**6.2 Mathematics for constructing curves**

The quality of a curve is related to the mathematics used for this representation. Advanced CAD systems can offer different possibilities to generate a curve, in order to allow modelling of complex geometric forms, smooth and seamless.

Using parametric polynomial equations, Bezier-based methodology, the developers responsible for the development of each CAD software creates specific mathematical resources in order to allow greater control and ease of work to the user. This development of each software house is confidential and may represent the differential of the CAD software. Thus, only generic formulations of curves are discussed in the user guides for each software. Therefore, we will discuss below, some methodologies that differ from the creation of curves in different CAD systems.

Other mathematical formulations based on Bezier method have been developed for the representation of curves and surfaces in computer systems. The B-spline curve is also used for a parametric polynomial equation with modifications of the proposal Bezier, allowing among other things, to represent a curve using a polynomial of low degree, facilitating computations, also allowing local changes of the curve.

The Spline NURBS (Non-Uniform Rational B-Spline) represents the state of the art for the representation of curves and surfaces in CAD system, providing better control of the curve, allowing local issues, and allows more efficient calculation. Basically, the methodology is based on NURBS B-Spline method, adding two main functions:

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a. **Degree of mathematical equation**. Some CAD systems allow changing the degree of the equation used to construct the curve. Using the points P1 (0, 0), P2 (50, 50), P3 (100, - 50), P4 (150, 0), curves were generated by changing the degree of the polynomial, as shown in Figure 21. Note that due to the number of control points, according to the mathematical rule are allowed to generate curves of maximum degree 3 (the degree of the equation is 1 minus the number of control points). Also note that a polynomial of

b. **Different equations**. Some CAD systems do not allow changing the degree of the polynomial used to construct the curve. However, many systems offer different

It is important to note that the same sequence of points can build distinguish ways in the same CAD system. Thus, it is difficult to build the same geometrical shape in two

c. **Influence of the control points (weight).** Some CAD systems allow changing the degree of influence of each control point of the polygon under the curve calculated

degree 1 generates a sequence of lines.

Fig. 21. Curves constructed by polynomials of different degrees

equations to build a curve, as illustrated in Figure 22.

Fig. 22. Curves for different mathematical built using the same points

different CAD systems using the same sequence of points.

(Figure 23).

Non-Uniform: The vectors (knot) that indicate which portion of the curve is affected by a single point of control are not necessarily uniform.

Rational: You can set the intensity (weight) that each control point "attracts" the curve (exemplified below). It also allows the representation of elementary geometric entities: cylinders, cones, and plans, as well as conic curves, such as circles, ellipses, parabolas and hyperbolas.

In summary, these features mean more control factors can be applied to the curve, so more complex surfaces can be represented with a smaller number of turns, maintaining the continuity of curvature and tangency. For these reasons, the NURBS method is the state of the art in the representation of complex curves and surfaces in CAD systems.

The most advanced CAD systems can offer users different methodologies to create a curve. These methods are implemented by each developer and may involve from the choice between different mathematical functions, such as Bezier curves, B-Spline curve, NURBS curve, etc. As well as to allow changing the degree of the equation used.

It should be noted that each software has individual characteristics. The options of mathematical CAD software are not the same as other CAD software. These options, as well as construction tools and quality analysis of the geometries built, may be important differences for specific applications. These differences may involve the continuity of a curve, chances of building and editing, as well as computational workload.

The construction of curves in CAD systems is divided into two main mathematical methods:


Fig. 20. Curves created by the methods of interpolation and approximation of points

Besides the method, interpolation or approximation, CAD systems can still provide other resources for modelling curves, such as:

Non-Uniform: The vectors (knot) that indicate which portion of the curve is affected by a

Rational: You can set the intensity (weight) that each control point "attracts" the curve (exemplified below). It also allows the representation of elementary geometric entities: cylinders, cones, and plans, as well as conic curves, such as circles, ellipses, parabolas and

In summary, these features mean more control factors can be applied to the curve, so more complex surfaces can be represented with a smaller number of turns, maintaining the continuity of curvature and tangency. For these reasons, the NURBS method is the state of

