Parametrizing a Hyperbola to Find Unit Tangent & Normal Vectors

In summary, we are given the hyperbola y^2-x^2=1 with y>0 and are asked to find a parameterization in vector form, R(t). We are also asked to calculate the unit tangent vector and the unit normal vector, as well as the curvature vector, all as functions of the parameter. The attempted method involved setting x=t and y= sqrrt(1+t^2), but this resulted in a complicated expression. It is suggested to try finding functions f(t) and g(t) such that f(t)^2-g(t)^2=1, similar to the approach of x^2+y^2=1. However, it is not clear if this problem is due soon.
  • #1
soccer*star
3
0
Consider the hyperbola y^2-x^2=1 (y>0)
a.) Find a parameterization for the curve and write it in vector form, R(t)
(b) Calculate the unit tangent vector as a function of the parameter.
(c) Calculate the unit normal vector and the curvature vector as a function of the parameter.
 
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  • #2


What did you try already??

You must find function f and g such that

[tex]f(t)^2-g(t)^2=1[/tex]
 
  • #3


I tried to set x=t and y= sqrrt(1+t^2) ...it comes out nasty and ugly, so ugly that i didn't even finish it..i'm not sure if there's a better way to do it.
 
  • #4


soccer*star, is this problem due tomorrow by any chance?
 
  • #5


If you would have [itex]x^2+y^2=1[/itex], then there's an easy choice:

[tex]\sin^2(t)+\cos^2(t)=1[/tex]

But now you have [itex]x^2-y^2=1[/itex]. Can you do something similar?
 

Related to Parametrizing a Hyperbola to Find Unit Tangent & Normal Vectors

1. What is a hyperbola?

A hyperbola is a type of conic section that is formed when a plane intersects a double cone at an angle. It has two branches that are symmetrical about the center and can be described using a set of parametric equations.

2. What does it mean to parametrize a hyperbola?

Parametrizing a hyperbola means expressing its coordinates in terms of one or more parameters, such as x = a cosh(t) and y = b sinh(t). This allows us to represent the hyperbola in a more simplified form and easily manipulate its properties.

3. How do I find the unit tangent vector of a hyperbola?

The unit tangent vector of a hyperbola can be found by taking the derivative of the parametric equations and then dividing it by its magnitude. This will give you a vector that is tangent to the curve at a specific point and has a magnitude of 1.

4. What is the normal vector of a hyperbola?

The normal vector of a hyperbola is a vector that is perpendicular to the tangent vector at a specific point on the curve. It can be found by taking the cross product of the unit tangent vector and the unit binormal vector, which is perpendicular to both the tangent and normal vectors.

5. How can I use parametrization to find the unit tangent and normal vectors of a hyperbola?

To find the unit tangent and normal vectors of a hyperbola, you can first parametrize the hyperbola using its parametric equations. Then, you can take the derivatives of these equations to find the tangent vector, and use the cross product to find the normal vector. Finally, you can normalize these vectors to get the unit tangent and normal vectors of the hyperbola.

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