1. Curvature & Radius of Curvature

Find the curvature and radius of curvature.
This question involves encapsulation.

y = 3x - 2 given x = a

r (t) = < t, y, 0 >

r (t) = < t, 3t - 2, 0 >

r (t) = (t)i + (3t - 2) j

r'(t) = i + 3 j

||r'(t)|| = root {10}

T(t) = (i + 3 j)/[root {10}]

I am stuck here.

2. Re: Curvature & Radius of Curvature

do you know what the formula for curvature of a space curve is?

3. Re: Curvature & Radius of Curvature

Originally Posted by romsek
do you know what the formula for curvature of a space curve is?
There are two formulas:

1. ||T'(t)||/||r'(t)||

2. ||r'(t)Xr"(t)||/||r'(t)||^3

I find that the second formula works best for polynomials. How do I rewrite the given equation as a vector r (t)? What are the steps? When do I plug x = a? Here "a" is constant, right?

4. Re: Curvature & Radius of Curvature

y = 3x - 2 given x = a

r (t) = < t, y, 0 >

r (t) = < t, 3t - 2, 0 >

r (t) = (t)i + (3t - 2) j

r'(t) = i + 3 j

r'(a) = 0

||r'(a)|| = root (0^2) = 0

T (t) = r'(a)/||r'(a)||

T(t) = 0/0 = 0

T'(t) = 0

||T'(t)|| = root (0) = 0

K = ||T'(t)||/||r'(t)||

K = 0/0

K = 0

Radius = undefined when k = 0.

Is this correct?

5. Re: Curvature & Radius of Curvature

$r(t) = (t, 3t-2,0)$

$r^\prime(t) = (1, 3, 0)$

$\| r^\prime(t) \| = \sqrt{10}$

$T(t) = \dfrac{r^\prime(t)}{\|r^\prime(t)\|} = \dfrac{(1,3,0)}{\sqrt{10}}$

$T^\prime(t) = (0,0,0)$

$\kappa=0$

as expected. A straight line has no curvature. And yes, the radius is infinite as you'd expect for a straight line.

you shouldn't plug $a$ until the very end and as seen in this example the curvature is constant.

6. Re: Curvature & Radius of Curvature

Originally Posted by romsek

$r(t) = (t, 3t-2,0)$

$r^\prime(t) = (1, 3, 0)$

$\| r^\prime(t) \| = \sqrt{10}$

$T(t) = \dfrac{r^\prime(t)}{\|r^\prime(t)\|} = \dfrac{(1,3,0)}{\sqrt{10}}$

$T^\prime(t) = (0,0,0)$

$\kappa=0$

as expected. A straight line has no curvature. And yes, the radius is infinite as you'd expect for a straight line.

you shouldn't plug $a$ until the very end and as seen in this example the curvature is constant.
We can also say that the T'(t) =(0,0,0) = zero vector, right?

If so, then the magnitude of T'(t) = 0.

K = 0/root{10}

K = 0

Because K = 0, the radius of curvature R = 1/K is undefined.

7. Re: Curvature & Radius of Curvature

Originally Posted by USNAVY
We can also say that the T'(t) =(0,0,0) = zero vector, right?

If so, then the magnitude of T'(t) = 0.

K = 0/root{10}

K = 0

Because K = 0, the radius of curvature R = 1/K is undefined.
yes, I didn't state the completely obvious. Apologies.

Very good.

9. Re: Curvature & Radius of Curvature

Did you not realize that this is a straight line so the curvature is 0 without having to do any computation?

10. Re: Curvature & Radius of Curvature

Originally Posted by HallsofIvy
Did you not realize that this is a straight line so the curvature is 0 without having to do any computation?
No.