Question

Compute the curvature k(t) of the clothoid r(t) = (x(t), y(t)), where

x(t)= ∫₀ᵗ sin (u¹³/) du, y(t) = ∫₀ᵗ cos (u¹³/13) du

(Use symbolic notation and fractions where needed.)
k(t)=

Answer 1

The curvature k(t) of the clothoid described by r(t) = (x(t), y(t)) is given by the expression k(t) = x'(t)y''(t) - y'(t)x''(t) / (x'(t)^2 + y'(t)^2)^(3/2), where x'(t) and y'(t) denote the first derivatives of x(t) and y(t) with respect to t, and x''(t) and y''(t) denote the second derivatives.

Step-by-step explanation:

To compute the curvature k(t) of the given clothoid, we need to find the first and second derivatives of x(t) and y(t) with respect to t. Starting with the given expressions for x(t) and y(t):

x(t) = ∫₀ᵗ sin(u¹³) du,

y(t) = ∫₀ᵗ cos(u¹³/13) du.

After finding x'(t) and y'(t), calculate x''(t) and y''(t). Finally, substitute these derivatives into the curvature formula:

k(t) = x'(t)y''(t) - y'(t)x''(t) / (x'(t)^2 + y'(t)^2)^(3/2).

This curvature formula provides a measure of how sharply the clothoid curves at any given point t. It involves the rates of change of x(t) and y(t), offering insights into the geometric properties of the clothoid. Calculating curvature is a common task in differential geometry, aiding in the analysis of curves and their behavior. The use of subscript/superscript style enhances the clarity and readability of mathematical expressions, ensuring accurate representation of the involved derivatives and exponents.