The most advanced CAD systems can offer users different methodologies to create a curve. These methods are implemented by each developer and may involve from the choice between different mathematical functions, such as Bezier curves, B-Spline curve, NURBS

It should be noted that each software has individual characteristics. The options of mathematical CAD software are not the same as other CAD software. These options, as well as construction tools and quality analysis of the geometries built, may be important differences for specific applications. These differences may involve the continuity of a curve,

The construction of curves in CAD systems is divided into two main mathematical methods: a. **Interpolation method**. interpolation is to create a curve that passes through the points provided by the user. The curve of Figure 20 shows the interpolation process to create a curve, using four points as an example: P1 (0, 0), P2 (50, 50), P3 (100, -50), P4 (150, 0).

The Hermite curve is a curve described by the example of interpolation method. b. **Approach method**. In this case, the points provided by the user are used to create the control polygon will represent the curve, as shown in Figure 20 curve b. The Bezier

curve is a curve described by the example of approximation method.

Fig. 20. Curves created by the methods of interpolation and approximation of points

resources for modelling curves, such as:

Besides the method, interpolation or approximation, CAD systems can still provide other

the art in the representation of complex curves and surfaces in CAD systems.

curve, etc. As well as to allow changing the degree of the equation used.

chances of building and editing, as well as computational workload.

single point of control are not necessarily uniform.

hyperbolas.

a. **Degree of mathematical equation**. Some CAD systems allow changing the degree of the equation used to construct the curve. Using the points P1 (0, 0), P2 (50, 50), P3 (100, - 50), P4 (150, 0), curves were generated by changing the degree of the polynomial, as shown in Figure 21. Note that due to the number of control points, according to the mathematical rule are allowed to generate curves of maximum degree 3 (the degree of the equation is 1 minus the number of control points). Also note that a polynomial of degree 1 generates a sequence of lines.

Fig. 21. Curves constructed by polynomials of different degrees

b. **Different equations**. Some CAD systems do not allow changing the degree of the polynomial used to construct the curve. However, many systems offer different equations to build a curve, as illustrated in Figure 22.

Fig. 22. Curves for different mathematical built using the same points

It is important to note that the same sequence of points can build distinguish ways in the same CAD system. Thus, it is difficult to build the same geometrical shape in two different CAD systems using the same sequence of points.

c. **Influence of the control points (weight).** Some CAD systems allow changing the degree of influence of each control point of the polygon under the curve calculated (Figure 23).

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This can partition the curve internally into segments known as patches, and it often is not

(b) Curves with continuity of position

(d) Curves with continuity of position,

tangency and curvature

In many cases it is necessary for modelling three-dimensional boundary conditions that go beyond the requirements of design and ergonomics. Surfaces are required in these cases with a high degree of continuity in order to maintain the smoothness in the shape of the product. The quality of a complex surface generated by a CAD system is related to the

a. The smoothness of a surface can be related to the continuity of the curve used for its

When you want to develop surfaces for products requiring strict-looking, such as appliances, the external body of a car, etc., small discontinuities in the surfaces of the 3D CAD model can be replicated in the final product, be it cast or stamped. When creating surfaces through the interpolation of two sections and non-planar curves with different radius of curvature, and the need to maintain continuity of the tangential surface created in which the directions of the surface change smoothly, the surface must also have continuity of curvature. Applying tests of light reflection on the product such geometric discontinuities

b. The connection between different areas may represent regions with impaired smooth. c. Areas of intersection with rays (Fillets) calculated "automatically" by the software.

noticed by the user.

(a) Non-continuous

and tangency

Fig. 24. Degree of continuity of continuity of a curve

(c) Curves with continuity of position

Curve 1 Curve 2

**6.4 Quality of surfaces generated by CAD systems** 

model used for its creation in three circumstances:

creation.

can be found.

Fig. 23. Changed the influence of the point P2 on the curve

These are resources that help advanced modelling of complex geometries containing forms.
