There is no hard and fast rule. It is simply a matter of convention.Master J said:In vector calculus, with a space curve C, there are the 3 vectors, tangent, normal, and binormal.
Are they always considered in their UNIT form, ie., divided by their length?
The TNB frame, also known as the tangent-normal-binormal frame, is a set of three vectors that describe a local coordinate system for a curve or surface in three-dimensional space. It is commonly used in vector calculus to analyze the behavior of a vector or function along a curve or surface.
The tangent vector, denoted as T, is a unit vector that lies along the direction of the curve or surface at a specific point. The normal vector, denoted as N, is a unit vector that is perpendicular to the tangent vector and points towards the center of curvature. The binormal vector, denoted as B, is a unit vector that is perpendicular to both the tangent and normal vectors and points in the direction of increasing curvature.
The TNB frame allows us to analyze the behavior of a vector or function along a curve or surface in a more intuitive and geometric way. It also helps us to understand the curvature and orientation of the curve or surface at a specific point, which is important in many applications such as physics, engineering, and computer graphics.
The tangent vector can be calculated by taking the derivative of the curve or surface with respect to the parameter that defines its path. The normal vector can be calculated by taking the derivative of the tangent vector with respect to the parameter. The binormal vector can then be calculated by taking the cross product of the tangent and normal vectors.
Yes, the TNB frame can be extended to higher dimensions by adding additional perpendicular vectors. For example, in four-dimensional space, we would have a T, N, B, and another vector that is perpendicular to all three, known as the normal binormal vector. This allows us to analyze the behavior of a vector or function along a higher-dimensional surface.