Why is a closed curve with a domain in 3D not always simply connected?

  • Thread starter daku420
  • Start date
  • Tags
    Curves
In summary, the conversation discusses the concept of simple connectivity in different types of domains, specifically in the cases of R^2 and R^3. While R^2 is simply connected, R^3 can be more complex due to domains with certain restrictions, such as when x=0 and y=0. The conversation then suggests visualizing a closed curve in R^3 with a removed line, and questioning how it can be contracted to a point without touching the line. This serves as a guide for understanding the concept of simple connectivity in R^3 domains.
  • #1
daku420
3
0
i know if every simple closed curve in D can be contracted to a point it is simply connected as in the case of|R^2 Domain or R^3 it is simply connected

but i am not feeling uncomfortable with |R^3 especially with the domains like when x =! 0 and y =! 0 why it is not simply connected?

how do I should visualize a closed curve with the domain in 3D?
 
Physics news on Phys.org
  • #2
So R^3 is just the regular three-dimensional space we're all used to. If your domain is {(x,y,z) : x != 0 and y != 0}, then what does it look like? It's R^3, but with a line taken out of it--because the points that we took out look like (0,0,z), for whatever value of z. In fact we've removed the z-axis, because that's exactly where x=y=0.

Now imagine a circle that goes around that line. The line is infinite in both directions. How can you shrink that circle to a point, while never touching the line?

This is not a proof of course, but hopefully it can guide your intuition.
 

1. What is "Simply connecting curves"?

"Simply connecting curves" is a mathematical concept that involves connecting two or more curves in a way that creates a single continuous curve without any intersections or self-intersections. It is often used in geometry and topology to study the properties of curves and surfaces.

2. How is "Simply connecting curves" different from regular curve connections?

Regular curve connections allow for intersections and self-intersections, while "Simply connecting curves" requires a single, continuous curve without any intersections. This means that the curves must be carefully connected or modified to meet this requirement.

3. What are some real-world applications of "Simply connecting curves"?

"Simply connecting curves" has a variety of applications in different fields. In computer graphics and animation, it is used to create smooth and continuous lines and shapes. In architecture and engineering, it is used to design structures with curved surfaces. It is also used in physics and biology to study the movement and behavior of particles and organisms.

4. What are some common challenges when dealing with "Simply connecting curves"?

One of the main challenges with "Simply connecting curves" is ensuring that the curves are connected in a way that meets the requirement of being single and continuous. This often requires careful planning and adjustments to the curves. Another challenge is dealing with curves that are complex or have many segments, which can make it difficult to determine the best way to connect them.

5. Are there any techniques or tools available to help with "Simply connecting curves"?

Yes, there are various techniques and tools that can assist with "Simply connecting curves". One common technique is using Bézier curves, which allow for smooth and precise connections between curves. There are also software programs specifically designed for this purpose, such as vector graphics software and CAD programs, which have features for connecting and modifying curves.

Similar threads

Replies
2
Views
2K
Replies
3
Views
219
Replies
20
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
118
Replies
3
Views
939
Replies
1
Views
1K
Replies
2
Views
985
Replies
4
Views
193
Back
Top