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# Geometry for Modeling and Design

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## 4.14 Spline Curves

Splines are used to describe complex, or freeform, curves. Many surfaces cannot be easily defined using simple curves such as circles, arcs, or ellipses. For example, the flowing curves used in automobile design blend many different curves into a smooth surface. Creating lifelike shapes and aerodynamic forms may require spline curves (Figure 4.40).

The word spline originally described a flexible piece of plastic or rubber used to draw irregular curves between points. Mathematical methods generate the points on the curve for CAD applications.

One way to create an irregular curve is to draw curves between each set of points. The points and the tangencies at each point are used in a polynomial equation that determines the shape of the curve. This type of curve is useful in the design of a ship’s hull or an aircraft wing. Because this kind of irregular curve passes through all the points used to define the curve, it is sometimes called an interpolated spline or a cubic spline. An example and its vertices are shown in Figure 4.41.

Other spline curves are approximated: they are defined by a set of vertices. The resulting curve does not pass through all the vertices. Instead, the vertices “pull” the curve in the direction of the vertex. Complex curves can be created with relatively few vertices using approximation methods. Figure 4.42 shows a 3D approximated spline curve and its vertices.

The mathematical definition for this type of spline curve uses the X- and Y- (and Z- for a 3D shape) coordinates and a parameter, generally referred to as u. A polynomial equation is used to generate functions in u for each point used to specify the curve. The resulting functions are then blended to generate a curve that is influenced by each point specified but not necessarily coincident with any of them.

Splines are drawn in CAD systems based on the mathematical relationships defining their geometry. Figure 4.46 shows an approximated spline drawn using AutoCAD. Figure 4.47 shows an interpolated spline drawn using SolidWorks. Both curves are drawn with a spline command, and both provide a dialog box that allows you to change properties defining the curve; however, the properties that are controlled vary by the type of spline being created by the software package. You should be familiar with the terms used by your modeling software for creating different types of spline curves.