Page: Role of Curves in Geometric Modeling
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Curves are used to draw a wireframe model, which consists of points and curves, the curves are utilized to generate surfaces by performing parametric transformations on them. A curve can be as simple as a line as a complex as a B-spline.

Curves can be classified (based on forms) as follows:

1) Analytical Curves (Known form curves):

This type of curve can be represented by a simple mathematical equation, such as circle or ellipse. They have a fixed form and cannot be modified to achieve a shape that violates the mathematical equations.

2) Synthetic curves (Free form curves):

The analytical curves are insufficient to meet the requirements of mechanical parts that have complex curved shapes, such as propeller blades, aircraft fuselage, automobile body, etc.

Design of curved boundaries and surfaces require curve representations that can be manipulated by changing data points, which will create bends and sharp turns in the shape of curve, these curves are called synthetic curves and the data points are called vertices or control points. If the curve passes through all the data points, it is called an interpolant (interpolated). Smoothness of the curve is the most important requirement of a synthetic curve.

Various continuity requirements at the data points can be specified to impose various degrees of smoothness of the curve. A complex curve may consist of several curve segments joined together. Smoothness of the resulting curve is assured by imposing one of the continuity requirements. A zero order continuity ($C^o$) assumes a continuous curve, first order continuity ($C^1$) assures a continuous slope, and a second order continuity ($C^2$) assures a continuous curvature as a shown below.

A cubic polynomial is the lowest degree polynomial that can guarantee a $C^2$ curve. Higher order polynomials are not used in CAD, because they try to oscillate about the control points and require large data storage. Major CAD/CAM system provide three types of synthetic curves:

a) Hermite Cubic Spline

b) Bezier Curves

c) B-Spline Curves

Hermite Cubic spline curve pass through all the data points and therefore they can be called as interpolated curves. Bezier and B-spline curves do not pass through all the data points. Both the cubic spline and Bezier curve have first-order continuity, where as B-spline curves have a second-order continuity.

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