## Abstract

The first digitization systems successfully used in curves with origin in mathematical calculations or experimental tests proved useless when they were the result of heuristic methods developed under aesthetic criteria. Renault engineer Pierre Bézier and Citroën engineer Paul de Casteljau found the solution by focusing on the period of ideation of shapes and not on their subsequent translation and integration into the digital domain, studying the working methods of designers and incorporating in the mathematical definition of their curves the own actions of drawing, understood as basic instrumental support in creative processes. This article traces and analyzes the strong relationship between Bézier’s original approach and the procedures driving the manual stroke as the main reason for the consolidation of his graphic proposal as the most effective model for the hand drawing of digital curves and its inclusion since that time and to the present day in the software used by architects.

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## Notes

The design teams adapted their working methods to the complexity of the curves and surfaces demanded by the new industrial image, extending the method developed by the General Motors engineer Harley J. Earl, based, in addition to the traditional preliminary sketches, on the use of clay models as a suitable means to facilitate the development of the continuous surfaces characteristic of the new image, far from the mechanistic sincerity.

The success of the proposal was based on the novelty of the approach and not on its technical complexity. Proof of this is that the management teams of the two companies questioned its feasibility because of the simplicity of the arguments on which it rested. In the case of Renault, given the simplicity of the method, the management team considered that, if valid, it would have already been developed by other teams. Rogers David quotes the famous phrase they used to address Bézier when dealing with the issue “if your system were that good, the Americans would have invented it first!” (Rogers 2001: 36). In Citroen, as reflected in his autobiography, at the time when de Casteljau presents the first results in parallel to those being developed in Renault, the criticisms from superiors and supervisors focus, among other issues, on considering his approach too simple to be a line of work worthy to follow (de Casteljau 1999).

As Alastair Townsend (2014: 52) states, there seems to be a relationship with the drawing templates, although we do not have direct information about the author´s intentions.

The compilation work carried out in (Bézier 1986) stands out among the published studies.

The mathematical development is trivial, not only from the current perspective but also for any mathematician of that time. Keep in mind that the basis used by Bézier and de Casteljau was enunciated by Bernstein in 1912, and that the mathematical definition of spline curves was developed by Isaac Jakob Schoenberg in 1946. Therefore, the mathematical concepts used were chosen from an existing material.

Tools such as flexible material strips were used for the forced plotting of curves not analytically defined. Their use allowed a clean stroke on curves already defined by the freehand stroke. They were instruments designed to force previous geometric relationships, implicit in the previous manual stroke, although they were not analytical relationships.

Bézier curves can be developed in any of the usual systems of equations, in their implicit, explicit or parametric form, but they are designed to work in the latter. Bézier makes direct reference to this formulation in his texts as a feature that makes the curve independent of spatial coordinates, facilitating its reading as a geometric object (Bézier 1971: 211; Bézier 1974: 130).

Functions of parameter

*t*: (1) Uniform support: equal time periods between segments:*U*_{i}=1. (2) Chord length support: different times in the plotting of each segment of the curve depending on the position of the end points, achieving a movement adapted to the dimensions of each segment, approximating spaces and times: \(U_{i} = \frac{{\left\| {P_{{\left( {i + 1} \right)n}} - P_{{\left( {i + 1} \right)0}} } \right\|}}{{\left\| {P_{in} - P_{i0} } \right\|}}\). (3) Centripetal support: it captures the “centripetal” force exerted by the artist to compensate for the centrifugal force caused by the drawings of the curve in the short segments, so that the trajectory opens somewhat less than in the case of a continuous movement based on a uniform support, but something more than the rigorous and excessively homogeneous trajectory generated by the chord length support. Although seldom used due to its high computational cost, this support function goes a step further in the interpretation of the digital curve as a drawing, simulating the control that the draftsman exerts on his hand through the rectification of the speed applied to the stroke and with it on its curvature: \(U_{i} = \frac{{\sqrt {\left\| {P_{{\left( {i + 1} \right)n}} - P_{{\left( {i + 1} \right)0}} } \right\|} }}{{\sqrt {\left\| {P_{in} - P_{i0} } \right\|} }}\).There is a different treatment of this issue in the 1974 and 1986 documents. In (Bézier 1974) he proposed the control of the curvature through the non-interpolated points, while in (Bézier 1986: 41) he seems to assume the modifications proposed by the new conditions of the spline curves, considering the equality of the curvatures in the links between curves.

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Díaz Moreno, F. Hand Drawing in the Definition of the First Digital Curves.
*Nexus Netw J* **22**, 755–775 (2020). https://doi.org/10.1007/s00004-019-00472-1

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DOI: https://doi.org/10.1007/s00004-019-00472-1

### Keywords

- Digital curves
- Pierre Bézier
- Paul de Casteljau
- Bézier curves
- Development of CAD