ARC (GEOMETRY): MEASURE, TYPES OF ARCHES, EXAMPLES - MATHS - 2024

The arc, in geometry, is any curved line that connects two points. A curved line, unlike a straight line, is one whose direction is different at each point on it. The opposite of an arc is a segment, since this is a straight section that joins two points.

The arc most frequently used in geometry is the arc of circumference. Other arches in common use are the parabolic arch, elliptical arch and the catenary arch. The arch form is also frequently used in architecture as a decorative element and a structural element. This is the case of the lintels of the doors and windows, as well as of the bridges and aqueducts.

Figure 1. The rainbow is a curved line that joins two points on the horizon. Source: Pixabay

The arch and its measure

The measure of an arc is its length, which depends on the type of curve that connects the two points and their location.

The length of a circular arc is one of the simplest to calculate, because the length of the complete arc or perimeter of a circumference is known.

The perimeter of a circle is two pi times its radius: p = 2 π R. Knowing this, if we want to calculate the length s of a circular arc of angle α (measured in radians) and radius R, a proportion is applied:

(s / p) = (α / 2 π)

Then, clearing s from the previous expression and substituting the perimeter p for its expression as a function of the radius R, we have:

s = (α / 2 π) p = (α / 2 π) (2 π R) = α R.

That is, the measure of a circular arc is the product of its angular opening times the radius of the circular arc.

For an arch in general, the problem is more complicated, to the point that the great thinkers of antiquity claimed that it was an impossible task.

It was not until the advent of differential and integral calculus in 1665 that the problem of measuring any arc was satisfactorily solved.

Before the invention of differential calculus, solutions could only be found by using polygonal lines or arcs of circumference that approximated the true arc, but these solutions were not exact.

Types of bows

From the point of view of geometry, arcs are classified according to the curved line that joins two points on the plane. There are other classifications according to its use and architectural form.

Circular arc

When the line connecting two points in the plane is a piece of circumference of a certain radius, we have a circular arc. Figure 2 shows a circular arc c of radius R connecting points A and B.

Figure 2. Circular arc of radius R that connects points A and B. Elaborated by Ricardo Pérez.

Parabolic arch

The parabola is the path followed by an object that has been thrown obliquely into the air. When the curve that joins two points is a parabola, then we have a parabolic arc like the one shown in figure 3.

Figure 3. Parabolic arc connecting points A and B. Elaborated by Ricardo Pérez.

This is the shape of the jet of water that comes out of a hose pointing upwards. The parabolic arc can be observed in the water sources.

Figure 4. Parabolic arch formed by water from a fountain in Dresden. Source: Pixabay.

Catenary arch

The catenary arch is another natural arch. The catenary is the curve that forms naturally when a chain or rope hangs loosely from two separate points.

Figure 5. Catenary arch and comparison with the parabolic arch. Prepared by Ricardo Pérez.

The catenary is similar to the parabola, but it is not exactly the same as can be seen in figure 4.

The inverted catenary arch is used in architecture as a high compressive strength structural element. In fact, it can be shown to be the strongest type of bow among all possible shapes.

To build a solid catenary arch, just copy the shape of a hanging rope or chain, then the copied shape is flipped to reproduce it on the door or window lintel.

Elliptical arch

An arc is elliptical if the curve that connects two points is a piece of ellipse. The ellipse is defined as the locus of points whose distance to two given points always adds up to a constant quantity.

The ellipse is a curve that appears in nature: it is the curve of the trajectory of the planets around the Sun, as demonstrated by Johannes Kepler in 1609.

In practice an ellipse can be drawn by pinning two struts to the ground or two pins in a piece of paper and tying a string to them. The rope is then tightened with the marker or pencil and the curve is traced. A piece of ellipse is an elliptical arc. The following animation illustrates how the ellipse is drawn:

Figure 5. Tracing an ellipse using a taut rope. Source: Wikimedia Commons

Figure 6 shows an elliptical arc connecting points G and H.

Figure 6. Elliptical arch connecting two points. Prepared by Ricardo Pérez.

Examples of arches

The following examples refer to how to calculate the perimeter of some specific arches.

Example 1

Figure 7 shows a window finished in a cut circular arc. Dimensions shown in figure are in feet. Find the length of the arc.

Figure 7. Calculation of the length of the circular arc of a window. (Own annotations - window image on Pixabay)

To obtain the center and radius of the circular arc of the window lintel, the following constructions are made on the image:

-The segment KL is drawn and its bisector is drawn.

-Then the highest point of the lintel is located, which we call M. Next, the KM segment is considered and its mediatrix is ​​traced.

The intercept of the two bisectors is point N and it is also the center of the circular arc.

-Now we must measure the length of the NM segment, which coincides with the radius R of the circular arc: R = 2.8 feet.

-To know the length of the arc in addition to the radius, it is necessary to know the angle that the arc forms. Which can be determined by two methods, either it is measured with a protractor, or alternatively it is calculated using trigonometry.

In the case shown, the angle formed by the arc is 91.13º, which must be converted to radians:

91.13º = 91.13º * π / 180º = 1.59 radians

Finally we calculate the length s of the arc using the formula s = α R.

s = 1.59 * 2.8 feet = 4.45 feet

Example 2

Find the length of the elliptical arc shown in figure 8, knowing the semi-major axis r and the semi-minor axis s of the ellipse.

Figure 8. Elliptical arch between GH. Prepared by Ricardo Pérez.

Finding the length of an ellipse was one of the most difficult problems in mathematics for a long time. You can get solutions expressed by elliptical integrals but to have a numerical value you have to expand these integrals in power series. An exact result would require infinite terms of those series.

Fortunately, the Hindu mathematical genius Ramanujan, who lived between 1887 and 1920, found a formula that very precisely approximates the perimeter of an ellipse:

The perimeter of an ellipse with r = 3 cm and s = 2.24 cm is 16.55 cm. However, the elliptical arc shown has half that value:

Length of the elliptical arch GH = 8.28 cm.

References

1. Clemens S. 2008. Geometry and Trigonometry. Pearson Education.
2. García F. Numerical procedures in Java. Length of an ellipse. Recovered from: sc.ehu.es
3. Dynamic geometry. Bows. Recovered from geometriadinamica.es
4. Piziadas. Ellipses and parabolas around us. Recovered from: piziadas.com
5. Wikipedia. Arch (geometry). Recovered from: es.wikipedia.com