Arc Length Parameterization for Unit Circle: Cos(s) & Sin(s)

In summary, the conversation discusses the arc-length parameterization of the unit circle, where the speed is constantly 1 but the second derivative does not give zero acceleration. It is noted that acceleration is defined as the rate of change of velocity per unit time and since the velocity vector is changing direction, there will be non-zero acceleration. It is also mentioned that a vector which is a function of t and has unit length for all t may not always be perpendicular to its integrated vector, but its derivative will be perpendicular.
  • #1
rabbed
243
3
(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?
 
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  • #2
THe speed is constant but you are constantly turning too right?

Acceleration is defined as the rate of change of velocity per unit time and since the velocity vector is changing direction then there's an acceleration that is not zero.
 
  • #3
Ah, right.. tricky
Thanks
 
  • #4
Does this mean that a vector which is a function of t and that vector has unit length for all t, will always be perpendicular to its integrated vector or can you only say that it's derivative will be perpendicular?
 

1. What is arc length parameterization for the unit circle?

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.

2. How is the arc length parameterization for the unit circle expressed using cos(s) and sin(s)?

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.

3. What is the range of values for the parameter s in arc length parameterization for the unit circle?

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.

4. How is arc length calculated using the arc length parameterization for the unit 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.

5. What are the advantages of using arc length parameterization for the unit circle?

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.

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