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 January 14th, 2014, 12:45 AM #1 Newbie   Joined: Jan 2014 Posts: 4 Thanks: 0 What's this shape called and how do I make it? There is a particular shape that I am trying to make, but I don't know how. If you were to call a donut a circular cylinder, then what I am making would be called a circular cone. The point of the cone meets the base of the cone. How can I design this shape? What do I use? Help!
 January 14th, 2014, 06:34 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,805 Thanks: 2150 I think there's no established name for the surface you want. First, devise parametric equations for the torus (donut) in which your surface fits. If the radius of the cylinder you mentioned is r, r is a constant, as it's the same no matter how far "along" the torus you go. For the surface you want to construct, r will become a constant multiple of some parameter that specifies how far along the torus you are (if you see what I mean).
January 14th, 2014, 10:17 AM   #3
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Math Focus: Electrical Engineering Applications
Re: What's this shape called and how do I make it?

Hi [color=#0040BF]quantman[/color],

I found the parametric equations mentioned by [color=#FF0040]skipjack[/color], here, and I modified them to (I think) create the shape that you want. Please see the diagrams below.

I am on my lunch break and I am in a hurry so I hope I did not make a mistake.
Attached Images
 torus_2.jpg (76.4 KB, 140 views) torus_1.jpg (91.1 KB, 140 views)

 January 15th, 2014, 01:09 AM #4 Newbie   Joined: Jan 2014 Posts: 4 Thanks: 0 Re: What's this shape called and how do I make it? Thanks skipjack and jks! I don't quite understand these equations, especially what "numpts" is and how you define the variables. What software do I use to plug them in? How would the equations look if the point of origin of the coordinates was where the point of the cone meets the base? Thanks a million!
January 15th, 2014, 07:22 PM   #5
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Re: What's this shape called and how do I make it?

I will try my best to explain. Feel free to reply in order to ask questions or to clarify what you need. For example, I am not sure exactly what you mean by:

Quote:
 I am trying to make
or
Quote:
 How can I design this shape? What do I use?
.

Anyway, here goes (note: variables in the text are italicized for ease of identification):

First, let's use the 'regular' torus and not the 'cone' torus. Picture a wagon wheel that, of course, has a center, and has a few spokes radiating out from the center to the rim. The rim is at a distance of c1 from the center. We can construct the torus by drawing circles of radius a1, centered on the points where the spokes meet the rim, with the plane of each circle in line with the corresponding spoke. I hope the following picture shows what I mean:

[attachment=6:t62jx3xb]torus_circles.jpg[/attachment:t62jx3xb]

In the reference that I gave above, the parametric equations use the variables v and u, and I used m1 and n1, respectively. One variable (n1) is used to work our way around the wagon wheel while the other variable (m1) is used to work our way around each of the circles of radius a1.

numpts is related to the number of points used to make the graph, and may be thought of as the number of spokes in the wagon wheel analogy. I used the same number of points in making the wagon wheel as I did in making the circles of radius a1 but this is not necessary as the number of points used may differ.

In order to help you visualize all of this, instead of a surface plot, here is a plot of the points used in making the surface plot with numpts=30:

[attachment=5:t62jx3xb]torus_pts_30.jpg[/attachment:t62jx3xb]

and with numpts=10:

[attachment=4:t62jx3xb]torus_pts_10.jpg[/attachment:t62jx3xb]

Here is a plot with n1 (the spokes of the wheel) going only 2/3 of the way around to show the effect of this variable:

[attachment=3:t62jx3xb]torus_n1_tq.jpg[/attachment:t62jx3xb]

Here is a plot with m1 (construction of the circles of radius a1) only going half way around (I changed the orientation for a better view) to show the effects of this variable:

[attachment=2:t62jx3xb]torus_m1_oh.jpg[/attachment:t62jx3xb]

Now for the 'cone' shape. We want the circles that we draw where the wagon wheel spokes meet the rim to increase in radius the further we go around the circle (or we could make the radius decrease. I chose increase). Since n1 is the variable that works our way around the wagon wheel, we will use it to linearly increase the radius a1 by multiplying a1 by $\ \frac{n1}{numpts}$ wherever a1 appears in the equations (see my first post).

Note that both m1 and n1 go from 0 to numpts.

Quote:
 How would the equations look if the point of origin of the coordinates was where the point of the cone meets the base?
For this particular orientation of the curve:

[attachment=1:t62jx3xb]torus_offset.jpg[/attachment:t62jx3xb]

I simply subtracted c1 (in this case = 5) from x1 so:

[attachment=0:t62jx3xb]torus_eqn_x1.jpg[/attachment:t62jx3xb]

(In the figure we are looking straight down the z axis).

Quote:
 What software do I use to plug them in?
Good question. Again, I think the answer depends upon exactly what it is that you want to do (and your available resources). The software that I am using is commercially available. I will not mention it by name on the forum but if you want to know what it is feel free to send a PM.

I use this software almost daily at work, so I am not very familiar with other software that may be available. I believe that there is software available for free that will do these types of plots (and more). We may want to start a topic in the Math Software section to see what people recommend but again I think it depends upon your resources and what you want to do.

I hope this helps. Please feel free to reply if you are so inclined.
Attached Images
 torus_circles.jpg (19.6 KB, 96 views) torus_pts_30.jpg (40.1 KB, 96 views) torus_pts_10.jpg (19.0 KB, 96 views) torus_n1_tq.jpg (31.7 KB, 96 views) torus_m1_oh.jpg (27.9 KB, 96 views) torus_offset.jpg (34.5 KB, 96 views) torus_eqn_x1.jpg (12.8 KB, 96 views)

 January 16th, 2014, 12:02 AM #6 Newbie   Joined: Jan 2014 Posts: 4 Thanks: 0 Re: What's this shape called and how do I make it? Thank you so much! That is incredibly helpful. Yes, I would like to know the software you used to build that. Please send me a PM (I would send you one, but I don't know how...)

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