# What is the parametric equation for a helix on a vertical, circular cylinder?

• dlacombe13
In summary, the given parametric equations x = cos t, y = sin t, z = 1/(1+t^2) represent a vertical, circular cylinder in the xy-plane with a counter-clockwise helix wrapping around it. The graph can be visualized as an infinitely long, infinitely thin slinky spring centered at (0,0,1) and approaching but never touching the plane z=0 as t increases.
dlacombe13

## Homework Statement

Match the parametric equations with the graphs.
In this case, I am stuck on this equation:
x = cos t
y = sin t
z = 1/(1+t^2)

## The Attempt at a Solution

So far I have:
x^2 + y^2 = cos ^2 t + sin ^2 t = 1
I know this is a circle in the xy-plane, and thus this yields a vertical, circular cylinder. I understand that there will be some form of helix going counter-clockwise around the cylinder. However, I do not understand how exactly to "graph" this using the parameter t. I know what the graph is and looks like, but I can't understand why it looks the way it does. I am having a hard time grasping this section in total.

Think of an infinitely long, infinitely thin slinky spring whose top is in the plane ##z=1##, centred on the point (0,0,1). As you follow the spring down, in the negative direction of the ##z## axis, the coils get closer and closer to one another so that it approaches but never quite touches the plane ##z=0##.

## 1. What are space curves?

Space curves are three-dimensional curves that exist in space. They are often represented using mathematical equations or parametric equations.

## 2. How do space curves differ from regular curves in two dimensions?

Space curves differ from regular curves in two dimensions in that they exist in three-dimensional space, instead of just on a two-dimensional plane. Space curves can also have more complex shapes and behaviors due to their additional dimension.

## 3. What are some real-life examples of space curves?

Space curves can be seen in many natural phenomena, such as the path of a planet orbiting the sun, the motion of a rollercoaster, or the shape of a DNA molecule. They are also used in many man-made structures, such as bridges and architectural designs.

## 4. How are space curves used in science and engineering?

Space curves are used in many fields of science and engineering, including physics, astronomy, and computer graphics. They are used to model and understand the motion and behavior of objects in space, as well as to design and create complex structures and animations.

## 5. Are there different types of space curves?

Yes, there are many different types of space curves, each with their own unique properties and equations. Some common types include helix curves, cycloid curves, and parabolic curves. Each type can be described using different mathematical equations and have different applications in science and engineering.

• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
899
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
8K
• Calculus and Beyond Homework Help
Replies
3
Views
443
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K