**2. A definition about free form geometries**

First, a definition of free form geometry (also known as sculptured surface) has to be set: it is a non-linear and curved shape, with neither constant nor pattern curvature radius along the geometry

Initially these geometric forms were employed exclusively in situations where they were to be used such as the aeronautical and naval industry. Currently complex geometric forms are becoming popular and can be found in various components in the automobile industry; in the consumption industry; toys, packaging, electronic products , esthetical products and ergonomics. This definition is important once that any manufacturing planning depends upon its shape. For instance, rotational parts have to be turned and prismatic parts have to be milled. Free form geometry has its own particularity to be manufactured.

Advanced Free Form Manufacturing by Computer Aided Systems – Cax 557

It is also important to distinguish that free form geometries do not have to be associated with complex products. Many complex products have no complex shapes, for instance, a machine centre. It has many components, high precision assembling, and relatively high cost. However, in general, its components have prismatic or rotation shapes. Instead a simple bottle of pop soda has no assembly, it does not require such precision, and it is not an expensive product. However its shape are quite complex, and most of the time to manufacture complex geometries are much more expensive than simple forms, once it

Due to its complexity, free form shapes cannot be machined by a conventional machines driven by human. It requires a CNC machine and the free form tool paths have to be calculated by software. After machined the geometry, geometric errors have to be accessed. Once again, due to its complexity it is not possible to make any inspection by ordinary metrology devices. Therefore, such cases require a measure machine coordinate (or any

So, the manufacturing steps of complex forms requires different CAx systems. Within these

CAD (Computer Aided Design/Drafting). Software to aid the Project , design, mould

The Figure 3 presents the integration of some CAx systems required to manufacture for free

 CAM (Computer Aided Manufacturing). Software to aid manufacturing activities. CAI (Computer Aided Inspection). Software to aid the inspection of geometric forms. CAE (Computer Aided Engineering). Software to aid the simulation of mechanical,

order technique to accessed geometric errors, like as laser or photometry).

Fig. 2. A Bézier curve plotted in graphic software

required high level of technology, like CAx.

systems the following stand out:

form geometries.

industrially design the products.

strength, temperature, pressure etc.

**3. Computer aided free form manufacturing** 

(b) Mould for plastic product

Fig. 1. Free form geometries

To represent computationally these sorts of geometry higher order polynomials are required. These equations were developed by Pierre Bézier, an French engineer who based on a Hermite proposal implemented the software Unisurf, in 1972. Such polynomials are known as Spline. The equation 1 is the Spline defined by Bézier. The curve is guided by a polygon (SOUZA et al, 2010). Using a mathematical software the Figure 2 was plotted.

$$\mathbf{p(u)} = \mathbf{p}\_0 \left( \mathbf{1} - 3\mathbf{u} \; \; \mathbf{u} + 3\mathbf{u}^2 - \mathbf{u}^3 \right) + \mathbf{p}\_1 \left( 3\mathbf{u} \; \; \; \mathbf{u} - 6\mathbf{u}^2 + 3\mathbf{u}^3 \right) \\ + \mathbf{p}\_2 \left( 3\mathbf{u}^2 - 3\mathbf{u}^3 \right) + \mathbf{p}\_3 \left( \mathbf{u}^3 \right) \tag{1}$$

where:

p: is the control point of the polygon

u: parameter 0-1

This is a brief approach to understanding how to model a free form curve. There are many other equations to create a Spline curve by computer, like NURBS (non-uniform rational Bspline) and others. A 3D surface is created analogous to a curve, but using one more axes.

(a) Product free form shape

(b) Mould for plastic product

To represent computationally these sorts of geometry higher order polynomials are required. These equations were developed by Pierre Bézier, an French engineer who based on a Hermite proposal implemented the software Unisurf, in 1972. Such polynomials are known as Spline. The equation 1 is the Spline defined by Bézier. The curve is guided by a polygon (SOUZA et al, 2010). Using a mathematical software the Figure 2 was plotted.

<sup>0</sup> 2 3 23 23 3 p u p 1 3u 3u u p 3u 6u 3u p 3u 3u p u 1 23 (1)

This is a brief approach to understanding how to model a free form curve. There are many other equations to create a Spline curve by computer, like NURBS (non-uniform rational Bspline) and others. A 3D surface is created analogous to a curve, but using one more axes.

Fig. 1. Free form geometries

p: is the control point of the polygon

where:

u: parameter 0-1

Fig. 2. A Bézier curve plotted in graphic software

It is also important to distinguish that free form geometries do not have to be associated with complex products. Many complex products have no complex shapes, for instance, a machine centre. It has many components, high precision assembling, and relatively high cost. However, in general, its components have prismatic or rotation shapes. Instead a simple bottle of pop soda has no assembly, it does not require such precision, and it is not an expensive product. However its shape are quite complex, and most of the time to manufacture complex geometries are much more expensive than simple forms, once it required high level of technology, like CAx.
