![]() Using the result for and integrating ( 2) with initial condition then yields the profile of the space curve. ![]() Provided and the initial conditions, , and for the Frenet triad, we can integrate ( 9) to determine the evolution of the triad as it moves along the curve. The vector, which has the equivalent representations ( 11) and ( 13), is often called the Darboux vector. Where is a rotation tensor whose axial vector Noting that the Frenet triad is a right-handed basis, the compact form ( 10) is a statement of the fact that, locally, the triad can be viewed as rigidly rotating along the curve. We can write these relations using the compact notation These relations are obtained by using the definitions ( 3) and ( 6) and by differentiating : The Serret-Frenet relations are compact expressions of the rate of change of the Frenet triad basis vectors expressed in the basis. Note the conventional division by in this definition. The most popular approach is to use the geometric torsion obtained from the Frenet triad to define the total torsion of a space curve, :įor a curve segment of length. Finally, we mention that there are several methods of defining the twist of a space curve. Notice that it is possible to define the curvature and geometric torsion without referring explicitly to the Frenet triad: A space curve is said to be right-handed if and left-handed if (see ). In this definition, the minus sign is a convention. Using the fact that the Frenet triad is orthonormal, we can define the geometric torsion of the space curve,, by the relation For a curve whose point locations also vary with time, i.e., when, we can still calculate the Frenet triad, but we must do so at each instant in time. Where is referred to as the unit binormal vector. We then use the unit tangent and normal vectors to construct an orthonormal and right-handed triad known as the Frenet triad: The unit vector that is tangent to the curve is given byĪnd the derivative of this vector defines the curvature and the unit normal vector : Therefore, we can express the position vector of any point on the curve as follows: Evolution of the Frenet triad for a general space curve. We assume the curve is parameterized by an arc-length parameter such that any point along the curve can be located relative to a fixed origin using a set of Cartesian coordinates with corresponding orthonormal, right-handed basis by specifying the value of at the point of interest.įigure 1. The Frenet triad, curvature, and torsionĬonsider a space curve in Euclidean three-dimensional space, as illustrated in Figure 1. The primary resource for the content we present here is the remarkable text by Kreyszig our discussion of the forthcoming examples of space curves is adapted from the textbook. Their work forms the basis for most treatments of space curves in modern texts on differential geometry (e.g., ). The notion of curvature of a plane curve was extended to a space curve by Serret in 1851 and by Frenet in 1852 (see ). The geometric features of space curves, or curves that are embedded in three-dimensional space, are a classic topic in differential geometry.
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