# Tangent Plane for a torus

• margaret23
In summary, the conversation involves finding an equation for the plane tangent to a torus at a specific point. The person is trying to solve for s and t in order to substitute them into the equations, but is getting two different answers for s. They are wondering what they are doing wrong and the solution involves using the values for t to solve for s.
margaret23
I know I am making a stupid mistake but I am not sure what it is...

Find an equation for the plane tangent to the torus X(s,t)=((5+2cost)coss, (5+2cost)sins, 2sint) at the point ((5-(3)^1/2)/(2)^1/2, (5-(3)^1/2/(2)^1/2, 1).

First I have to find what s and t are in order to sub them in for dT/ds and dT/dt. So first I solved for t using 2sint=1 and get t=pi/6. However, when I attempt to sub it into one of the other equations to solve for s I get 2 different answers for s when using each of the different equations. What am I doing wrong?

sin(t)= 1/2 means either t= $\pi/6$, in which case cos(t)= $\sqrt{3}/2$ or t= $5\pi/6$, in which case cos(t)= $-\sqrt{3}/2$. Obviously, since x= (5+ 2cos(t))cos(s)) and y= (5+ 2cos(t))sin(s), if x= y, as you have here, then sin(s)= cos(s)= $\pm1/\sqrt{2}$. Then 5+ 2cos(t)= 5-sqrt(3) so that cos(t)= $-\frac{\sqrt{3}}{2}$. From sin2(t)+ cos2(t)= 1 that gives immediately sin(t)= 1/2 as needed. The given point has t= $5\pi/6$, s= $\pi/4$.

## 1. What is a tangent plane for a torus?

A tangent plane for a torus is a flat surface that touches the torus at only one point. It is a geometric representation of the slope or direction of the torus at that point.

## 2. How is a tangent plane for a torus calculated?

A tangent plane for a torus can be calculated by finding the normal vector at the point of tangency and using it to create an equation for the plane. This involves using calculus and vector operations.

## 3. Why is the tangent plane for a torus important?

The tangent plane for a torus is important because it helps us understand the local behavior of the surface at a specific point. It can also be used to approximate the surface and make calculations for engineering and design purposes.

## 4. What is the relationship between the tangent plane and the normal vector for a torus?

The normal vector for a torus is perpendicular to the tangent plane at the point of tangency. This means that the normal vector is a crucial component in calculating the equation for the tangent plane.

## 5. Can a tangent plane for a torus intersect the torus at more than one point?

No, a tangent plane for a torus can only intersect the torus at one point. This is because the tangent plane represents the direction of the torus at a specific point, and if it were to intersect at multiple points, it would not be a true representation of the slope at that point.

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