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rabbed
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(cos(s), sin(s)) gives an arc-length parameterization of the unit circle so that the speed is constantly 1, but the second derivative doesn't give zero acceleration which should be the case with constant speed?
Arc length parameterization is a way of representing points on a curve, in this case the unit circle, using a single variable (s) to represent the distance along the curve from a starting point. This allows for easier calculations and better visualization of the curve.
The arc length parameterization for the unit circle is expressed as (cos(s), sin(s)), where s represents the distance along the circle from the starting point (1, 0) in a counter-clockwise direction.
The range of values for the parameter s in arc length parameterization for the unit circle is 0 ≤ s ≤ 2π, which represents one full revolution around the circle.
Arc length can be calculated using the formula: L = ∫√(cos²(s) + sin²(s)) ds, where s represents the distance along the curve and L represents the total arc length.
Arc length parameterization allows for easier calculations and a more intuitive understanding of curves. It also allows for consistent measurements of distance along the curve, regardless of the curve's shape or orientation. Additionally, it is useful for applications such as computer graphics and animation.