Scott Harris
Cofounder
SolidWorks Corp.

Edited by Leslie Gordon

SolidWorks Corp.
Concord, Mass.
SolidWorks.com
In brief

• A tangent line (first derivitive) is useful for seeing if a curve is level.
• The second derivitive shows the precise shape of a curve.
• The third derivitive describes the rate of change of a curve.

Edited by Leslie Gordon
leslie.gordon@penton.com

The creation of attractive products typically involves modeling beautiful, aesthetically pleasing shapes on a computer. Unfortunately, too many CAD programs speak in terms of mathematics rather than the language of artists. Understanding mathematical concepts and how they apply to CAD can help you design products with a visual appeal that, although difficult to describe, evokes an emotional response that is impossible to deny.

The most important aspect of any design geometry affecting how “good” something looks is its surfaces. Smoothness, visual lines, and the reflective properties of a shape’s exterior, as well as how various surfaces connect to each other contribute to the aesthetic quality of the object’s form.

The creation of “good-looking” surfaces requires the application of curves. Think back to your college calculus classes. Recall that taking the derivatives at points along a curve tells a lot about the shape of the curves. Derivative properties are often referred to as continuity.

First, second, and third derivatives of curves
The first derivative of a curve is the tangent. Think of it as a straight line drawn along the curve that tells the direction of the curve at that point. The first derivative is useful for seeing if the curve is level or matches another curve’s direction at a particular point. The curve is rippled when the tangent direction oscillates as you move along it.

The second derivative provides the curvature at any given point, useful for knowing the precise shape of the curve. Where curvature changes from positive to negative or concave to convex there is an inflection. An example is the middle of an “S.” Imagine curvature represented by lines radiating straight out from a curve. The lines change direction at the inflection point. Also, the longer the line, the higher the curvature. Curvature and inflections are key factors in determining the aesthetics of a surface because they influence how light is reflected to the eye.

The third derivative describes the acceleration or the rate of change of curvature along the curve. Even attributes as subtle as how smooth the changes are in the second or third derivative can have an effect on the curve and overall aesthetics. Smooth changes are more desirable as design attributes. For example, most automobile companies avoid abrupt curvature changes because they appear as subtle yet generally ugly features.

Derivatives of surfaces
The same concepts that apply to curves apply to surfaces. Surface shape is characterized by size and position, as well as the way the surface flows, the rate at which its curvature changes, and the manner in which surfaces connect to each other. Think of a square surface with lines drawn across from boundary to corresponding points on the opposite boundary, similar to a tic-tac-toe game. The lines are called isoparametric curves.

Surfaces can be evaluated in several ways but the simplest is looking at the derivatives along the isoparametric curves. Recall the first derivative tells the direction of the tangent at any given point. Imagine placing a ruler on the surface parallel to one of the isoparametric curves. The ruler sits in the direction of the surface tangent at the point of contact. This information is useful for determining if the surface is level at that point.

The second derivative is also commonly used because it shows the degree of curvature at a particular point in relation to the direction of the isoparametric curve. Looking at the second derivatives along a surface tells where the surface is flat and where it is curvy. Again, when the second derivative changes from positive to negative, or vice versa, there is an inflection. Simply put, the surface changes from concave to convex.

These attributes greatly affect how light reflects from surfaces and the overall attractiveness of a design. In fact, automotive designers, for example, spend a lot of time working out curvatures and inflections of car bodies to get a desired aesthetic. They might, for instance, evaluate the transition of second derivatives and inspect third derivatives to help detect uneven transitions, which the human eye generally finds less attractive.

Geometric continuity
Another important design element involves how surfaces meet at a common edge. A helpful method uses the surface normal tool, a vector that starts at a point on a surface and moves directly away or “normal” to the surface. Comparing surface normals on points along a common edge makes it easy to see how well two surfaces match. How surfaces meet is mathematically defined as Gn continuity or geometric continuity, with n being the measure of smoothness.

For example, consider a box. The way the faces meet at sharp corners is classified as G0. Surfaces that connect smoothly with a filleted or rounded edge are called G1. Here, surface normals along the common edge are parallel. These surfaces are tangent and they produce nice-looking shapes, such as those used for cast and injection-molded products. But because the human eye can easily detect a nonsmooth curvature change, designers often blend surfaces together so they match curvature along the common edge. Mathematicians call this G2. Along with even higher-derivative matching, G2 is used in automotive and product design to produce continuous, smooth, flowing, and aesthetically pleasing shapes.

Other common terms are C0, C1, and C2, which are called first, second, and third-level parametric continuities, respectively. (For more information on the details of continuity, visit http://en.wikipedia.org/wiki/NURBS.)

The human eye is generally sensitive to how light reflects from objects. Well-designed shapes just look right, whereas objects with curve and surface-continuity problems can look odd. It can be difficult to locate the source of such problems, but certain tools help. One method maps surface curvature to a color. Distinct lines show where there are abrupt changes in curvature. Another analysis tool called Zebra Stripes shows how light will reflect from the surface in an exaggerated manner.

Industrial designers spend a lot of time and energy making reflection lines look good so that consumers will find the final product aesthetically pleasing. An understanding of these mathematical concepts and how they apply to CAD can help designers master the art of creating good looking products. The next time you see a beautiful car with great lines, remember that someone designed and analyzed every curve and surface for overall form and how the form reflects various kinds of light at different angles.

Isoparametric lines on a surface go from one boundary to the corresponding point on the opposite boundary.

A surface with a nonsmooth curvature transition has harsher reflection lines than does a smooth surface (top).

The abrupt change in curvature (left) has noncontinuous 2nd derivatives, while the smooth change in curvature (right) has continuous second derivatives.

The first derivative of a curve is a tangent vector, a straight line showing the direction of the curve at a certain point (above). A rippled curve has tangents that oscillate in direction (right).

Two surfaces meet with G0 corner (top), G1 tangent (left), and G2 curvature-matching (right) continuity.

The Zebra Stripes map shows how light will reflect from the surface in an exaggerated manner.

The color map (inset) shows a distinct line where there is an abrupt change in curvature. The map (right) shows smooth curvature change as it goes from low curvature towards the bend apex where there is high curvature.