Final answer:
The question involves demonstrating that a curve is unit speed and computing its Frenet-Serret basis. The unit speed condition is verified by the norm of the velocity vector being one. The Frenet-Serret basis comprises the tangent, normal, and binormal vectors to characterize the spatial geometry of the curve.
Step-by-step explanation:
The question pertains to the mathematical concept of curves in space, specifically regarding unit speed curves and the Frenet-Serret frame, which includes the tangent, normal, and binormal vectors. To show that a curve α(s) is unit speed, we must compute the derivative with respect to 's' (known as the speed), and show that its magnitude is constantly 1 for all 's' within the given domain. The Frenet-Serret basis consists of the tangent vector (T), the normal vector (N), and the binormal vector (B). These vectors describe the local geometry of the curve and are used to analyze its properties.
To calculate the Frenet-Serret basis, we use the formula: T = α'(s), N = T'(s)/|T'(s)|, and B = T × N. Though the specific expressions for α(s) given in the message are not mathematically coherent, the process would involve taking derivatives and computing the necessary magnitudes and cross products to find the Frenet-Serret basis if the actual function of α(s) were provided.