Equations for normal, osculating, and rectifying planes

In summary, we are being asked to find the equations for the normal, osculating, and rectifying planes for the given equation at t=pi/4. We have already found the tangent vector, T(pi/4), and the normal vector, N(pi/4), but we are unsure of how to find the equations for these planes. Using the Frenet-Serret formulas, we can easily obtain the equations of these planes since we have all three vectors and a point on the planes (the point at r(pi/4)).
  • #1
cue928
130
0
We are being asked to find the eq's for normal, osculating, and rectifying planes for the following equation:
r(t) (cos t)i + (sin t)j - k @ t=pi/4
I have already found the following:
T(pi/4) = (-√2/2)i + (√2/2)j = 0k

N(pi/4) = (-√2/2)i + (-√2/2)j + 0k

B = 0i + 0j + k

But, I don't know where to start for those three equations.
 
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  • #2
cue928 said:
We are being asked to find the eq's for normal, osculating, and rectifying planes for the following equation:
r(t) (cos t)i + (sin t)j - k @ t=pi/4
Should be r(t) = ...
cue928 said:
I have already found the following:
T(pi/4) = (-√2/2)i + (√2/2)j = 0k
Should be T(pi/4) = (-√2/2)i + (√2/2)j + 0k

Note that I didn't check your math, just obvious typos.

cue928 said:
N(pi/4) = (-√2/2)i + (-√2/2)j + 0k

B = 0i + 0j + k

But, I don't know where to start for those three equations.
Try this link to a wikipedia page that discusses these planes- http://en.wikipedia.org/wiki/Frenet–Serret_formulas. You have all three vectors, so getting an equation of any of the three planes should be easy, since you have a normal to the plane and can easily get the point that's on all three planes (the point at r(pi/4)).
 

FAQ: Equations for normal, osculating, and rectifying planes

1. What is the purpose of having equations for normal, osculating, and rectifying planes?

The equations for normal, osculating, and rectifying planes are used to describe the curvature and orientation of a curve in three-dimensional space. They are important in fields such as differential geometry and physics, where the study of smooth curves is essential.

2. How are the equations for normal, osculating, and rectifying planes derived?

The equations for these planes are derived from the parametric equations of a curve, using derivatives to determine the curvature and orientation at a specific point on the curve.

3. What is the difference between the normal, osculating, and rectifying planes?

The normal plane is perpendicular to the tangent line at a point on the curve and contains the normal vector. The osculating plane is a unique plane that touches the curve at a point and has the same curvature as the curve at that point. The rectifying plane is defined as the plane that contains the tangent line and the normal vector at a point on the curve.

4. Can these equations be used for any type of curve?

Yes, the equations for normal, osculating, and rectifying planes can be used for any type of smooth curve in three-dimensional space.

5. How are these equations applied in real-world situations?

These equations have various applications in fields such as engineering, computer graphics, and robotics. They are used to describe the motion and behavior of physical objects or systems that follow a smooth curve.

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