How Do You Solve Complex Parametrization Problems in Mathematics?

[LostInSpace]

Hi all! I'm having some problems with parametrization. I read somewhere that you should locate circles, ellipses, hyperboloides, paraboloides etc and use these elements to express a parametric function.

But someone must have figured out how to do it! The way I see it, there's nothing logical about
<br /> x^2+y^2-z^2=1 \Leftrightarrow f(t,\varphi)=\left(\sqrt{t^2+1}\cos\varphi,\sqrt{t^2+1}\sin\varphi,t\right)<br />

Yes, I understand that
<br /> (\sqrt{t^2+1}\cos\varphi)^2 + (\sqrt{t^2+1}\sin\varphi)^2 - t^2 = 1<br />
but how do you actually get to that conclusion (without plotting the function)?

Consider this problem (which may be simple):
Determine the intersecting curve (parametric function) between the surfaces z^2=x^2+y^2 and z=x+y. How do you approach that?

Thanks in advance!

 

[HallsofIvy]

I have no idea what is meant by "you should locate circles, ellipses, hyperboloides, paraboloides etc and use these elements to express a parametric function." Certainly standard figures such as circles,ellipses, etc. (hyperboloids and paraboloids are surfaces, not curves) and have standard parameterizations but if your curve, surface, or function doesn't happen to be one, it doesn't help.

It does help to think about some "standard" parameterizations and perhaps try to adapt them.

First, you do understand that there exist an infinite number of different parameterizations for any curve, don't you?

If I were trying to find a parametrization for x^2+y^2-z^2=1, then first thing I would notice is that x^2+y^2 and think sin^2(\theta)+cos^2(\theta)= 1 (or some people might think "circle!"- it's the same thing.) I see immediately that if I let x= cos(&theta;) and y= sin(&theta;), I will have x²+ y²= 1. Now I have to take care of that "-z²" term. It would be nice if I could just set it equal to 0 but I can't!. I might think: If instead I set x= R cos(&theta;), y= R sin(&theta;) (Yes, I confess: I wrote "R" instead of "A", for example, because I really thinking "circle"!). Now x²+y²= R² and I have R²- z²= 1. Okay, how about if I take R²= 1+ z²? That is: set R= &radic;(1+ z²)? It's not illegal to use z itself as a parameter but, since "t" is more common, let z= t and the parametrization becomes

x= &radic;(1+t²)cos(&theta;)
y= &radic;(1+t²)sin(&theta;)
z= t.


Finding the intersection of z²= x²+ y² and z= x+ y is, in fact, relatively straightforward (in theory at least). Saying that (x,y,z) is on the intersection means that the same values for x,y,z satisfy both equations or that
z= x²+y²= x+ y.

The intersection of two surfaces (2 dimensional) is a curve (1 dimensional) so I would expect this to depend on a single variable.

One thing I could do is think of x²+y²= x+ y as a quadratic equation for x:x²- x+(y²- y)= 0 and use the quadratic formula to find x as a function of y. Use y itself as the parameter (or write y= t) so the solution is a parametric equation for x. We could use z= x+y to then get an equation for z. The only problem with that is that "+/-" in the quadratic formula.

Actually, you should be able to look at x²+ y²= x+ y and immediately think "circle!" (Your first sentence is making more and more sense, at least for these problems!). A standard way of rewriting an equation for a circle is to complete the square (remember way back when you first learned to do that?). Write this as
x²-x + y²- y= 0 and think "In order to make both x and y terms perfect squares, I have to add 1/4 to each term". That would of course, add 1/2 to the right:
x²-x+ 1/4 + y²- y+ 1/4= 1/2 or
(x- 1/2)²+ (y- 1/2)²= 1/2.

That's a circle (in the x,y plane) with center at (1/2, 1/2) and radius &radic;(1/2). More importantly I see that if I set
x= cos(&theta;)+ 1/2 and y= sin(&theta;)+ 1/2, then
(x- 1/2)²+ (y- 1/2)²= cos²(&theta;)+sin²(&theta;)= 1.

To account for the "1/2" on the right side, just multiply each by 1/&radic;(2):

x= (1/&radic;(2))(cos(&theta;)+ 1/2)
y= (1/&radic;(2))(sin(&theta;)+ 1/2) and
z= x+ y= (1/&radic;(2)(cos(&theta;)+ sin(&theta;)+ 1).

 

[M98Ranger]

Parametrization can definitely be a challenging concept to grasp at first, but with some practice and understanding, it can become a useful tool in solving problems involving curves and surfaces. In response to your concerns, I would like to offer some clarification and tips on how to approach parametrization.

Firstly, it is important to understand that parametrization is not a one-size-fits-all solution. Different types of curves and surfaces may require different approaches to parametrization. For example, circles and ellipses can be easily parametrized using trigonometric functions, while hyperboloids and paraboloids may require more complex expressions.

In the case of the equation x^2+y^2-z^2=1, the key is to recognize the familiar form of a circle in the x-y plane, which can be expressed as x^2+y^2=r^2. This allows us to use the familiar parametric equations for a circle, x=r\cos\theta and y=r\sin\theta, and substitute in the expression for r, which in this case is \sqrt{t^2+1}. The z component is simply t, as it can take on any value along the z-axis. This approach may not be immediately apparent, but with practice and familiarity with different types of curves, it becomes easier to identify the appropriate parametric equations.

As for the intersecting curve problem, it is important to first visualize the two surfaces and understand how they intersect. In this case, the surfaces intersect along a line, meaning the resulting parametric function will be a straight line passing through the point of intersection. To find the parametric equations, we can set one variable (in this case, z) equal to the other (x+y) and solve for the remaining variables. This results in the parametric equations x=t and y=t. The z component is simply the expression we set equal to x+y, which is t.

In summary, parametrization may not always be a straightforward process, but with practice and understanding of different types of curves and surfaces, it becomes easier to identify appropriate parametric equations. It is also important to visualize the problem and understand how the curves or surfaces intersect in order to determine the appropriate approach. Hopefully these tips will help you in your future parametrization problems. Good luck!