**1. Introduction**

The topic of this article is the subject of numerous publications [1–3], which demonstrates the interest therein and follows the line of research on the recovery of technical historical heritage [4–9], in particular that of antique clocks [10]. The sun and water were, in ancient times, the first references to the measurement of time. After these types of clocks, perfected by Egyptians and Greeks, hourglasses appeared, and subsequently, clocks with cog mechanisms were invented in the 13th century. In the 15th century, the first spring-powered timepieces were developed in Germany [11,12].

In the 16th century, major geographical discoveries provoked an interest in the precise determination of longitude at sea. Therefore, the lunar clock was invented, as were astronomic clocks [13,14]. By the 17th century, the cog-regulating pendulum of Christiaan Huygens had appeared, as had the discoveries of Robert Hooke with respect to the law of elasticity; these mechanical improvements contributed towards the invention of the first balance-spring-regulated pocket chronometer from Hooke and Huygens, which constituted a milestone in precision horology.

In the 18th century, John Harrison solved the problem of longitude, and certain elements were perfected, such as the profile of the teeth of the cogs and the compensation of the pendulum; these aspects contributed towards improving accuracy. Finally, in the 19th century, with the industrial revolution, mass production and mechanization appeared. These improvements in manufacturing technologies constituted major contributions towards the improvement of the precision of watchmaking mechanisms.

Christiaan Huygens (The Hague, Netherlands, 1629–1695) was one of the central characters of the scientific revolution in the 17th century alongside Francis Bacon, René Descartes, Galileo Galilei, and Isaac Newton. With training in physics, mathematics, and astronomy, he contributed notably by perfecting Kepler's telescope, thereby discovering Titan and the Orion Nebula. In 1656, he invented the pendulum clock, after adding a pendulum to a clock driven by weights, so that the clock would keep the pendulum moving and regulate its progress [15]. After this discovery, he stated the pendulum theory employed to measure the acceleration of gravity and its variations with altitude and latitude [16–18].

He subsequently developed, with little success, a portable chronometer to make it easier for seafarers to determine geographical longitude at sea, based on the double suspension of the pendulum between cycloidal blades. Interest in his theories is still shown in various studies [19,20].

Huygens published several books, among which *Horologium* [21,22] stands out, in which he defines the first pendulum clock, and *Horologium Oscillatorium* [23–25], his masterpiece, where he describes the isochronous pendulum clock and studies the properties of the cycloid and of geometric curves in general.

Huygens's pendulum clock is a notable example of technical historical heritage, although there are many projects for the recovery of cultural heritage in general. A good sample of these include the mechanical lion of Leonardo Da Vinci [26], the initiative of the Canarian Orotava Foundation of History of Science to study the machines invented by the distinguished Spanish engineer Agustín de Betancourt [27], the virtual reconstruction of the device by Juanelo Turriano to raise water from the Tagus river to the city of Toledo, Spain [28], the clock of the Cathedral of Santiago de Compostela, Spain [29], the Church of Santa María de Melque in Toledo, Spain [30], and the Church of San Agustín de la Laguna in Tenerife, Spain [31].

The objective pursued in this historical investigation consists of obtaining a reliable three-dimensional model and the simulation of its movement through the parametric software CATIA V5 [32] of the isochronous pendulum clock, as designed by Christiaan Huygens and published in 1673, to verify its correct operation and underline its exactitude. Its selection is due to the significant technological advance that it represented as the first reliable time meter, especially thanks to the use of the properties of the cycloid, and also for its historical interest, since the whole theory of pendulum movement that Huygens enunciated remains in force.

The contribution of this research lies in the fact that there is currently no 3D CAD model of this historical invention with any degree of detail, which would help in the comprehension of its general operation, which is anything but simple. This operation has been verified through simulations from its virtual recreation, generating videos to check the properties of the cycloid (isochronous movement), and therefore, the accuracy of its functioning.

Another objective of this 3D model is educational, as it is designed for display in a History of Technology museum or interpretation centre. The impact of this research depends on the future uses of the model, which include:

