2011 drawing techniques of cars - automatic drawing up thinkdesign Luciano Fabal Tecniche design cars
automatic drawing up thinkdesign
Luciano Fabal
One of the first concepts that must learn the industrial designer, in order for the correct operation of the instruments of representation of own ideas, is to know the difference between vector and bitmap graphics.
Bezier curves are used in vector graphics. Objects: circles, polygons, rectangles, triangles, lines, points, curves and text are described on the worksheet with their geometric properties. Ie il programma (vettoriale) utilizza dei vettori per rappresentare gli oggetti, quindi per il rettangolo i vettori direzione base per altezza, per un cerchio il vettore del raggio che traccia la circonferenza e così via.
La grafica vettoriale permette alle moderne stampanti, siano esse di tipo laser oppure a getto d’inchiostro, di ridurre ed ingrandire la pagina senza che questo alteri la nitidezza dei caratteri tipografici. Ciò significa che i caratteri stessi non sono memorizzati come immagini definite per punti ( bitmapped ), ma piuttosto sono memorizzate le equazioni delle curve che descrivono i caratteri stessi. Dunque la definizione del carattere dipende dalla qualità della stampante usata, regardless of magnification or reduction of the page.
The mathematics behind this is relatively simple, and its understanding requires knowledge of the third degree polynomials, including the derivation of these polynomials, and also the knowledge of the basics of vector calculus in the plane.
like to sum up the advantages of vector graphics describe images (in the form of objects) with mathematical functions, making them independent of the resolution (so they can be played on any device always getting the best quality). In contrast to vector graphics programs are less intuitive than raster graphics (which work much like hand-drawn). They organize objects on the work surface (sheet) as independent entities overlap. Two objects can not be on the same (imaginary) part, so they must be merged together. There is no eraser tool, they can only be deleted.
The vector graphics program "CorelDraw" (well known to designers in particular those who specialize in publishing graphic) in the toolbox called the point at which the geometric shape is flat "Bezier tool."
Who uses or has used this program will be disclosed sooner or later the history of French and will be pleasantly surprised that the name of the instrument, used so many times, both linked to the name of the developers of this technology rather than the technology itself. Not all programs of graphics and industrial design makes this tribute to the inventor of the "curves for control points."
So we find that we are going to describe curves that derive their name from Pierre Bezier (1910-1999), a mechanical engineer at the Renault factory.
Bézier The problems that had to do was not represented graphically, but concerned rather the management of CNC machines that cut the sheet metal parts with which to make car bodies. Bézier same must be said that, very honestly admitted that he had come to the same curves also his colleague Paul De Casteljau Citroen, but the constraints of secrecy imposed by this factory meant that these curves are linked to his name.
curves were constructed in 1959 using the algorithm of De Casteljau. Established a way to create Bezier curves started from two points and a vector line exactly.
Bézier, assuming that each point on the curve should be represented by the following parametric function :
B (t) = ( n / i ) P the (1 - t ) ni t the , t [0, 1].
characteristic of Bezier curves, resulting from the parametric formulation, is to allow the definition of a closed curve (Fig. 1)
1 - closed Bezier curve
addition, as shown by the mathematical formulation, they are entirely manageable through the so-called, control polygon.
In particular, the curve is tangential to the control polygon and the control of the shape of the curve is obtained by changing the position of the control points (Fig. 2)
2 - cubic curve Bézier
This event is very convenient in the implementation of algorithms for modeling, because it reduces drastically the number of variables needed for the management of the whole geometry.
An observation to be made about the degree of the resulting curve.
The possibility of local control of a Bezier curve, can be considerably increased by the increased number of control points.
The increase of these, however, requires an increase in the degree of the curve with a consequent increase the complexity of its calculation.
addition, moving a control point produces a global effect, which involves changing the shape of the entire curve (Fig. 3)
3 - effect of moving control points on a Bezier curve
This depends on the mathematical form of a curve Bézier , from which shows that the functions depend upon the mixing of all control points.
This is, indeed, the main limitation of Bezier curves that, in a perspective of overcoming the above problem, find their natural extension B-Spline curves.
B-spline curves are the natural evolution of models Bézier.
The main difference between the two types of curve is, from the practical point of view, the ability to manage the "local" shape of the curve.
In particular, determining the level and number of control points, calculate the appropriate number of components of the vector of the nodes using the following equation:
m = number of appropriate components of the vector nodes
p = degree della curva
Le curve B-spline , per quanto abbiano una forma matematica pressoché identica alle curve di Bézier , si differenziano decisamente da queste ultime, nella formulazione delle funzioni di miscelamento, le quali, in generale, hanno grado indipendente dal numero dei punti di controllo.
(tutte le illustrazioni che seguono sono state realizzate dall'autore di questa tesina)
Dati i punti P 0 e P 1 , una curva Bézier lineare è una linea retta che li attraversa.
B (t) = (1- t ) P 0 + t P 1, t [0, 1].
curva Bézier lineare la The curve with the same degree of interpolation points
A quadratic Bézier curve is the path traced by the function B (t ) data points P 0, P 1, 2 and P ,
B (t) = (1 - t ) 2 P 0 + 2 t (1- t ) P 1 + t 2 P 2, t [0, 1].
curve the same quadratic Bezier curve interpolation points with grade II
I fonts TrueType usano le spline Bézier composte da curve Bézier quadratiche.
I quattro punti P 0 , P 1 , P 2 e P 3 nel piano o in uno spazio tridimensionale definiscono una curva Bézier cubica. La curva ha inizio in P 0 si dirige verso P 1 and ends in P 3 coming from the direction of P 2. In general, it does not pass these points P 1 or 2 P and they shall be required only to give directional information to the curve. The distance between P 0 and P 1 determines how the curve moves in the direction of P 2 before heading P 3 .
La forma parametrica della curva è:
B (t) = (1- t ) 3 + 3 P 1 t (1- t ) 2 + 3 P 2 t 2 (1- t ) + P 3 t 3 , t [0, 1].
cubic Bezier curve and the same curve with third-degree interpolation points
Examples of application of the Bezier curves
I want to expose my personal experience as a designer in the use of Bezier curves. Of course I think that experience similar to that of the multitude of professionals who work with vector graphics and modeling. I do not think that the following exhibits have a particular technical or scientific value, but only a personal example of practical application.
In particular, Bezier curves, or curves can be managed with the control handles which are not handles interpolation, are particularly useful in the design tridimensionale.
Premetto che molto raramente utilizzo delle curve superiori al quarto grado (con più di cinque punti di controllo), cerco sempre di definire il mio disegno con delle curve di secondo e terzo grado sul piano (fig. 4) e del terzo e quarto grado se devo modificarle nello spazio tridimensionale.
Per ridurle ai minimi termini, ed anche per utilizzarle nella creazione di superfici con quattro lati (le migliori), le spezzetto dandogli la continuità di tangenza (all’esigenza le riunisco).
Se la curva l’ho generata dalla proiezione ortogonale nello spazio di due curve piane e soddisfa le mie esigenze (quindi non a touch more) the approximate minimum number of points does not change its shape. This is where I make use of curves above the fourth degree (Fig. 11).
4 - Bezier curves of the second and third degree
The decision to handle the curves with few control points is dictated by the fact that a curve of a second or third degree is clean, no changes in direction that sticks in the design and is much more manageable in the inevitable changes which will be submitted in the further design phase. In particular, within the 3D modeling curves are used to generate surfaces on which the imperfections of the curves will be amplified (Fig. 5 and 6).
5 - Bezier curves of different degree compared
6 - surfaces generated by Bezier curves of different grade
Sometimes imposed curves with many points of control, but then begins to move, to delete or add up to the desired pattern, and this is one of the great advantages that the Bezier curves give the designer can change the degree of the curve at any time (Fig. 7).
7 - set above curve with many control points and low points in the essential satisfying the same initial design
Drawing a curve in three dimensional space
imposed on the curves almost always on a plane, so if you would like to form a curve that has different directions in the first two corners on the design space plane x, y, then one of the tilting plane y, z by the orthogonal projection of the two-dimensional curve creates a curve (Fig. 8, 9 and 10).
8 - two curves drawn on the plane x, y
Fig. 9 – ribaltamento di una delle due curve sul piano z,y
Fig. 10 – curva risultante 3D generata dalla proiezione ortogonale nello spazio delle due curve piane
La curva così creata avrà un numero eccessivo di punti di controllo, per facilitarmene la gestione la approssimo the minimum number of points that allows me to keep the design quite similar to the original curve (Fig. 11).
11 - resulting 3D curve approximated to a smaller number of control points
Now this curve can be used to construct a back of a chair type Thonet.
will be the pin on which rotates the surface. To vary the thickness of the surface insert tre cerchi con diametro differente. Essi saranno le sezioni della superficie nei punti di minima e massima espansione (fig. 12 e 13).
Fig. 12– inserimento di cerchi sulla curva ortogonalmente alla stessa spostando il piano di lavoro sulla curva
13 - construction of the surface, mirror and top view
In the following catches (Fig. 15) as set before I actually designed the Thonet No 8 with the support of orthogonal projections original designer of the chair (Fig. 14).
14 - original design of the Thonet chair n. 8
This result is not a game purely virtual, but with specific indications can be realized in the numerical control machines. And this is what led by Ing. To develop its Bézier curves, needs to communicate with the shearing industry in the Renault as they should be cut sheets that would become the plates from the cars.
Fig. 15 - a sequence of modeling Thonet Chair No. 8 with thinkdesign 3D
Obviously the curves are curves, and the points system control (Bézier) or interpolation points are just ways to control them (a third degree curve has four control points if you use the system and two Bézier control points when using the interpolation, but will always remain same curve of third degree), is the operator from time to time decide how to proceed based on the needs that arise. Of course I can generate the same curve and shape it with Bezier curves and then to other intersecting curves or modify with the system of interpolation points or vice versa.
In the drawings below shows how I made the bodywork of a car fa_B 112 (my personal new version dell'Autobianchi 112) curves drawn by using only the control points (Bezier), in absolute points and interpolation points. The lines you see are only the references of the tangents of the curves, cutting surfaces and management elements of the design.
Fig. 16 - lines, curves and points of construction of the 3D model of fa_B112, the points are points on the curves flow and absolute were included in subsequent function snap in order to allow the intersections between the curve where he served
17 - the surfaces created by the lines and points of previous figure
18 - final result, fa_B112 my personal interpretation of the famous Autobianchi 112
Fig. 19 – rendering fa_B112 (Accento12)
Fig. 20 – rendering fa_B112 (Accento12)
Di esempi sull'utilizzo delle curve di Bézier o delle curve in generale ogni disegnatore potrebbe scriverci un libro, io ho ritenuto interessante spiegare come le uso quando devo generare curve tridimensionali da curve bidimensionali (la base della mia tecnica di disegno).
Ogni disegnatore con l'esperienza avrà sviluppato delle tecniche che sono frutto del particolare campo di applicazione della propria professionalità, della propria intelligenza e della propria personale conoscenza della geometria.
Una buona conoscenza della matematica con cui i software di disegno generano le curve sicuramente aiuterà a migliorare le prestazioni.
The virtual prototype model
fa_B112 is published on the following sites:
Julius C. Barozzi - University of Bologna
Wikipedia, the free encyclopedia
Advanced Computer Graphics Technology
KAEMaRT Group - Department of Mechanics Politecnico di Milano
The mathematics of parametric curves: the curves of the Lagrange Nurbs
dispensa ad uso degli studenti del corso preparata e redatta dall’ Ing. Antonio Mancuso
Giulio C. Barozzi – Università di Bologna
Wikipedia, l'enciclopedia libera
Corso di Grafica Computazionale Tecnica
KAEMaRT Group - Dipartimento di Meccanica Politecnico di Milano
Geometria computazionale;
The mathematics of parametric curves: the curves of the Lagrange Nurbs
pantry for use by students of the course prepared and written by 'Mr. Anthony Mancuso
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