10 junio, 2024

Oval (geometric figure): characteristics, examples, exercises

He oval Symmetric is defined as a flat and closed curve, which has two perpendicular axes of symmetry -one greater and one less- and is made up of two arcs of circumference equal two to two.

In this way it can be traced with the help of a compass and some reference points on one of the axes of symmetry. In any case, there are several ways to draw it, as we will see later.

It is a very familiar curve, since it is recognized as the contour of an ellipse, this being a particular case of the oval. But the oval is not an ellipse, although sometimes it looks very similar to it, since its properties and layout differ. For example, the ellipse is not built with a compass.

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Characteristics

The oval has very varied applications: architecture, industry, graphic design, watches and jewelry are just some areas where its use stands out.

The most outstanding characteristics of this important curve are the following:

-It belongs to the group of technical curves: it is traced forming circumferential arcs with the help of a compass.

All its points lie on the same plane.

-Lacks curves or ties.

-Its layout is continuous.

-The curve of the oval should be smooth and convex.

-When drawing a line tangent to the oval, all of it is on the same side of the line.

-An oval only admits two parallel tangents at most.

examples

There are several methods for constructing ovals that require the use of a ruler, square, and compass. Next we will mention some of the most used.

Construction of an oval using concentric circles

Figure 2, above, shows two concentric circles with centers at the origin. The major axis of the oval measures the same as the diameter of the outer circumference, while the minor axis corresponds to the diameter of the inner circumference.

-An arbitrary radius is drawn up to the outer circumference, which intersects both circles at points P1 and P2.

-Next, point P2 is projected on the horizontal axis.

-In a similar way, the point P1 is projected on the vertical axis.

-The intersection of both projection lines is the point P and belongs to the oval.

-All the points of this section of the oval can be plotted in this way.

-The rest of the oval is traced with the analogous procedure, carried out in each quadrant.

Exercises

Next, other ways of building ovals will be examined, given a certain initial measurement, which will determine their size.

– Exercise 1

Trace with the help of a ruler and compass an oval, known its major axis whose length is 9 cm.

Solution

In figure 3, shown below, the resulting oval appears in red. Pay special attention to dashed lines, which are the auxiliary constructions needed to draw an oval whose major axis is specified. We will indicate all the necessary steps to reach the final drawing.

Step 1

Trace the 9 cm segment AB with a ruler.

Step 2

Trisect the segment AB, that is, divide it into three segments of equal length. Since the original segment AB measures 9 cm, the segments AC, CD, and DB must each measure 3 cm.

Step 3

With the compass, centering at C and opening CA, an auxiliary circle is drawn. Similarly, the auxiliary circle with center D and radius DB is drawn with the compass.

Step 4

The intersections of the two auxiliary circles built in the previous step are marked. We call them points E and F.

step 5
With the ruler, draw the following rays:[FC)[FD)[EC)[ED)[FC)[FD)[EC)[ED)
step 6

The rays of the previous step intersect the two auxiliary circles at points G, H, I, J respectively.

step 7

With the compass the center is made in F and with opening (or radius) FG the arc is drawn GH. Similarly, making the center in E and with radius EI the arc is traced IJ.

step 8

The union of the arches G.J., JI, IH and H.G. They form an oval whose major axis is 9 cm.

step 9

We proceed to delete (hide) the points and auxiliary lines.

– Exercise 2

Draw an oval with a ruler and compass, whose minor axis is known and its measure is 6 cm.

Solution

The figure above (figure 4) shows the final result of the construction of the oval (in red), as well as the intermediate constructions necessary to reach it. The steps followed to build the oval with a minor axis of 6 cm were the following:
Step 1

The 6 cm long segment AB is drawn with the ruler.

Step 2

With the compass and the ruler, draw the perpendicular bisector to the segment AB.

Step 3

The intersection of the bisector with the segment AB, results in the midpoint C of the segment AB.

Step 4

With the compass draw the circle with center C and radius CA.

step 5

The circle drawn in the previous step intersects the bisector of AB at points E and D.

step 6

Rays[AD)[AE)[BD)and[BE)aredrawn[AD)[AE)[BD)y[BE)

step 7

With the compass draw the circles with center A and radius AB and the circle with center B and radius BA.

step 8

The intersections of the circles drawn in step 7, with the rays built in step 6, determine four points, namely: F, G, H, I.

step 9

With center in D and radius DI the arc IF is drawn. In the same way, with center in E and radius EG the arc GH is drawn.

step 10

The union of the arcs of circumference FG, GH, HI and IF determine the sought oval.

References

Plastic Ed. Technical curves: ovals, ovoids and spirals. Recovered from: dibujonavarres.wordpress.com.
Mathematische Bastelien. Egg Curves and Ovals. Retrieved from: mathematische-basteleien.
University of Valencia. Conical and Flat Technical Curves. Retrieved from: ocw.uv.es.
Wikipedia. Oval. Recovered from: es.wikipedia.org.
Wikipedia. Oval. Retrieved from: en.wikipedia.org.

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