![]() GeoGebra is a versatile and powerful tool. ![]() Here they are from a different point of view: Now, we add its negative: z = − y / x 2 (the one in green): Next, we'll look at z = − x / y 2 by itself: Here's what it looks like, along with our original surface: Let's now add the negative of the above graph, that is z = − y / x 2 We can see this shape in our 3D graph above. ![]() If we fix y = 1 then the curve z = 1 / x 2 (it's in 2 dimensions) looks like this: Let's consider this curve for a bit (using GeoGebra to help, of course), and investigate it from different angles. Those graphs are similar to the following ones, which also involve an asymptote: I used some different software for those ones (which has a problem where asymptotes appear as vertical "walls" - but shouldn't be there at all. One reader asked how to draw some graphs involving interesting asymptotes. Next, here are some 3D graphs that were suggested by some comments on the article How to draw y^2 = x - 2? Here's a water droplet-like shape, whose equation is: Let's draw a few 3D surfaces using GeoGebra. Of course, you can still create 2D graphs as before, and the interface is largely unchanged for that aspect. You also get a set of empty 3-D coordinate axes, like this: When you first open GeoGebra now, you are greeted with this choice of Perspectives:Ĭhoosing "3D Graphics", you get several new panels, which allow you to create 3D objects like a line perpendicular to a plane, a plane intersecting a cone, a plane through 3 points, a sphere, and so on: GeoGebra is a powerful and free graphing tool that anyone learning - or teaching - mathematics would find useful.įor me, the best feature of the new version is the ability to create 3D graphs. They also helped me understand the intuition behind the formulas for finding the volume of such solids.GeoGebra released version 5 a few months back. $a$ and $b$ can be switched on or off and colors changed to give a better intuition of the solid.Īnd of course, the graph can be moved around and rotated to give a better sense of what the solid really looks like! I found these tools really helpful for visualizing solids of revolution, thus helping me solve problems involving these better. For example, "The solid formed by rotating the area bound by the functions $x^2$ and $\sqrt$) must be replaced. Here's a quick overview of how solids of revolution can be graphed using either software: WolframAlphaĮxplain the figure in as precise a language as possible. GeoGebra and WolframAlpha (thanks seem to work well for me right now. I just found Math3D, so I haven't really had a chance to uncover its pros and cons. GeoGebra seems to have a very poor documentation, and Manim might be too complicated for simple graphs. Perhaps a tip for that? Finding a good 3D graphing software would generally be useful though, since I'll probably need it anyways for future courses. I understand that parametric surfaces give a 3D vector in terms of $u$ and $v$, but I can't figure out how to convert solids of revolution to a parametric surface. I have to use parametric surfaces or something similar, which is difficult for me to do since I haven't really learnt about those. GeoGebra, Math3D, and Manim all seem to be good, but I don't really see an easy option to graph such solids of revolution. Is there any good software that I can use to visualize such solids formed from an arbitrary curve? Preferably even solids formed from an intersection of two such curves. Some solids of revolution are also formed by rotating the area under a 2D curve about the $y$-axis: In particular, we're learning about shapes called solids of revolution, which are formed by rotating the area under a 2D curve about the $x$-axis. I'm taking AP Calc with AoPS, and we're learning how to find the volumes of some 3D solids using integration methods.
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