf

(discussing the historical context but as well technical details like the degree of similitude with ellipses having the same major and minor axes). See as well :

http://faculty.evansville.edu/ck6/ellipse.pdf