How Do You Calculate the Curvature K(t) of a Given Curve?

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In summary, the curvature K(t) of the curve r(t) = (-4sin(t)) i + (-4sin(t)) j + (5cos(t)) k can be found using the formula K(t) = |r'(t) x r"(t)| / |r'(t)|3. The cross product of r'(t) and r"(t) is [20cos(t)+20sin(t)]i - [20cos(t) + 20sin(t)]k +0j, which simplifies to 2. Therefore, the final formula for K(t) is 2 / [sqrt(16cos2(t) + 16
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shards5
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Homework Statement


Find the curvature K(t) of the curve r(t) = (-4sin(t)) i + (-4sin(t)) j + (5cos(t)) k.

Homework Equations


K(t) = |r'(t) x r"(t)| / |r'(t)|3

The Attempt at a Solution


r'(t) = (-4cos(t))i + (-4cos(t))j + (-5sin(t))k
r"(t) = (4sin(t))i + 4sin(t))j + (-5cos(t))k
|r'(t)| = sqrt(16cos2(t) + 16cos2(t) + 25sin2(t))
r'(t) x r"(t) = [20cos(t)+20sin(t)]i - [20cos(t) + 20sin(t)]k +0j
|r'(t) x r"(t)| = sqrt([20cos(t)+20sin(t)]^2 + [-20cos(t) - 20sin(t)]^2)
Answer should be:
sqrt([20cos(t)+20sin(t)]^2 + [-20cos(t) - 20sin(t)]^2)/[sqrt(16cos2(t) + 16cos2(t) + 25sin2(t))]3
But it isn't, so I am confused as to what I am doing wrong.
 
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  • #2
think you added a minus... though i didn't check the original cross product
r'(t) x r"(t) = [20cos(t)+20sin(t)]i - [20cos(t) + 20sin(t)]k +0j
so shouldn't it become
|r'(t) x r"(t)| = sqrt([20cos(t)+20sin(t)]^2 + [20cos(t) + 20sin(t)]^2)
|r'(t) x r"(t)| = sqrt(2[20cos(t)+20sin(t)]^2)
 
  • #3
I don't think that would be it since the cross product formula says that for the middle number it is subtraction. Besides, it is being squared, even if it is negative the answer which results will be positive.
 
  • #4
r(t) = (-4sin(t)) i + (-4sin(t)) j + (5cos(t)) k.
r'(t) = (-4cos(t)) i + (-4cos(t)) j + (-5sin(t)) k.
r''(t) = (4sin(t)) i + (4sin(t)) j + (-5cos(t)) k.

ok redoing the cross product you get
|r'(t) x r"(t)| = i(20c^2 + 20s^2) + j(-20s^2 - 20c^2) + k(-16cs +16cs)

which becomes
|r'(t) x r"(t)| = i(1) + j(-1) + k(0)
 

Related to How Do You Calculate the Curvature K(t) of a Given Curve?

1. What is curvature?

Curvature is a measure of the amount by which a curve deviates from being a straight line. It is defined as the rate of change of the tangent angle with respect to the arc length of the curve at a given point.

2. How is curvature calculated?

Curvature can be calculated using the formula K(t) = |dT(t)/ds|, where t is the parameter of the curve and s is the arc length. This formula essentially measures the rate of change of the unit tangent vector with respect to the arc length of the curve.

3. What does a positive or negative curvature value indicate?

A positive curvature value indicates that the curve is bending towards the direction of the unit normal vector, while a negative curvature value indicates that the curve is bending away from the direction of the unit normal vector.

4. What is the significance of curvature in science?

Curvature is an important concept in various scientific fields, such as mathematics, physics, and engineering. It is used to describe the shape of objects, the trajectory of moving particles, and the properties of surfaces, among other applications.

5. Are there any limitations to using the curvature K(t) calculation?

While the curvature K(t) calculation is a useful tool for analyzing curves, it has some limitations. It may not accurately reflect the curvature of curves with sharp corners or cusps, and it may be difficult to calculate for highly complex or irregular curves.

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