Parametric equation of the intersection between surfaces

In summary, to find a parametric equation of the intersection between surfaces S and T, change the equation of the intersection to x^2 + 2y^2 = 1 and use the "standard" parameterization of an ellipse: x = cos(t), y = sin(t)/√2. This yields z = cos^2(t) + (1/2)sin^2(t) or z = 1 - (1/2)sin^2(t), and t = 0..2π. This results in a single curve, an ellipse projected onto the xy-plane.
  • #1
BilalX
7
0
[SOLVED] Parametric equation of the intersection between surfaces

Homework Statement



Given the following surfaces:
S: z = x^2 + y^2
T: z = 1 - y^2

Find a parametric equation of the curve representing the intersection of S and T.

Homework Equations



N/A

The Attempt at a Solution



The intersection will be:
x^2 + y^2 = 1 - y^2
x = (1 - 2y^2)^0.5

At this point, I plug in the following parametrization:
y = sin(t)

Which yields:

x = (1 - 2(sin(t))^2)^0.5
y = sin(t)
z = 1-(sin(t))^2 (from the equation for T)

with t = 0..2*Pi.

Judging from a Maple plot this seems to make sense; the curve is a projected ellipse, but due to the x term I have to split it into two separate segments. I'm pretty sure I should be able to use a more elegant solution with a single curve, but I haven't been able to figure it out - any help would be welcome.

Thanks-
 
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  • #2
In a situation like that it is better not to solve for one of the variables.

Instead, change x2+ y2= 1- y2 to x2+ 2y2= 1, the equation of an ellipse. Then use the "standard" parameterization of an ellipse: x= cos(t), y= sin(t)/[itex]\sqrt{2}[/itex]. Then, of course, you can have either [itex]z= cos^2(t)+ (1/2)sin^2(t)[/itex] or [itex]z= 1- (1/2)sin^2(t)[/itex].
 
  • #3
Great, thank you.
 

Related to Parametric equation of the intersection between surfaces

1. What is a parametric equation of the intersection between surfaces?

A parametric equation of the intersection between surfaces is a set of equations that describe the coordinates of points where two or more surfaces intersect in three-dimensional space. These equations use parameters (usually denoted by u and v) to represent the coordinates of the points on the surfaces and can be used to determine the points of intersection between the surfaces.

2. How is a parametric equation of the intersection between surfaces different from a regular equation?

A regular equation typically describes a relationship between x and y coordinates on a two-dimensional plane. A parametric equation of the intersection between surfaces, on the other hand, describes a relationship between x, y, and z coordinates in three-dimensional space. Additionally, parametric equations use parameters to represent the coordinates, while regular equations do not.

3. What information is needed to create a parametric equation of the intersection between surfaces?

To create a parametric equation of the intersection between surfaces, you will need to know the equations of the individual surfaces and the points where they intersect. These points can then be used to determine the parameters and create the parametric equations.

4. Can a parametric equation of the intersection between surfaces be used to solve real-world problems?

Yes, parametric equations of the intersection between surfaces can be used to solve real-world problems in various fields such as engineering, physics, and computer graphics. They can be used to determine the points of intersection between objects or to create complex shapes and curves.

5. Are there any limitations to using parametric equations of the intersection between surfaces?

While parametric equations of the intersection between surfaces can be useful in solving problems, there are some limitations. These equations may not be able to accurately describe the intersections of highly complex or irregular surfaces, and they may require a significant amount of computation to solve. Additionally, they may not be suitable for all applications, and other methods may be more appropriate.

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