Sketching 3d shapes with looking at an equation

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Discussion Overview

The discussion revolves around methods for visualizing 3D shapes represented by equations, specifically focusing on the equation x² + y² - z² = 1. Participants explore various techniques for graphing or conceptualizing these shapes without the use of computer software.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant describes a method of setting variables to zero to obtain cross-sections of the function in different planes (xy, xz, yz) to visualize the shape.
  • Another participant suggests using rotational symmetry to simplify the visualization by transforming the equation into a 2D curve that can be rotated about the z-axis.
  • There is a discussion about the differences between horizontal slices (constant z) and vertical slices, with one participant arguing that horizontal slices may complicate understanding the overall shape.
  • One participant proposes sketching cross-sections in the coordinate planes and combining them to form a visual representation of the 3D shape, mentioning specific shapes like circles and hyperbolas that arise from the equation.
  • Another participant expresses uncertainty about the effectiveness of the proposed methods and asks for clarification on the underlying principles and their applicability to other cases.

Areas of Agreement / Disagreement

Participants express differing views on the effectiveness of various methods for visualizing the shape of the function. There is no consensus on a single best approach, and some participants question the applicability of certain methods to other equations.

Contextual Notes

Participants acknowledge that the methods discussed may not universally apply to all equations, indicating limitations in their approaches. There are also unresolved questions regarding the simplification of variables and the conditions under which certain methods are effective.

Who May Find This Useful

This discussion may be useful for students or individuals interested in mathematical visualization techniques, particularly in the context of graphing 3D shapes from equations in physics or engineering.

korr2221
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i was wondering what are some more popular ways to graph or get an idea of how a shape of an function would turn out to. the way I've learned it is to set certain variable to zero to get a certain plane. for example if i wanted the the xy plane of a function such as x^2+y^2-z^2 = 1 i would need to set the z = 0 and then solve for the y and solve for the x plugging random numbers to see how the shape is for that plane and do the same for the other two planes. which is the xz and yz. is there any other way to graph such an equation w/o using a computer and maple or something?
 
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Welcome to PF!

Hi korr2221! Welcome to PF! :smile:

Yes … this equation obviously has rotational symmetry …

so put x2 + y2 = r2, and you get:

r2 - z2 = 1 …

that's an easy 2D curve, which you can then rotate about the z-axis. :wink:
 


tiny-tim said:
Hi korr2221! Welcome to PF! :smile:

Yes … this equation obviously has rotational symmetry …

so put x2 + y2 = r2, and you get:

r2 - z2 = 1 …

that's an easy 2D curve, which you can then rotate about the z-axis. :wink:

sorry if i am incorrect, but isn't this just about the same as what I've said? i don't see any difference... unless you're trying to make a point that in this case because it's shape has symmetry the positive xy plane and -xy plane is th same because of it's shape but sometimes that isn't always the case? sorry i don't understand where you were going with you reply above...
 
korr2221 said:
sorry if i am incorrect, but isn't this just about the same as what I've said?

Hi korr2221! :smile:

No … your method, putting z = constant (i assume that's what you meant :wink:), gives you horizontal slices, which as you remarked still leaves you a problem in deciding the shape.

My method uses a single vertical slice (well, half-slice), which you can just rotate. :smile:
 
tiny-tim said:
Hi korr2221! :smile:

No … your method, putting z = constant (i assume that's what you meant :wink:), gives you horizontal slices, which as you remarked still leaves you a problem in deciding the shape.

My method uses a single vertical slice (well, half-slice), which you can just rotate. :smile:

ah yes... i sort of get what you've said now... but i still don't fully understand it... what 'phrase' should i google to find more about this method? what are you basically doing? turning the z plane into r^2? does your method always work? or only for this special case?
 
One method I like to use is to sketch their cross sections in the coordinate planes. For example, in the xy-plane, z= 0 so x^2+y^2-z^2 = 1 becomes x^2+ y^2= 1, the unit circle. In the xz-plane, y= 0 so the equation becomes x^2- z^= 1, a hyperbola. Finally, in the yz-plane, x= 0 so the equation becomes y^2- z^2= 1, a hyperbola.

Sketch those on three separate pieces of paper and put them together to get an idea of the shape.

That together with drawing the "level curves":
When z= 0, that is x^2+ y^2= 1, a circle with center at (0,0,0) radius 1
When z= 1, that is x^2+ y^2- 1= 1 or x^2+ y^2= 2, two circles with centers at (0, 0, 1) and (0, 0, -1) with radius [itex]\sqrt{2}[/itex]
etc.
which gives you a "geodesic map" idea of the surface should give a very good picture of the surface.
 
korr2221 said:
ah yes... i sort of get what you've said now... but i still don't fully understand it... what 'phrase' should i google to find more about this method? what are you basically doing? turning the z plane into r^2? does your method always work? or only for this special case?

Hi korr2221! :smile:

I can't think of anything to google …

it's really just a matter of simplifying …

we started off with three variables (x,y,z), which gives a 3D image that is difficult to visualise …

I reduced it to two variables (r,z), which gives a familiar 2D image …

I suppose the general rule would be to try to reduce the number of variables while keeping the whole original equation! (which you didn't … you sliced it! :wink:).

And no … sorry … it won't always work! :redface:

(you knew that didnt you? :smile:)
 

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