Thread: Osculating Circle for a Curve

1. Osculating Circle for a Curve

I lost trying to find the osculating circle for the curve defined by r(t) = <sint, cost, -pi + t> at the point t=pi.

Ive calculated the radius of the circle to be 1 and I know I need to somehow use the unit normal vector at point t=pi, which is <0, 1/sqrt(2), 0> to solve for the osculating circle's centre, but I do not know how to do that.

I lost trying to find the osculating circle for the curve defined by r(t) = <sint, cost, -pi + t> at the point t=pi.

Ive calculated the radius of the circle to be 1 and I know I need to somehow use the unit normal vector at point t=pi, which is <0, 1/sqrt(2), 0> to solve for the osculating circle's centre, but I do not know how to do that.

That is not a unit vector.

CB

3. Right...i calculated that wrong...

The unit normal vector is actually [0, -1, 0] from the point where t=pi.

How can I use this to calculate the center of the osculating circle if possible?

I lost trying to find the osculating circle for the curve defined by r(t) = <sint, cost, -pi + t> at the point t=pi.

Ive calculated the radius of the circle to be 1 and I know I need to somehow use the unit normal vector at point t=pi, which is <0, 1/sqrt(2), 0> to solve for the osculating circle's centre, but I do not know how to do that.

1. The stationary vector of point P when $\displaystyle t = \pi$ is

$\displaystyle \vec p=(0,-1,0)$.

2. I've got the normal unit vector as $\displaystyle \vec n = (0,1,0)$

3. The sum of these two vectors will yield the stationary vector of the center of the circle:

$\displaystyle \vec p + \vec n = (0,0,0)$

4. The tangent at the curve in P and the normal vector produce a plain in which the osculating circle must be placed. The equation of the tangent is:

$\displaystyle t: \overrightarrow{r(t)}=(0,-1,0)+t \cdot (-1,0,1)$

5. In addition to my previous post I've attached a drawing containing

the curve: $\displaystyle \overrightarrow{r(t)}=(\sin(t), \cos(t), -\pi+t)$

the tangent: $\displaystyle t: \overrightarrow{r(t)}=(0,-1,0)+t \cdot (-1,0,1)$

the circle: $\displaystyle \overrightarrow{r(t)}=(\sin(t), \cos(t), -\sin(t))$